\section{Introduction}

The applications of the bootstrap in principal components analysis (PCA) go back almost as far as the advent of the bootstrap itself~\citep{Diaconis:1983}, and over the years such applications have become part of standard practice in  multivariate analysis~\citep{Davison,Jolliffe,Olive}.
With regard to theory, there is also a well-established set of asymptotic results showing that the bootstrap generally works in the context of PCA with low-dimensional data~\citep{BeranS85,EatonT91}.
Furthermore, in aberrant situations where the bootstrap is known to encounter difficulty in low dimensions, such as in the case of tied population eigenvalues, various remedies have been proposed and analyzed~\citep{BeranS85,Dumbgen:1993,Hall:2009}.

 However, in the context of PCA with high-dimensional data, the relationship between theory and practice is quite different.  On one hand, bootstrap methods are popular among practitioners for solving inference problems related to high-dimensional PCA~\citep[e.g.][]{Wagner:2015,Fisher:2016,Webb:2017,Terry:2018,Ralph:2019,Holmes:2019,Stewart:2019}.
Yet, on the other hand, the theory for describing these methods is relatively incomplete.

As a way to develop a more precise understanding of the bootstrap in this context, we focus on the fundamental problem of approximating the joint distribution of the leading eigenvalues $\lambda_1(\hat\Sigma),\dots,\lambda_k(\hat\Sigma)$ of a sample covariance matrix $\hat\Sigma\in\mathbb{R}^{p\times p}$, where $k<p$. (Precise defnitions will be given later.) Because the fluctuations of these eigenvalues are relevant to many inference tasks, this problem plays a central role in multivariate analysis, and is also of broad interest in other areas,
such as signal processing~\citep{Debbah:Couillet:2011}, and finance~\citep{Ruppert:2015}.
For concreteness, we summarize below some examples of inference tasks involving sample eigenvalues. In addition, these tasks are illustrated with real-data examples based on stock market returns in Section~\ref{sec:stocks} of the supplementary material.

\begin{itemize}
\item  \textit{Selecting principal components.} A key step that occurs in any implementation of PCA is to choose the number of principal components, and many established techniques for making this choice are informed by the distributions of eigenvalue-based statistics. Examples of these statistics include eigengaps $\lambda_j(\hat\Sigma)-\lambda_{j+1}(\hat\Sigma)$, and the proportions of explained variance  $(\lambda_1(\hat\Sigma)+\cdots+\lambda_k(\hat\Sigma))/\operatorname{tr}(\hat\Sigma)$, as well as the componentwise proportions $\lambda_j(\hat\Sigma)/\operatorname{tr}(\hat\Sigma)$ for $j=1,\dots,k$. In addition, other selection rules are based on confidence intervals for the eigenvalues $\lambda_1(\Sigma),\dots,\lambda_k(\Sigma)$ of the population covariance matrix $\Sigma\in\mathbb{R}^{p\times p}$, and the construction of such intervals is directly linked to the distribution of the eigenvalues of $\hat\Sigma$.
For a general overview of selection rules, we refer to~\cite{Jolliffe}.

\item \textit{Quantifying uncertainty.}
The eigenvalues of a population covariance matrix arise as unknown parameters of interest in many situations beyond the selection of principal components.
For instance, these parameters govern the performance of statistical methods for covariance estimation, regression, and classification~\citep{Ledoit:2012,Hsu:2014,Dobriban:2018}. Also, these parameters have domain-specific meaning in applications ranging from portfolio selection to ecology~\citep{Fabozzi,Ecology}.
Consequently, it becomes necessary to quantify the uncertainty associated with the population eigenvalues, such as in constructing confidence intervals for them---and again, this leads to the use of distributional approximation results for the sample eigenvalues.

\end{itemize}

Although there is an extensive literature on distributional approximations for sample eigenvalues, this body of work primarily work focuses on asymptotic results involving analytical formulas. Roughly speaking, the bulk of the literature can be divided into two parts, dealing either with classical asymptotics where $p$ is held fixed as $n\to\infty$~\citep{Anderson}, or high-dimensional asymptotics where $p/n$ converges to a positive constant as $p$ and $n$ diverge simultaneously~\citep{BaiSilverstein}. In either case, an essential limitation is that asymptotic results do not usually quantify how close the limiting distribution is to the finite-sample distribution. In more practical terms, this means that it is often hard to know if tests statistics and confidence intervals are well calibrated (i.e.~if their actual levels and coverage probabilities are close to the nominal values). A second limitation is that approximations based on analytical formulas are often tied to specific model assumptions, which can make it difficult to adapt such formulas outside of a given model.

With regard to the second limitation, bootstrap methods have an advantage insofar as they do not rely on formulas, and hence can be applied in a more flexible manner.  Nevertheless, the existing work on bootstrap methods for PCA still tends to suffer from the first limitation above, since the results are generally asymptotic~\citep{BeranS85,EatonT91,ElKarouiP19}. From this standpoint, a key motivation for our work is to provide results that explicitly quantify the accuracy of bootstrap approximation in terms of the sample size $n$ and the effective rank of $\Sigma$. (For example, our results can be used to quantify how close the coverage probabilities of bootstrap confidence intervals are to the nominal values.)  Another motivation is based on the fact that, until quite recently, most of the literature on bootstrap methods for PCA has been limited to low-dimensional settings. Consequently, it is of general interest to establish a more complete theoretical description of bootstrap methods for high-dimensional PCA---a point that was highlighted in a recent survey on this topic~\citep[][\textsection X.C]{JohnstoneP18}.

\subsection{Contributions}\label{sec:contrib}
Let $X_1,\dots,X_n\in\mathbb{R}^p$ be centered i.i.d.~observations with population covariance matrix $\Sigma=\mathbb{E}[X_1X_1^{\top}]$. Also, let $\hat\Sigma=\textstyle\sum_{i=1}^nX_iX_i^{\top} / n$ denote the associated sample covariance matrix, and let $\hat\Sigma^{\star}=\textstyle\sum_{i=1}^nX_i^{\star}(X_i^{\star})^{\top} / n$ be its bootstrap version, formed from random vectors $X_1^{\star},\dots,X_n^{\star}$ that are sampled with replacement from the observations. In addition, let the eigenvalues of a symmetric matrix $A\in\mathbb{R}^{p\times p}$ be denoted as $\lambda_1(A)\geq\cdots\geq\lambda_p(A)$, and let $\boldsymbol \lambda_k(A)=(\lambda_1(A),\dots,\lambda_k(A))$ for a fixed integer $k<p$.

In this notation, our goal is to establish non-asymptotic bounds on the multivariate Kolmogorov distance
\small
$$ \Delta_n= \sup_{t \in \mathbb{R}^k } \Bigg| \P \Big( \sqrt{n} \big( {\boldsymbol \lambda}_k (\hat{\Sigma}) - {\boldsymbol \lambda}_k (\Sigma) \big) {\  \preceq t} \Big) - \P \Big( \sqrt{n} \big( {\boldsymbol \lambda}_k (\hat{\Sigma}^\star) - {\boldsymbol \lambda}_k (\hat\Sigma) \big) {\  \preceq t} \, \Big| \, X \Big) \Bigg|,$$
\normalsize
where the relation $v\preceq w$ between two vectors $v,w\in\mathbb{R}^k$ means $v_j\leq w_j$ for all $j=1,\dots,k$, and  $\P(\cdot\,|X)$ refers to probability that is conditional on $X_1,\dots,X_n$. Under certain conditions, our central result (Theorem~\ref{THM:ALL}) shows that the dimension-free bound
\begin{equation}\label{eqn:informal}
\Delta_n \ \leq \ \frac{C_n{\,\tt{r}}(\Sigma)}{\sqrt n}
\end{equation}
holds with high probability, where $C_n>0$ is a polylogarithmic function of $n$, and the quantity ${\tt{r}}(\Sigma)$ is the effective rank of $\Sigma$, defined by ${\tt{r}}(\Sigma)=\operatorname{tr}(\Sigma) / \lambda_1(\Sigma)$.

There are several aspects of the bound~\eqref{eqn:informal} and the parameter ${\tt{r}}(\Sigma)$ that are worth noting. First, the effective rank satisfies \smash{$1\leq {\tt{r}}(\Sigma)\leq p$} whenever $\Sigma$ is nonzero, and can be interpreted as a proxy for the number of ``dominant'' principal components of $\Sigma$. Hence, even in very high-dimensional settings where $n\ll p$, the bound~\eqref{eqn:informal} shows that the bootstrap can perform well if the number of dominant components is not too large, which is precisely the situation where high-dimensional PCA is of greatest interest.
Meanwhile, even in situations where ${\tt{r}}(\Sigma)$ is moderately large, e.g.~${\tt{r}}(\Sigma)\to\infty$ with ${\tt{r}}(\Sigma)=o(\sqrt n)$, the bound~\eqref{eqn:informal} is still able to quantify the accuracy of the bootstrap. Indeed, both of these points are borne out by our numerical experiments in Section~\ref{sec:experiments}, which confirm that the performance of the bootstrap is governed more by ${\tt{r}}(\Sigma)$ than $p$, and that the bootstrap can still be accurate when ${\tt{r}}(\Sigma)$ is moderately large. More generally, it should also be mentioned that the theoretical role of effective rank in many other aspects of high-dimensional PCA has attracted considerable attention in recent years~\citep[e.g.][]{Lounici14, BuneaX15, KoltchinskiiL17b, JungA18, NaumovSU19, KoltchinskiiLN20}.

As an alternative to approximating the distribution of $\sqrt{n} \big( {\boldsymbol \lambda}_k (\hat{\Sigma}) - {\boldsymbol \lambda}_k (\Sigma) \big)$ by bootstrapping in a direct manner, it can be advantageous to use a \emph{transformation} prior to bootstrapping, which is a fundamental topic in the bootstrap literature~\citep[e.g.][]{Diciccio:1984, Tibshirani:1988,Konishi:1991,Diciccio:1996,Davison,Chernick}. To be more specific, let $h$ be a univariate scalar function, referred to as a transformation,
and for any symmetric matrix $A\in\mathbb{R}^{p\times p}$, let $\boldsymbol h(\boldsymbol\lambda_k(A))=(h(\lambda_1(A)),\dots,h(\lambda_k(A)))$. Then, the conditional distribution of $\sqrt n(\boldsymbol h(\boldsymbol\lambda_k(\hat\Sigma^{\star}))-\boldsymbol h(\boldsymbol\lambda_k(\hat\Sigma)))$ given the observations can be used to approximate the distribution of $\sqrt n(\boldsymbol h(\boldsymbol\lambda_k(\hat\Sigma))-\boldsymbol h(\boldsymbol\lambda_k(\Sigma)))$.
(Additional discussion is provided in Sections~\ref{sec:main} and~\ref{sec:experiments}.) For instance, a classical choice of transformation is $h(x)=\log(x)$, because it is known to be variance-stabilizing under certain conditions when $n\to\infty$ with $p$ held fixed~\citep{BeranS85}.
With this in mind, a second contribution our analysis is an extended version of the bound~\eqref{eqn:informal} that can accommodate the use of certain transformations (Theorem~\ref{THM:TRANS_GENERAL}).

From a more methodological standpoint, our numerical experiments also shed new light on the role of transformations in bootstrap methods for high-dimensional PCA. Although we confirm that the classical logarithm transformation can be beneficial in low dimensions, we show that it is less effective when ${\tt{r}}(\Sigma)$ is moderately large. Consequently, we explore some alternative transformations and provide numerical results demonstrating that there are opportunities to improve upon $h(x)=\log(x)$ in high dimensions. To put such empirical findings into perspective, we are not aware any prior work investigating how transformations can be used to enhance  bootstrap methods in this context.

\subsection{Related work}
Quite recently, there has been an acceleration in the pace of
research on bootstrap methods for high-dimensional sample covariance matrices, as evidenced in the papers~\cite{HanXZ18,JohnstoneP18,ElKarouiP19,LopesBA19,LopesEM19,NaumovSU19}. Among these, the most relevant to our work is \cite{ElKarouiP19}, which examines both the successes and failures of the bootstrap in doing inference with the leading eigenvalues of $\hat\Sigma$. In the negative direction, that paper focuses  on a specialized model with $\lambda_1(\Sigma)>1$ and $\lambda_2(\Sigma)=\cdots=\lambda_p(\Sigma)=1$, which corresponds to a very large effective rank ${\tt{r}}(\Sigma)\asymp p$ that makes dimension reduction via PCA inherently difficult. In the positive direction, that paper deals with a different situation where $\Sigma$ is assumed to have a near low-rank structure of the form
\begin{equation}\label{eqn:blockassume}
\Sigma=\begin{pmatrix} A & B \\ B^{\top} & C(\eta) \end{pmatrix},
\end{equation}
\normalsize
where $A$ is of size $k\times k$ with $k\asymp 1$, and the diagonal blocks satisfy $\lambda_1(A)\asymp 1$, and  $\lambda_1(C(\eta))\lesssim n^{-\eta}$ for a fixed parameter $\eta>1/2$. Working under an elliptical model, the paper~\citep{ElKarouiP19} shows that the bootstrap consistently approximates the distribution of $\sqrt{n} \big( {\boldsymbol \lambda}_k (\hat{\Sigma}) - {\boldsymbol \lambda}_k (\Sigma) \big)$ in an asymptotic framework where $p/n\lesssim 1$. In relation to our work, the most crucial distinction is that our results quantify the accuracy of the bootstrap with non-asymptotic \emph{rates of approximation}. To illustrate the significance of this, note that our bound~\eqref{eqn:informal} provides an explicit link between the size of ${\tt{r}}(\Sigma)$ and the accuracy bootstrap, whereas in an asymptotic setup, the effect of ${\tt{r}}(\Sigma)$ is hidden---because it ``washes out in the limit''. Our numerical experiments will also confirm that different sizes of ${\tt{r}}(\Sigma)$ can have an appreciable effect on the finite-sample accuracy of the bootstrap. In this way, our work indicates that the quantity ${\tt{r}}(\Sigma)/\sqrt n$ serves as a type of conceptual diagnostic for assessing the reliability of the bootstrap in high-dimensional PCA.

Beyond these points of contrast with~\cite{ElKarouiP19}, there are several distinctions with regard to model assumptions. First, we work in a dimension-free setting where there are no restrictions on the size of $p$ with respect to $n$. Second, the model based on~\eqref{eqn:blockassume} implicitly requires that $\lambda_j(\Sigma)\lesssim n^{-\eta}$ for all $j\geq k+1$, whereas this constraint on $\Sigma$ is not used here.
Third, it is straightforward to check that in the model based on~\eqref{eqn:blockassume}, the condition $\eta>1/2$ \emph{implies} ${\tt{r}}(\Sigma)=o(\sqrt n)$, which means that our bound~\eqref{eqn:informal} ensures bootstrap consistency in models that subsume the one based on~\eqref{eqn:blockassume}.
(As an example, if $p\asymp e^{m(n)}$ for some sequence of integers satisfying $m(n)=o(\sqrt n)$  and if $\lambda_j(\Sigma) \asymp j^{-1}$, then the bound~\eqref{eqn:informal} implies bootstrap consistency, whereas this is not guaranteed by the previous result even when $p\asymp n$.)

Other works on bootstrap methods related to high-dimensional sample covariance matrices have dealt with models or statistics that are qualitatively different from those considered here. The papers~\cite{HanXZ18,LopesEM19} look at bootstrapping the operator norm error $\sqrt n\|\hat\Sigma-\Sigma\|_\textup{op}$, as well as variants of this statistic, such as $\sup_{u\in\mathcal{U}}\sqrt n |u^{\top}(\hat\Sigma-\Sigma) u|/u^{\top} \Sigma u$, where $\mathcal U$ is a set of sparse vectors in the unit sphere of $\mathbb{R}^p$.
In a different direction, the paper~\citep{LopesBA19} focuses on ``linear spectral statistics'' of the form $\sum_{j=1}^p f(\lambda_j(\hat\Sigma)) / p$, where $f:[0,\infty)\to \mathbb{R}$ is a smooth function. In that paper, it is shown that a type of parametric bootstrap procedure consistently approximates the distributions of such statistics when $p/n$ converges to a positive limit. Lastly, the paper~\cite{NaumovSU19} deals with bootstrapping statistics related the eigenvectors of $\hat\Sigma$. \\[-0.2cm]

\noindent\textbf{Notation.}
For a random variable $X$ and an integer $q\in\{1,2\}$, define the $\psi_q$-Orlicz norm as $\| X \|_{\psi_q} = \inf \: \{ t > 0 \ | \ \mathbb{E} [\exp( |X|^q / t^q)] \leq 2 \}$. The random variable $X$ is said to be sub-exponential if $\|X\|_{\psi_1}$ is finite, and sub-Gaussian if $\|X\|_{\psi_2}$ is finite. In addition, for any $q\geq 1$, the $L_q$ norm of $X$ is defined as $\| X \|_q = ( \mathbb{E} [ |X|^q ] )^{1/q}$.
For any vectors $u,v\in\mathbb{R}^p$, their inner product is $\langle u,v\rangle = \sum_{j=1}^p u_jv_j$.
For any real numbers $a$ and $b$, the expression $a\ll b$ is used in an informal sense to mean that $b$ is much larger than $a$. Also, we use the notation $a\vee b=\max\{a,b\}$ and $a\wedge b=\min\{a,b\}$. If $\{a_n\}$ and $\{b_n\}$ are two sequences on non-negative numbers, then the relation $a_n \lesssim b_n$ means that there is a positive constant $c$ not depending on $n$ such that $a_n \leq c \, b_n$ holds for all large $n$. When both of the conditions $a_n\lesssim b_n$ and $b_n\lesssim a_n$ hold, we write $a_n\asymp b_n$.

\vspace{0.1cm}

\section{Main results}\label{sec:main}
We consider a sequence of models indexed by $n$, in which all parameters may depend on $n$ except when stated otherwise.  In particular, the dimension $p = p(n)$ is allowed to have arbitrary dependence on $n$. Likewise, if a parameter does not depend on $n$, then it is understood not to depend on $p$ either. One of the few parameters that will be treated as fixed with respect to $n$ is the positive integer $k<p$.

\begin{assumption}[Data-generating model]\label{model_assumptions}  \noindent \\[-0.4cm]
\begin{enumerate}
\item[\namedlabel{A1}{(a)}.]
There is a non-zero positive semidefinite matrix $\Sigma\in\mathbb{R}^{p\times p}$, such that the $i$th observation is generated as  $X_i = \Sigma^{1/2} Z_i$ for all $i = 1, \dots, n,$ where $Z_1,\dots,Z_n\in\mathbb{R}^p$ are i.i.d.~random vectors with $\mathbb{E}[Z_1]=0$, and $\mathbb{E}[Z_1Z_1^{\top}]=I_p$. \\[-0.2cm]

\item[\namedlabel{A2}{(b)}.] The eigenvalues of $\Sigma$ satisfy
$\displaystyle\min_{1\leq j\leq k} \textstyle\big(\lambda_j(\Sigma) - \lambda_{j+1}(\Sigma)\big) \gtrsim  \lambda_1(\Sigma) $. \\[-0.2cm]

\item[\namedlabel{A3}{(c)}.]
Let $u_j\in\mathbb{R}^p$ denote the $j$th eigenvector of $\Sigma$, and let $\Gamma \in\mathbb{R}^{k \times k}$ have entries given by $\Gamma_{jj'}=\mathbb{E} [(\langle u_j, Z_1 \rangle^2 - 1)(\langle u_{j'}, Z_1 \rangle^2 - 1)]$ for all $1\leq j,j'\leq k$. Then, the matrix $\Gamma$ satisfies $\lambda_k(\Gamma)\gtrsim 1$.
\end{enumerate}
\end{assumption}
\noindent In connection with the model described by  Assumption~\ref{model_assumptions}, our results will make reference to a moment parameter defined as
$\beta_q = \max_{1 \leq j \leq p} \| \langle u_j, Z_1 \rangle^2 \|_q$
for any $q\geq 1$. \\

\noindent \textbf{Remarks.} Regarding Assumption~\ref{model_assumptions}.\ref{A2}, it ensures that there is some degree of separation between the leading eigenvalues of $\Sigma$.  In less compact notation, the assumption states that there is a fixed constant $c>0$ such that the inequality \smash{$\lambda_j(\Sigma)-\lambda_{j+1}(\Sigma)\geq c \lambda_1(\Sigma)$} holds for all $j=1,\dots,k$, and all large $n$. (There is no restriction on the size of $c$.) In general, a separation condition on the leading eigenvalues is unavoidable, because it is known both theoretically and empirically that the bootstrap can fail to approximate the distribution of $\sqrt{n} \big( {\boldsymbol \lambda}_k (\hat{\Sigma}) - {\boldsymbol \lambda}_k (\Sigma) \big)$ if the leading population eigenvalues are not distinct~\citep{Beran:correction,Hall:2009}. In more technical terms, the source of this issue can be explained briefly as follows: If $\mathcal{S}^{p\times p}$ denotes the space of  real symmetric $p\times p$ matrices, and if $\lambda_j(\cdot)$ is viewed as a functional from $\mathcal{S}^{p\times p}$ to $\mathbb{R}$, then $\lambda_j(\cdot)$ becomes non-differentiable at $\Sigma$ in the case when  $\lambda_j(\Sigma)$ is a repeated eigenvalue (i.e.~with multiplicity larger than 1). In turn, this lack of smoothness makes it difficult for the bootstrap to approximate the distribution of $\sqrt n(\lambda_j(\hat\Sigma)-\lambda_j(\Sigma))$.

To interpret Assumption~\ref{model_assumptions}.\ref{A3}, the matrix $\Gamma$ serves a technical role as a surrogate for the correlation matrix of $\sqrt{n} \big( {\boldsymbol \lambda}_k (\hat{\Sigma}) - {\boldsymbol \lambda}_k (\Sigma) \big)$. Hence, the lower bound $\lambda_k(\Gamma)\gtrsim 1$ can be viewed as a type of non-degeneracy condition for the distribution of interest. The  proposition below gives examples of well-established models in which Assumption \ref{model_assumptions}.\ref{A3} holds. Namely, parts $(i)$ and $(ii)$ below respectively correspond to \emph{Mar\v{c}enko-Pastur models} and \emph{elliptical models}. The latter case also illustrates that the entries of the vector $Z_1$ are not required to be independent.

\begin{proposition} \label{PROP1}
\noindent \namedlabel{PROP1_i}{(i)} (Mar\v{c}enko-Pastur case). Suppose that Assumption~\ref{model_assumptions}.\ref{A1} holds. In addition, suppose that the entries of $Z_1$ are independent, and there is a constant $\kappa>1$ not depending on $n$ such that $\min_{1\leq j\leq p }\mathbb{E}[Z_{1j}^4]\geq \kappa$. Then, Assumption \ref{model_assumptions}.\ref{A3} holds.

\noindent \namedlabel{PROP1_ii}{(ii)}  (Elliptical case). Let $V$ \!be a random vector that is uniformly distributed on the unit sphere of $\mathbb{R}^p$,
and let $\xi $ be a non-negative scalar random variable independent of $V$ that satisfies $\mathbb{E}[\xi ^2]=p$ and $\mathbb{E}[\xi^4]<\infty$. Under these conditions, if $Z_1$ has the same distribution as $\xi V$, then
Assumption \ref{model_assumptions}.\ref{A3} holds.
\end{proposition}

\noindent The proof of Proposition~\ref{PROP1} is given in Section~\ref{appendix_G} of the supplementary material. \\

\noindent \textbf{Bootstrap approximation.}
The following theorem is the central result of the paper, and quantifies the accuracy of the bootstrap when it is used to approximate the distribution of $\sqrt{n}( {\boldsymbol \lambda}_k (\hat{\Sigma})- {\boldsymbol \lambda}_k (\Sigma) )$.

\begin{theorem} \label{THM:ALL}
Suppose that Assumption \ref{model_assumptions} holds and let $q = 5\log(kn)$. Then, there is a constant $c > 0$ not depending on $n$ such that the event
\scriptsize
\begin{equation}\label{eqn:thm:all}
 \sup_{t \in \mathbb{R}^k } \Bigg| \P \Big( \sqrt{n} \big( {\boldsymbol \lambda}_k (\hat{\Sigma}) - {\boldsymbol \lambda}_k (\Sigma) \big) {\  \preceq t} \Big) - \P \Big( \sqrt{n} \big( {\boldsymbol \lambda}_k (\hat{\Sigma}^\star) - {\boldsymbol \lambda}_k (\hat\Sigma) \big) {\  \preceq t} \, \Big| \, X \Big) \Bigg| \
\leq \ \frac{c \, \log(n) \,  \beta_{3q}^3\,{\tt{r}}(\Sigma) }{\sqrt n}
\end{equation}
\normalsize
holds with probability at least $1 - c/n$.
\end{theorem}

\noindent \textbf{Remarks.} The proof of Theorem~\ref{THM:ALL} is given in Section~\ref{appendix_A} of the supplemenatary material. It is possible to provide a more concrete understanding of the bound~\eqref{eqn:thm:all} by looking at how the factors ${\tt{r}}(\Sigma)$ and $\beta_{3q}$ behave in some well-known situations. For instance, consider the class of matrices $\Sigma$ whose eigenvalues have a polynomial decay profile of the form $\lambda_j(\Sigma)\asymp j^{-\gamma}$, for some fixed constant $\gamma>0$. This class offers a convenient point of reference, because it interpolates between models that have low-dimensional structure and those that do not. Specifically, the effective rank can be related to $\gamma$ as
$$ {\tt{r}}(\Sigma) \asymp \begin{cases} & \!\!1 \text{ \ \ \  \ \ \ \  \ \   if } \gamma >1  \\
&\!\!\log(p) \  \text{ \ \  if } \gamma = 1  \\
& \!\!  p^{1-\gamma} \  \text{ \ \  \ \   if } \gamma<1.
\end{cases}
$$

With regard to the parameter $\beta_{3q}$, its dependence on $q$ is simple to describe in some commonly considered cases.
If the entries of $Z_1$ are i.i.d.~and sub-Gaussian, then $\beta_{3q}$ grows at most linearly in $q$, with $\beta_{3q}\lesssim q \|Z_{11}\|_{\psi_2}^2$. Alternatively, if the entries of $Z_1$ are i.i.d.~and sub-exponential, then $\beta_{3q}$ grows at most quadratically in $q$, with $\beta_{3q}\lesssim q^2 \|Z_{11}\|_{\psi_1}^2$. (See Chapter 2 of~\cite{Vershynin:2018} for further details.) Hence, a direct consequence of Theorem~\ref{THM:ALL} in such cases is that bootstrap consistency holds when $\gamma>1/2$, $p\asymp n$ and $\|Z_{11}\|_{\psi_1}\lesssim 1$. Likewise, when $\gamma>1$, the bound in Theorem~\ref{THM:ALL} nearly achieves the \emph{parametric rate} $n^{-1/2}$ and is not influenced by the size of $p$ at all. This conclusion also conforms with the numerical results that we present in Section~\ref{sec:experiments}.

From a more practical standpoint, it is possible to gauge the size of ${\tt{r}}(\Sigma)$ in an empirical way, by either estimating ${\tt{r}}(\Sigma)$ directly, or estimating upper bounds on it. Some examples of upper bounds on ${\tt{r}}(\Sigma)$ for which straightforward estimation methods are known to be effective in high dimensions include $\operatorname{tr}(\Sigma)/\max_{1\leq j\leq p}\Sigma_{jj}$ and $\operatorname{tr}(\Sigma)^2/\|\Sigma\|_F^2$. (Although guarantees can be established for direct estimates of ${\tt{r}}(\Sigma)$ in high-dimensions, such results can involve a more complex set of considerations than the upper bounds just mentioned.) \\

\noindent \textbf{Transformations.} To briefly review the idea of transformations, they are often used to solve inference problems involving a parameter $\theta$ and an estimator $\hat\theta$ for which the distribution of $(\hat\theta-\theta)$ is difficult to approximate. In certain situations, this difficulty can be alleviated if there is a monotone function $h$ for which the distribution of $(h(\hat\theta)-h(\theta))$ is easier to approximate. In turn, this allows for more accurate inference on the ``transformed parameter'' $h(\theta)$, and then the results can be inverted to do inference on $\theta$.
In light of this, our next result shows that the rates of bootstrap approximation established in Theorem~\ref{THM:ALL} remain essentially unchanged when using the class of fractional power transformations from $[0,\infty)$ to $[0,\infty)$. This class will be denoted by $\mathcal{H}$, so that if $h\in\mathcal{H}$, then $h(x)=x^{a}$ for some $a\in (0,1]$.

Beyond the class of transformations just mentioned, the bootstrap can be combined with another type of transformation known as \emph{partial standardization}~\citep{LopesLM18}. Letting $h\in\mathcal{H}$ be a given function, and letting  $\varsigma_j^2 = \operatorname{var} \big( h( \lambda_j (\hat\Sigma)) \big)$ for each $j=1,\dots,p$, this technique is well suited to bootstrapping ``max statistics'' of the form
\begin{equation}\label{eqn:maxstat}
M = \max_{1\leq j\leq k}  \frac{ h(\lambda_j (\hat{\Sigma})) - h(\lambda_j (\Sigma)) }{\varsigma_j^\tau},
\end{equation}
where $\tau\in[0,1]$ is a parameter that can be viewed as a degree of standardization. The ability to approximate the distribution of $M$ is relevant to the construction of simultaneous confidence intervals for $\lambda_1(\Sigma),\dots,\lambda_k(\Sigma)$. It also turns out that the choice of $\tau$ encodes a trade-off between the coverage accuracy and the width of such intervals, and that choosing an intermediate value $\tau\in (0,1)$ can offer benefits in relation to $\tau=0$ and $\tau=1$. This will be discussed in greater detail later in Section~\ref{sec:experiments}.

In order to state our extension of Theorem~\ref{THM:ALL} in a way that handles both partial standardization and transformations $h\in\mathcal{H}$ in a unified way, we need to introduce a bit more notation. First, when considering the bootstrap counterpart of a partially standardized statistic such as~\eqref{eqn:maxstat}, the vector $\boldsymbol{\varsigma}_k^{\tau} = ( \varsigma_1^{\tau}, \dots, \varsigma_k^{\tau})$ can be replaced with the estimate $\boldsymbol{\hat\varsigma}_k^{\tau} = ( \hat\varsigma_1^{\tau}, \dots, \hat\varsigma_k^{\tau})$, whose entries are defined by $\hat\varsigma_j^2 = \operatorname{var} \big( h(\lambda_j (\hat\Sigma^\star) )\big| X \big)$ for all $j=1,\dots,p$. Second,
the expression $v/u$ involving vectors $v$ and $u$ denotes the vector obtained by entrywise division,  $(v/u)_j=v_j/u_j$. (To handle the possibility zero denominators, events of the form $\{V/\hat{\boldsymbol \varsigma}_k^{\tau}\preceq t\}$ are understood as $\{V\preceq t\odot \hat{\boldsymbol \varsigma}_k^{\tau}\}$, where $V\in\mathbb{R}^k$ is random, $t\in\mathbb{R}^k$ is fixed, and $\odot$ is entrywise multiplication. Lemma~\ref{lemma:trans_var_lower_bound} in the supplementary material also shows that such cases occur with negligible probability.) Lastly, recall that we write $\boldsymbol h (v) = (h(v_1), \dots, h(v_k))$ for a $k$-dimensional vector $v$ and \smash{transformation $h$.}

\begin{theorem} \label{THM:TRANS_GENERAL}
Suppose that Assumption \ref{model_assumptions} holds. Fix a transformation $h\in\mathcal{H}$ and a constant $\tau\in[0,1]$ with respect to $n$, and let $q = 5\log(kn)$. Then, there is a constant $c > 0$ not depending on $n$, such that the event
\scriptsize
\begin{align*}
\sup_{t \in \mathbb{R}^k } \bigg| \P \Bigg( \frac{\boldsymbol h ({\boldsymbol \lambda}_k(\hat\Sigma)) -\boldsymbol h( {\boldsymbol \lambda}_k(\Sigma))}{\boldsymbol\varsigma_k^{\tau}} {\  \preceq t} \bigg) %
- \P \bigg( \frac{\boldsymbol h ({\boldsymbol \lambda}_k(\hat\Sigma^{\star})) -\boldsymbol h( {\boldsymbol \lambda}_k(\hat\Sigma))}{\boldsymbol{\hat{\varsigma}}_k^{\tau}} {\  \preceq t}\,\bigg|X \bigg) \Bigg|
& \ \leq \
\frac{c\, \log(n)\, \beta_{3q}^{5}\, {\tt{r}}(\Sigma) }{\sqrt n}
\end{align*}
\normalsize
holds with probability at least $1 - c/n$.
\end{theorem}
\noindent \textbf{Remarks.}  The proof of Theorem~\ref{THM:TRANS_GENERAL} is given in Section \ref{appendix_D} of the supplementary material. To comment on the technical relationship between Theorems~\ref{THM:ALL} and \ref{THM:TRANS_GENERAL}, it is important to call attention to the differences between asymptotic and non-asymptotic analysis. When using asymptotics, the process of showing that bootstrap consistency for $\sqrt{n} \big( {\boldsymbol \lambda}_k (\hat{\Sigma}) - {\boldsymbol \lambda}_k (\Sigma) \big)$ implies the same for $(\boldsymbol h ({\boldsymbol \lambda}_k(\hat\Sigma)) -\boldsymbol h( {\boldsymbol \lambda}_k(\Sigma)))/\boldsymbol\varsigma_k^{\tau} $ can typically be handled with a brief argument, based on the delta method and the consistency of the estimate $\boldsymbol{\hat{\varsigma}}_k^{\tau}$. However, when taking a non-asymptotic approach, this process is much more involved. For instance, it is necessary to establish fine-grained error bounds for $\boldsymbol{\hat{\varsigma}}_k^{2}$, such as in showing that the uniform relative error $\|(\boldsymbol{\hat{\varsigma}}_k^{2}-\boldsymbol{\varsigma}_k^{2})/\boldsymbol\varsigma_k^2\|_{\infty}$ is likely to be at most of order $n^{-1/2}\beta_{2q}^3\lambda_1(\Sigma)^2{\tt{r}}(\Sigma)$ up to logarithmic factors.

\vspace{0.2cm}

\section{Numerical Experiments}\label{sec:experiments}

In this section, we focus on the application of constructing simultaneous confidence intervals for $\lambda_1(\Sigma),\dots,\lambda_k(\Sigma)$. This will be done in a variety of settings, corresponding to different values of $n$ and $p$, as well as different values of effective rank, and different choices of transformations. In a nutshell, there are two overarching conclusions to take away from the experiments: {\textbf{(1)}} In situations where $n\ll p$ and ${\tt{r}}(\Sigma)\asymp 1$, the bootstrap generally produces intervals with accurate coverage, which provides a confirmation of our theoretical results. {\textbf{(2)}} The classical log transformation mostly works well in low dimensions, but it can lead to coverage that is substantially below the nominal level when ${\tt{r}}(\Sigma)$ is moderately large. Nevertheless, we show that it is possible to find transformations that offer more reliable coverage in this challenging case.
More generally, this indicates that alternative transformations are worth exploring in high-dimensional settings. \\[-0.3cm]

\subsection{Simulation settings}
The eigenvalues of the population covariance matrix $\Sigma$ were chosen to have two different decay profiles:
\begin{enumerate}
\item[(a)] A polynomial decay profile $\lambda_j(\Sigma) = j^{-\gamma}$ for all $j = 1, \dots, p$, with $\gamma \in \{ 0.7, 1.0, 1.3 \}$. \\[-0.2cm]
\item[(b)] An exponential decay profile $\lambda_j(\Sigma) = \delta^j$ for all $j = 1, \dots, p$, with $\delta \in \{ 0.7, 0.8, 0.9 \}$.
\end{enumerate}
As a clarification, it is important to note that the effective rank of $\Sigma$ increases for larger values of $\delta$, but decreases for larger values of $\gamma$. For the purposes of simulations, the choices (a) and (b) have the valuable property that the eigenvalues are parameterized in the same way for every choice of $p$, which facilitates the comparison of results across different dimensions. The matrix of eigenvectors for $\Sigma$ was drawn uniformly from the set of $p\times p$ orthogonal matrices.
The dimension $p$ was taken from $\{10, 50, 100, 200\}$, and the sample size $n$ ranged from 50 to 500.  For each triple $(n, p, \gamma)$ or $(n,p,\delta)$, the data $X_1,\dots,X_n$ were generated in an i.i.d.~manner with the following choices for the distribution of $X_1$:
\begin{enumerate}
\item[(i)] The vector $X_1=\Sigma^{1/2}\xi V$ was generated with $V$ being uniformly distributed on the unit sphere of $\mathbb{R}^p$, and $\xi^2$ being an exponential random variable independent of $V$ with $\mathbb{E}[\xi^2]=p$. \\[-0.2cm]
\item[(ii)] The vector $X_1$ was generated from the Gaussian distribution $N(0,\Sigma)$.
\end{enumerate}
For each parameter setting, we generated 1000 realizations of the dataset $X_1,\dots,X_n$, and for each such realization, we generated $B:=1000$ sets of bootstrap samples of size $n$.
When constructing simultaneous confidence intervals for $\lambda_1(\Sigma),\dots,\lambda_k(\Sigma)$, the value of $k$ was set to $5$.

\subsection{Bootstrap confidence intervals}\label{sec:bootci}

For any $\alpha\in (0,1)$, we aim to construct approximate versions of ideal random intervals $\mathcal{I}_1,\dots,\mathcal{I}_k$ that satisfy
\begin{equation}\label{eqn:desiredCI}
\P \Bigg( \bigcap_{j = 1}^k \big\{ \lambda_j (\Sigma) \in \I_j \big\} \Bigg) \ \geq \ 1 - \alpha.
\end{equation}

\noindent
To this end, consider the following max and min statistics, based on any choice of partial standardization parameter $\tau\in[0,1]$ and transformation $h$,
\begin{align*}
M & \ =\ \max_{1 \leq j \leq k}  \frac{ h(\lambda_j (\hat{\Sigma})) - h(\lambda_j (\Sigma)) }{\varsigma_j^\tau}\\
L & \ =\ \min_{1\leq j\leq k} \frac{h(\lambda_j (\hat{\Sigma})) - h(\lambda_j (\Sigma)) }{\varsigma_j^\tau}.
\end{align*}
Letting $q_{M}(\alpha)$ and $q_L(\alpha)$ denote the respective $\alpha$-quantiles of $M$ and $L$ for any $\alpha\in (0,1)$, it follows that the desired condition~\eqref{eqn:desiredCI} holds if each interval  $\mathcal{I}_j$ is defined as
\begin{equation}\label{eqn:idealintervals}
\mathcal{I}_j =
h^{-1}\bigg(\bigg[ h(\lambda_j(\hat\Sigma))-\varsigma_j^{\tau}q_M(1-\textstyle\frac{\alpha}{2}) \ , \ h(\lambda_j(\hat\Sigma))-\varsigma_j^{\tau}q_L(\textstyle\frac{\alpha}{2}) \bigg]\bigg),
\end{equation}
with $h^{-1}([a,b])$ being understood as the preimage of $[a,b]$ under $h$.

To construct bootstrap intervals $\hat{\mathcal{I}}_1,\dots,\hat{\mathcal{I}}_k$ based on~\eqref{eqn:idealintervals}, it is only necessary to replace $q_M(1-\frac{\alpha}{2})$, $q_{L}(\frac{\alpha}{2})$, and $\varsigma_1,\dots,\varsigma_k$ with estimates.
In detail, each $\varsigma_j$ is first estimated using the sample variance of  $B$ bootstrap replicates of the form $h(\lambda_j(\hat\Sigma^{\star}))$. Next, the empirical $1-\frac{\alpha}{2}$ quantile of $B$ bootstrap replicates of the form $M^{\star}=\max_{1 \leq j \leq k}  [h(\lambda_j (\hat{\Sigma}^{\star})) - h(\lambda_j (\hat\Sigma))]/\hat\varsigma_j^{\tau} $ is taken as an estimate of $q_M(1-\frac{\alpha}{2})$, and similarly for $q_L(\frac{\alpha}{2})$.

Regarding the use of transformations, the following three options were included in the experiments:
\begin{itemize}
\item \emph{log transformation}: $h(x)=\log(x)$ with $\tau=0$.\\[-0.2cm]
\item \emph{standardization}: $h(x)=x$ with $\tau=1$.\\[-0.2cm]
\item \emph{square-root transformation}: $h(x)=x^{1/2}$ with $\tau\in[0,1]$ chosen data-adaptively.
\end{itemize}
In the case of the log transformation, the choice of $\tau=0$ corresponds to the way that this transformation has been used in the classical literature~\citep{BeranS85}, while in the case of standardization, the choice of $\tau=1$ is definitional. For the square-root transformation, the use of a data-adaptive selection rule for $\tau\in[0,1]$ is more nuanced, and can be informally explained in terms of the following ideas developed previously in~\citep{LopesLM18, LinLM2020}.

In essence, this choice can be understood in terms of a trade-off between two competing effects that occur in the extreme cases of $\tau=1$ and $\tau =0$.
When using $\tau=1$, the random variables $[\lambda_j(\hat{\Sigma})^{1/2} - \lambda_j (\Sigma)^{1/2}]/\varsigma_j$ with $\varsigma_j^2=\operatorname{var}(\lambda_j(\hat\Sigma)^{1/2})$ and $j=1,\dots,k$ are on approximately ``equal footing'', which makes the behavior of the statistic $M$ sensitive to their joint distribution (and likewise for $L$).
By contrast, when $\tau=0$ is used, the variables $[\lambda_j(\hat{\Sigma})^{1/2} - \lambda_j (\Sigma)^{1/2}]$ will tend to be on different scales, and the variable on the largest scale, say $j^{\prime}$, will be the maximizer for $M$ relatively often. In this situation, the statistic $M$ is governed more strongly by the marginal distribution of $[\lambda_{j^{\prime}}( \hat{\Sigma})^{1/2} - \lambda_{j^{\prime}} (\Sigma)^{1/2}]$. So, from this heuristic point of view, the choice of $\tau=0$ can simplify the behavior of $M$ relative to the case of $\tau=1$, making the distribution of $M$ easier to approximate.
However, the choice of $\tau=0$ also has the drawback that it can lead to simultaneous confidence intervals that are excessively wide, because the widths are no longer adapted to the different values $\varsigma_1,\dots,\varsigma_k$ (since $\varsigma_1^0=\cdots=\varsigma_k^0=1$).

To strike a balance between these competing effects, we used the following simple rule to select $\tau$ in the case of the square-root transformation. For a candidate value of $\tau$, let $\hat{\mathcal{I}}_1(\tau),\dots,\hat{\mathcal{I}}_k(\tau)$ denote the associated bootstrap intervals defined beneath equation~\eqref{eqn:idealintervals} (so that the dependence on $\tau$ is explicit), and let $|\hat\I_1(\tau)|,\dots,|\hat\I_k(\tau)|$ denote their widths. Also define $\hat\mu(\tau) =  \sum_{j=1}^k |\hat\I_j(\tau)| / k$ and $\hat\sigma(\tau)^2 = \sum_{i = 1}^k (|\hat\I_j(\tau)| - \hat\mu(\tau) )^2 / k$. In this notation, we selected the value of $\tau$ that minimized $\hat\mu(\tau)+\hat\sigma(\tau)$ over the set of candidates $\{0.0, 0.1,\dots,0.9, 1.0\}$. Different variants of this type of criterion minimization rule have also been observed to be effective in other contexts~\citep{LopesLM18,LinLM2020}. \\[-0.3cm]

\begin{figure}[H]
\vspace{0.5cm}

	\quad\quad\quad
	\begin{overpic}[width=0.29\textwidth]{new_plots/power_elliptical_sqrtexp/power_elliptical_sqrtexp_sim_tau=00_log_lvl=095_plt_coverage_beta=07.eps}
\put(25,80){ \ul{\ \ \  \ $\gamma=0.7$ \ \ \ \    }}
		\put(-20,-5){\rotatebox{90}{ {\small \ \ \ log transformation  \ \ }}}
\end{overpic}
	~
	\DeclareGraphicsExtensions{.png}
	\begin{overpic}[width=0.29\textwidth]{new_plots/power_elliptical_sqrtexp/power_elliptical_sqrtexp_sim_tau=00_log_lvl=095_plt_coverage_beta=10.eps}
		\put(25,80){ \ul{\ \ \  \ $\gamma=1.0$ \ \ \ \    }}
	\end{overpic}
	~
	\begin{overpic}[width=0.29\textwidth]{new_plots/power_elliptical_sqrtexp/power_elliptical_sqrtexp_sim_tau=00_log_lvl=095_plt_coverage_beta=13.eps}
		\put(25,80){ \ul{\ \ \  \ $\gamma=1.3$ \ \ \ \    }}
	\end{overpic}
\end{figure}

\vspace{-0.7cm}

\begin{figure}[H]

	\quad\quad\quad
	\begin{overpic}[width=0.29\textwidth]{new_plots/power_elliptical_sqrtexp/power_elliptical_sqrtexp_sim_tau=10_power_10_lvl=095_plt_coverage_beta=07.eps}
	\put(-20,-1){\rotatebox{90}{ {\small \ \ \ standardization \  \ \ }}}
	\end{overpic}
	~
	\DeclareGraphicsExtensions{.png}
	\begin{overpic}[width=0.29\textwidth]{new_plots/power_elliptical_sqrtexp/power_elliptical_sqrtexp_sim_tau=10_power_10_lvl=095_plt_coverage_beta=10.eps}
	\end{overpic}
	~
	\begin{overpic}[width=0.29\textwidth]{new_plots/power_elliptical_sqrtexp/power_elliptical_sqrtexp_sim_tau=10_power_10_lvl=095_plt_coverage_beta=13.eps}
	\end{overpic}
\end{figure}

\vspace{-0.7cm}

\begin{figure}[H]

	\quad\quad\quad
	\begin{overpic}[width=0.29\textwidth]{new_plots/power_elliptical_sqrtexp/power_elliptical_sqrtexp_sim_select_power_05_lvl=095_plt_coverage_beta=07.eps}
		\put(-21,1){\rotatebox{90}{\ $\sqrt{ \ \ }$}}
	    \put(-20,-3){\rotatebox{90}{  { \ \ \ \ \ \ \small transformation \ \ } }}

	\end{overpic}
	~
	\DeclareGraphicsExtensions{.png}
	\begin{overpic}[width=0.29\textwidth]{new_plots/power_elliptical_sqrtexp/power_elliptical_sqrtexp_sim_select_power_05_lvl=095_plt_coverage_beta=10.eps}
	\end{overpic}
	~
	\begin{overpic}[width=0.29\textwidth]{new_plots/power_elliptical_sqrtexp/power_elliptical_sqrtexp_sim_select_power_05_lvl=095_plt_coverage_beta=13.eps}
	\end{overpic}
	\vspace{+.2cm}
	\caption{(Simultaneous coverage probability versus $n$ in simulation model (i) with a polynomial decay profile). In each panel, the $y$-axis measures~$\P(\cap_{j=1}^5 \{\lambda_j(\Sigma)\in\hat{\mathcal{I}}_j\})$ based on a nominal value of 95\%, and the $x$-axis measures $n$. The colored curves correspond to the different values of $p$, indicated in the legend. }\label{fig1}

\end{figure}

\subsection{Discussion of coverage}

Figure \ref{fig1} contains nine panels displaying the results for the simultaneous coverage probability $\P(\cap_{j=1}^5 \{\lambda_j(\Sigma)\in\hat{\mathcal{I}}_j\})$, based on a nominal value of 95\% (i.e.~$\alpha=0.05$) in the case of the simulation model (i) with a polynomial decay profile for the population eigenvalues. The figure summarizes a large amount of information, because it shows how the coverage depends on $n$, $p$, the eigenvalue decay parameter $\gamma$, and the three transformations described above. For each panel, the $x$-axis measures $n$, and the $y$-axis measures $\P(\cap_{j=1}^5 \{\lambda_j(\Sigma)\in\hat{\mathcal{I}}_j\})$. Results corresponding to the dimensions $p=10, 50, 100, 200$ are plotted with colored curves that are labeled in the legend. The three rows of panels from top to bottom correspond to the log transformation, ordinary standardization, and the square-root transformation. The three columns of panels from left to right correspond to the eigenvalue decay parameters $\gamma=0.7, 1.0, 1.3$.  In addition, Figure~\ref{fig3} displays analogous results for exponentially decaying population eigenvalues in model (i). Lastly, results for model (ii), as well as for a nominal value of 90\% (instead of 95\%), are provided in Section~\ref{appendix_last} of the supplementary material.

\begin{figure}[H]
\vspace{0.5cm}

	\quad\quad\quad
	\begin{overpic}[width=0.29\textwidth]{new_plots/exp_elliptical_sqrtexp/exp_elliptical_sqrtexp_sim_tau=00_log_lvl=095_plt_coverage_beta=07.eps}
\put(25,80){ \ul{\ \ \  \ $\delta=0.7$ \ \ \ \    }}
		\put(-20,-5){\rotatebox{90}{ {\small \ \ \ log transformation  \ \ }}}
\end{overpic}
	~
	\DeclareGraphicsExtensions{.png}
	\begin{overpic}[width=0.29\textwidth]{new_plots/exp_elliptical_sqrtexp/exp_elliptical_sqrtexp_sim_tau=00_log_lvl=095_plt_coverage_beta=08.eps}
		\put(25,80){ \ul{\ \ \  \ $\delta=0.8$ \ \ \ \    }}
	\end{overpic}
	~
	\begin{overpic}[width=0.29\textwidth]{new_plots/exp_elliptical_sqrtexp/exp_elliptical_sqrtexp_sim_tau=00_log_lvl=095_plt_coverage_beta=09.eps}

		\put(25,80){ \ul{\ \ \  \ $\delta=0.9$ \ \ \ \    }}

	\end{overpic}
\end{figure}
\vspace{-0.7cm}

\begin{figure}[H]

	\quad\quad\quad
	\begin{overpic}[width=0.29\textwidth]{new_plots/exp_elliptical_sqrtexp/exp_elliptical_sqrtexp_sim_tau=10_power_10_lvl=095_plt_coverage_beta=07.eps}
	\put(-20,-1){\rotatebox{90}{ {\small \ \ \ standardization \  \ \ }}}
	\end{overpic}
	~
	\DeclareGraphicsExtensions{.png}
	\begin{overpic}[width=0.29\textwidth]{new_plots/exp_elliptical_sqrtexp/exp_elliptical_sqrtexp_sim_tau=10_power_10_lvl=095_plt_coverage_beta=08.eps}
	\end{overpic}
	~
	\begin{overpic}[width=0.29\textwidth]{new_plots/exp_elliptical_sqrtexp/exp_elliptical_sqrtexp_sim_tau=10_power_10_lvl=095_plt_coverage_beta=09.eps}

	\end{overpic}
\end{figure}

\vspace{-0.7cm}

\begin{figure}[H]

	\quad\quad\quad
	\begin{overpic}[width=0.29\textwidth]{new_plots/exp_elliptical_sqrtexp/exp_elliptical_sqrtexp_sim_select_power_05_lvl=095_plt_coverage_beta=07.eps}
		\put(-21,1){\rotatebox{90}{\ $\sqrt{ \ \ }$}}
	    \put(-20,-3){\rotatebox{90}{  { \ \ \ \ \ \ \small transformation \ \ } }}

	\end{overpic}
	~
	\DeclareGraphicsExtensions{.png}
	\begin{overpic}[width=0.29\textwidth]{new_plots/exp_elliptical_sqrtexp/exp_elliptical_sqrtexp_sim_select_power_05_lvl=095_plt_coverage_beta=08.eps}
	\end{overpic}
	~
	\begin{overpic}[width=0.29\textwidth]{new_plots/exp_elliptical_sqrtexp/exp_elliptical_sqrtexp_sim_select_power_05_lvl=095_plt_coverage_beta=09.eps}

	\end{overpic}
	\vspace{+.2cm}
	\caption{(Simultaneous coverage probability versus $n$ in simulation model (i) with an exponential decay profile). The plotting scheme is the same as described in the caption of Figure~\ref{fig1}, except that the three columns correspond to values of the eigenvalue decay parameter $\delta$.}
	\label{fig3}
\end{figure}

There are several notable patterns in Figures~\ref{fig1} to discuss. The first is that faster rates of decay tend to lead to better coverage accuracy---as anticipated by our theoretical results. In particular, when the eigenvalue decay parameter is set to $\gamma=1.3$, the coverage is rather accurate even when $n\ll p$. Furthermore, the accuracy is essentially unaffected by the dimension $p$ in this situation, as indicated by the overlap of the four colored curves. On the other hand, as the decay parameter becomes smaller,
the three transformations perform in different ways. For instance, when $\gamma=0.7$, $p=200$, and $n<200$, the log transformation yields coverage that clearly falls short of the nominal level. By contrast, the standardization and square-root transformations tend to err more safely in the conservative direction when $\gamma=0.7$. To give some indication of the difficulty of $\gamma=0.7$, it should be noted that if $\gamma$ were decreased slightly to 0.5 with $p\gtrsim n$, this would imply $\textstyle {\tt{r}}(\Sigma) / \sqrt{n} \asymp \sqrt{p/n}\gtrsim 1$, in which case bootstrap consistency would not be guaranteed. When considering all three cases $\gamma=0.7,0.8,0.9$ collectively, the square-root transformation seems to yield the best overall coverage results if conservative errors are viewed as preferable to anti-conservative ones.

Turning to the coverage results for exponential spectrum decay, the log and square-root transformations continue to follow the pattern that faster decay improves coverage accuracy. Also, the log transformation maintains its tendency to err in the anti-conservative direction, while the square-root transformation maintains its tendency to err in the conservative direction. Meanwhile, ordinary standardization yields larger errors in the anti-conservative direction than it did in the previous context.

\begin{figure}[H]
\vspace{0.5cm}

	\quad\quad\quad
	\begin{overpic}[width=0.29\textwidth]{new_plots/power_elliptical_sqrtexp/power_elliptical_sqrtexp_sim_tau=00_log_lvl=095_plt_width_beta=07.eps}
	%
		\put(25,80){ \ul{\ \ \  \ $\gamma=0.7$ \ \ \ \    }}
		\put(-20,-5){\rotatebox{90}{ {\small \ \ \ log transformation  \ \ }}}
	\end{overpic}
	~
	\DeclareGraphicsExtensions{.png}
	\begin{overpic}[width=0.29\textwidth]{new_plots/power_elliptical_sqrtexp/power_elliptical_sqrtexp_sim_tau=00_log_lvl=095_plt_width_beta=10.eps}
		\put(25,80){ \ul{\ \ \  \ $\gamma=1.0$ \ \ \ \    }}
	\end{overpic}
	~
	\begin{overpic}[width=0.29\textwidth]{new_plots/power_elliptical_sqrtexp/power_elliptical_sqrtexp_sim_tau=00_log_lvl=095_plt_width_beta=13.eps}
		%
		\put(25,80){ \ul{\ \ \  \ $\gamma=1.3$ \ \ \ \    }}
	\end{overpic}
\end{figure}

\vspace{-0.7cm}

\begin{figure}[H]

	\quad\quad\quad
	\begin{overpic}[width=0.29\textwidth]{new_plots/power_elliptical_sqrtexp/power_elliptical_sqrtexp_sim_tau=10_power_10_lvl=095_plt_width_beta=07.eps}
	    \put(-20,-1){\rotatebox{90}{ {\small \ \ \ standardization \  \ \ }}}
	\end{overpic}
	~
	\DeclareGraphicsExtensions{.png}
	\begin{overpic}[width=0.29\textwidth]{new_plots/power_elliptical_sqrtexp/power_elliptical_sqrtexp_sim_tau=10_power_10_lvl=095_plt_width_beta=10.eps}
	\end{overpic}
	~
	\begin{overpic}[width=0.29\textwidth]{new_plots/power_elliptical_sqrtexp/power_elliptical_sqrtexp_sim_tau=10_power_10_lvl=095_plt_width_beta=13.eps}
	\end{overpic}
	%

\end{figure}

\vspace{-0.7cm}

\begin{figure}[H]

	\quad\quad\quad
	\begin{overpic}[width=0.29\textwidth]{new_plots/power_elliptical_sqrtexp/power_elliptical_sqrtexp_sim_select_power_05_lvl=095_plt_width_beta=07.eps}
	    \put(-21,1){\rotatebox{90}{\ $\sqrt{ \ \ }$}}
	    \put(-20,-3){\rotatebox{90}{  { \ \ \ \ \ \ \small transformation \ \ } }}

	\end{overpic}
	~
	\DeclareGraphicsExtensions{.png}
	\begin{overpic}[width=0.29\textwidth]{new_plots/power_elliptical_sqrtexp/power_elliptical_sqrtexp_sim_select_power_05_lvl=095_plt_width_beta=10.eps}
	\end{overpic}
	~
	\begin{overpic}[width=0.29\textwidth]{new_plots/power_elliptical_sqrtexp/power_elliptical_sqrtexp_sim_select_power_05_lvl=095_plt_width_beta=13.eps}
		%
	\end{overpic}
	%
	\vspace{+.1cm}
	\caption{(Average width versus $n$ in simulation model (i) with a polynomial decay profile). In each of the nine panels, the $y$-axis measures the average width $\mathbb{E}[|\hat{\I}_1|+\cdots+|\hat{\I}_{5}|] / 5$, and the $x$-axis measures $n$. The colored curves correspond to the different values of $p=10$, $50$, $100$, $200$, indicated in the legend. The three rows and three columns correspond to labeled choices of transformations and values of the eigenvalue decay parameter $\gamma$.} \label{fig5}
\end{figure}

\subsection{Discussion of width}
Beyond coverage probability, interval width is another important factor to consider when appraising confidence intervals. In Figures~\ref{fig5}-\ref{fig_width_delta_ii}, the average width~$\mathbb{E}[|\hat{\mathcal{I}}_1|+\cdots+|\hat{\mathcal{I}}_k|] / k$ is plotted on the $y$-axis as a function of the sample size $n$ on the $x$-axis, with the underlying parameter settings being organized in the same manner as in Figures~\ref{fig1}-\ref{fig4}. (Corresponding results for settings based on model (ii) and a nominal value of 90\% are presented in  Section~\ref{appendix_last} of the supplementary material.) With regard to the three transformations, they produce intervals that have roughly similar widths across most parameter settings. However, at a more fine-grained level, the results in the case of polynomial spectrum decay show that the log transformation tends to yield slightly shorter widths than the square-root transformation, which in turn, tends to yield slightly shorter widths than ordinary standardization. In the case of exponential spectrum decay with $\delta=0.9$, the same pattern is also apparent, while for smaller values of $\delta$, there is not much difference among the transformations.

Aside from the transformations, there are two other general trends to notice. Within each of the 18 panels of Figures~\ref{fig5}-\ref{fig_width_delta_ii}, there is a monotone relationship between width and the dimension $p$, with the width generally increasing as the dimension increases. Similarly, the width generally also increases as the effective rank ${\tt{r}}(\Sigma)$ increases. \\[-0.2cm]

\begin{figure}[H]
\vspace{0.5cm}

	\quad\quad\quad
	\begin{overpic}[width=0.29\textwidth]{new_plots/exp_elliptical_sqrtexp/exp_elliptical_sqrtexp_sim_tau=00_log_lvl=095_plt_width_beta=07.eps}
\put(25,80){ \ul{\ \ \  \ $\delta=0.7$ \ \ \ \    }}
		\put(-20,-5){\rotatebox{90}{ {\small \ \ \ log transformation  \ \ }}}
\end{overpic}
	~
	\DeclareGraphicsExtensions{.png}
	\begin{overpic}[width=0.29\textwidth]{new_plots/exp_elliptical_sqrtexp/exp_elliptical_sqrtexp_sim_tau=00_log_lvl=095_plt_width_beta=08.eps}
		\put(15,80){ \ul{\ \ \  \ $\delta=0.8$ \ \ \ \    }}
	\end{overpic}
	~
	\begin{overpic}[width=0.29\textwidth]{new_plots/exp_elliptical_sqrtexp/exp_elliptical_sqrtexp_sim_tau=00_log_lvl=095_plt_width_beta=09.eps}
		\put(15,80){ \ul{\ \ \  \ $\delta=0.9$ \ \ \ \    }}
	\end{overpic}
\end{figure}

\vspace{-0.5cm}

\begin{figure}[H]

	\quad\quad\quad
	\begin{overpic}[width=0.29\textwidth]{new_plots/exp_elliptical_sqrtexp/exp_elliptical_sqrtexp_sim_tau=10_power_10_lvl=095_plt_width_beta=07.eps}
	\put(-20,-1){\rotatebox{90}{ {\small \ \ \ standardization \  \ \ }}}
	\end{overpic}
	~
	\DeclareGraphicsExtensions{.png}
	\begin{overpic}[width=0.29\textwidth]{new_plots/exp_elliptical_sqrtexp/exp_elliptical_sqrtexp_sim_tau=10_power_10_lvl=095_plt_width_beta=08.eps}
	\end{overpic}
	~
	\begin{overpic}[width=0.29\textwidth]{new_plots/exp_elliptical_sqrtexp/exp_elliptical_sqrtexp_sim_tau=10_power_10_lvl=095_plt_width_beta=09.eps}

	\end{overpic}
\end{figure}

\vspace{-0.5cm}

\begin{figure}[H]

	\quad\quad\quad
	\begin{overpic}[width=0.29\textwidth]{new_plots/exp_elliptical_sqrtexp/exp_elliptical_sqrtexp_sim_select_power_05_lvl=095_plt_width_beta=07.eps}
		\put(-21,1){\rotatebox{90}{\ $\sqrt{ \ \ }$}}
	    \put(-20,-3){\rotatebox{90}{  { \ \ \ \ \ \ \small transformation \ \ } }}

	\end{overpic}
	~
	\DeclareGraphicsExtensions{.png}
	\begin{overpic}[width=0.29\textwidth]{new_plots/exp_elliptical_sqrtexp/exp_elliptical_sqrtexp_sim_select_power_05_lvl=095_plt_width_beta=08.eps}
	\end{overpic}
	~
	\begin{overpic}[width=0.29\textwidth]{new_plots/exp_elliptical_sqrtexp/exp_elliptical_sqrtexp_sim_select_power_05_lvl=095_plt_width_beta=09.eps}

	\end{overpic}
	\vspace{+.2cm}
	\caption{(Average width versus $n$ in simulation model (i) with an exponential decay profile). The plotting scheme is the same as described in the caption of Figure~\ref{fig5}, except that the three columns correspond to values of the eigenvalue decay parameter $\delta$.}
	\label{fig7}
\end{figure}


\section{Introduction}

Standard Model relics do not have sufficient abundance to explain
the existence of dark matter in the universe, apart from  (sterile) neutrinos which erase small-scale fluctuations
via their free-streaming. To avoid excessive power suppression,  investigations have
generally focused on particles generated via extensions of the Standard Model and, in particular, from
minimal supersymmetric theories. So far, no signal of supersymmetry has
been found in experiments performed with the Large Hadron Collider \cite{arcadi}
and possible results from direct
search experiments are unconfirmed \cite{bernabei, aprile}.

Negative results have also been
obtained via indirect searches related to the detection of high energy neutrinos,  leptons, hadrons. and photons  that could have
been produced by the annihilation of dark matter particles  predicted in diverse environments \cite{srednicki}, including
the sun \cite{choi},  the solar neighborhood \cite{cirelli},  the Galactic Center \cite{leane}
or  in nearby dwarf galaxies  \cite{abdalla,hoof}.

Confronted with these difficulties, a revival of the idea that primordial black holes (PBHs) could be dark
matter particles  has flourished in the recent literature. Although early investigations on the formation of PBHs have
focused on the radiation-dominated phase just after reheating, some studies  have  raised the possibility that  the
scalar fields that drive inflation may be affected by gravitational instabilities (similar to the Jeans instability) that may lead to
black hole formation \cite{khlopov1,khlopov2,malomed} at the end of inflation when

the fields (or field) driving the expansion oscillate(s) around a local minimum of the potential before
decaying into matter/radiation. During this short oscillatory phase, the energy density varies approximately as
$\rho_{\phi} \propto a^{-3}$, i.e., it behaves like a "dust'' fluid. This short matter-dominated  era is advantageous for black hole formation
since pressure gradients are practically absent in density fluctuations, contrary to  the situation in  the post-reheating
phase. The formation of these PBHs depends not only on the amplitude of the perturbations but also on the form of the field potential.
Moreover, as discussed below,  a broad mass spectrum of PBHs is produced that results in substantial growth of the most massive PBHs by accretion of small PBHs before they have time to Hawking evaporate.

Two principal scenarios have been envisaged: in the first, only one field is responsible for driving the inflation, while in the second, an
additional ''spectator'' field (or several such fields) may be also present. In the former scenario, the scalar field can fragment into lumps via gravitational
instability and form black holes under certain conditions \cite{cotner}. In the second picture, one or more spectator fields are present.
These fields are characterized by the fact that their energy densities are never dominant in comparison with that of the main field. However, their
curvature power spectra dominate at small scales, favoring black hole formation \cite{tenkanen}. PBHs originating from these mechanisms
have, in general, masses larger than $10^{15}~\rm g$ and hence they survive the Hawking evaporation process.

Here we are concerned with smaller mass PBHs  that would evaporate on short timescales, potentially leaving
a Planck-sized remnant. The possible existence of these relics and their possible stability are the main purpose of the present investigation.
It is worth mentioning that PBHs of small masses  ($M \sim 10^{6}~g$) are expected to be formed in a hybrid inflationary scenario
as discussed by \cite{garcia,chen}. Also in a single
field scenario, small mass  black holes  form via a resonant instability that develops during the oscillatory phase for
wavelengths that exit the Hubble radius near the end of inflation \cite{martin}. This mechanism is able to produce PBHs with masses
around $\sim 10^{3}~\rm g$.

We envisage two alternative scenarios for these PBHs. They
may  evaporate but leave stable  Planck mass remnants,
accounting  for a substantial part of dark matter \cite{martin}. Alternatively, if the mass range  is broad, accretion of many low mass  PBHs by the most massive PBHs will produce long-lived PBHs that can equally account for most of the dark matter by populating the
only available PBH window, in the mass range $10^{17}-10^{22} $ g.

Here we consider a scenario in which PBHs  formed
by the aforementioned mechanisms, leaving extremal rotating remnants. Our motivation is based on recent
investigations suggesting that PBHs formed in such a particular phase of the universe might have acquired
significant angular momentum \cite{harada,kuhnel}.

Surviving  Planck-scale objects can be described by a particle-like formalism \cite{nicolini1,nicolini2} that permits the existence of an extremal configuration associated with the ground state.  The PBHs formed  by accretion must lie above the accretion limit imposed by the diffuse gamma ray background.

Our paper is organized as follows: Planck-sized relics and their stability are discussed in Section 2 and
the particle-like formalism describing the black hole will be presented in Section 3. It will be shown that
the resulting horizon area is equally spaced in agreement with the early findings by
Bekenstein \cite{bekenstein} and that the Bohr correspondence principle is verified or, in other words, General
Relativity is recovered at large quantum numbers.  A following section describes the PBH accretion scenario. Our  main conclusions are
given in Section 5.
\section{Planck remnants}

The Hawking evaporation process has always been an obstacle for explaining the survival of PBHs with
masses smaller than $\sim 10^{15}~g$. To avoid such a difficulty, MacGibbon \cite{macgibbon} postulated that Planck-sized black holes could be surviving
remnants of such a process and could eventually constitute a substantial fraction of the dark matter. The existence
of relics of the Hawking process is still a matter of debate since it  is related not only to the possibility of having Planck-scale black holes as dark matter but also to the disappearance of the singularity \cite{sabine,bonanno}.

A possible resolution derived from string theory leads to a generalization of the
Heisenberg uncertainty principle (or simply GUP). This  suggests that  Hawking evaporation ends when the black hole mass attains a
minimal value of the order of the Planck mass \cite{chen,adler,faizal}. Such remnants are expected to not emit radiation/particles
despite  having an extremely high horizon temperature.
Black holes with an inner Cauchy horizon could be a possible resolution of  this problem, especially since in the extremal case, i.e., when the two
horizons coincide, the associated temperature is zero and Hawking emission is suppressed.

In this situation, extremal
regular black holes described either by the Bardeen or the Hayward metrics \cite{bardeen,hayward} are able to produce Planck-sized
objects if their Riemann curvature invariant is comparable to that derived in terms of the volume operator in Loop Quantum Gravity
\cite{pacheco2,modesto}. A similar approach was adopted by Chamseddine et al. \cite{chamseddine} who
assumed that General Relativity is modified at curvatures near the Planck scale. As a result, they have obtained stable (zero Hawking
temperature) remnants with masses determined by the inverse limiting curvature.

We note that
in loop gravity the Hawking evaporation time scales as $m_{bh}^2,$ as opposed to $m_{bh}^3$ in  Einstein gravity. This potentially affects the mass range of PBHs that survive evaporation.

Extremal singular charged or rotating black holes described respectively by the Reissner-Nordstrom or the Kerr metrics have been also
considered in the literature. In the former case,  debate about discharge by spontaneous pair production remains open. This
process affects large black holes ($M \sim 10^{17}~g$) while other processes also affect  the charge evolution of smaller ones as
emphasized by \cite{gibbons,sorkin}. The main difficulty in describing the discharge mechanism is to find an adequate
transition from a space-time with a Cauchy horizon to another in which the inner horizon is absent \cite{sorkin2,herman}.
More recently, Lehmann et al. \cite{lehmann}, searching for detectable effects of primordial Planck mass black holes, explored the idea
that a fraction of them could be charged, regardless of their origin and stability.

\section{Quantum black holes}

A quantum description of a black hole  necessarily requires a complete quantum theory of gravitation, still non-existent. In the
meantime, it is possible to adopt an approach in which either the mass-energy spectrum or the horizon surface of the black hole
may assume discrete values. In fact,
in a seminal paper, Bekenstein \cite{bekenstein} considered that the horizon area of non-extremal black holes behaves as a classical adiabatic invariant, corresponding to a quantum operator with a discrete spectrum, according to the Ehrenfest principle. His point of departure was the
squared-irreducible mass relation for a Kerr-Newman black hole with which quantum mechanical operators are associated. Another point of view was
developed by Carr \cite{carr}, who assumed a smooth transition between the Schwarzschild and Compton scales associated with a black hole of
mass $M$. To some extent , these approaches suggest that BHs can be assimilated into elementary particles with Planck masses. Although these ideas can be pushed
a little further, one would expect that any formal quantum particle description of a black hole should permit the recovery
of General Relativity at larger quantum numbers. More recently, a particle-like black hole formalism was adopted \cite{nicolini2} to describe
a Reissner-Nordstrom (RN) black hole. This approach permits
the existence of extremal configurations that correspond to a minimal mass below which no horizon forms. The Hawking temperature
in this extremal state is zero, and objects deviating from extremal configurations radiate  well-defined quanta
of energy in the form of  radiation and particles. Thus, one should expect that a quantum rotating black hole in an excited state
will also spontaneously decay into lower spin states since the asymmetric emission of neutrino quanta will carry away angular momentum
according to Leahy \& Unruh \cite{unruh}.

Here we will focus our interest in the Kerr space-time since it includes a Cauchy horizon permitting extremal solutions, stable from a
thermodynamic point of view. However, for the moment, the question concerning the existence of a ground-state solution that at the same time coincides with the extremal case remains open. It should be emphasized that black holes must also be considered as relativistic objects
in the sense that they are subject to strong gravitational fields. Contrary to the approach by \cite{bekenstein}, who adopted the squared irreducible mass of a Kerr-Newman black hole as the departure point, here the relation
defining  the inner ("$-$") and outer ("$+$") horizons of a Kerr black hole will be used, that is
\begin{equation}
\label{horizonkerr}
r_{\pm}(M,J) = \frac{GM}{c^2}\left[1 \pm \sqrt{1 - \left(\frac{Jc}{GM^2}\right)^2}\right]
\end{equation}
In the above equation, $M$ and $J$ are respectively the mass and the spin of the black hole and the other symbols have their usual meaning.
In a second step, it is assumed that the black hole spin is quantized by the usual rules, i.e., $J = m\hbar$, where $m = 0,1,2,...$.
In the particle-like description, the size of the black hole (or its outer horizon) should never be  smaller than the associated
Compton wavelength. Hence, following \cite{nicolini2,pankovic}, one assumes that
\begin{equation}
\label{quantum1}
r_+(M,J) = n\frac{\hbar}{Mc}
\end{equation}
where $n = 1,2,3,...$ Notice that up to this point, nothing is imposed concerning the mass of the black hole, but as we
shall see, the adopted quantum
conditions will lead naturally to Planckian values. Replacing the horizon radius given by eq.\ref{quantum1} into eq.\ref{horizonkerr}
permits us  to obtain an equation defining the possible values for the black hole masses, that is
\begin{equation}
\label{mass}
\frac{M}{M_P} = \sqrt{\frac{(n^2+m^2)}{2n}}
\end{equation}
where $M_P$ is the Planck mass. It is interesting to remark that the equation above, which gives discrete values for the black hole
mass, coincides with that derived by Bekenstein \cite{bekenstein} if the electric charge in his relation is put equal to zero. Now
insert  this result into eq.\ref{horizonkerr} and after some straightforward algebra, one obtains
\begin{equation}
\label{horizon1}
r_{\pm} = \ell_P\sqrt{\frac{(n^2+m^2)}{2n}}\left[1 \pm \sqrt{1-\frac{4m^2n^2}{(n^2+m^2)^2}}\right]
\end{equation}
where $\ell_P$ is the Planck length.

It can be shown trivially from eq.\ref{horizon1} that extremal black holes can be obtained if both quantum numbers are
equal, i.e.,  $n = m$. In this case, the horizon radius and the mass are respectively
\begin{equation}
r_H = \sqrt{n} \ell_P \,\,\,\, and \,\,\,\, M = \sqrt{n} M_P
\end{equation}

If such an extremal rotating black hole is in the ''ground'' state ($n = 1$), then the degenerate horizon $r_H$ and the mass
correspond precisely to the Planck length and mass with a spin equal to $\hbar$. Eq.\ref{horizon1} can be expanded and explicit
relations for the Cauchy and the event horizons  obtained, that is
\begin{equation}
\label{horizon2}
r_{-} = \sqrt{\frac{2m^4}{n(n^2+m^2)}}~\ell_P
\end{equation}
and
\begin{equation}
\label{horizon3}
r_{+} = \sqrt{\frac{2n^3}{(n^2+m^2)}}~\ell_P
\end{equation}

General Relativity imposes no restrictions on the mass spectrum of black holes, contrary to the discrete values resulting
from eq.\ref{mass}. If  Bohr's correspondence principle is applicable, one should expect to recover General Relativity
in the limit of large quantum numbers. Performing the ratio between eqs.\ref{horizon3} and \ref{mass}, one obtains after some
algebra
\begin{equation}
\frac{c^2r_{+}}{2GM} = \frac{n^2}{(n^2+m^2)}
\end{equation}
It can be easily verified that when  $n > m$ and $n >> 1$, the right side of the above equation  goes to unity and General Relativity
is recovered since the event horizon goes to $2GM/c^2$ as expected.

Another important aspect of the particle-like picture concerns the horizon area or the black hole entropy. Bekenstein, in his
pioneering work \cite{bekenstein}, suggested that the black hole area should be discrete and equally spaced
with a spectrum $A_n = \alpha\ell_P^2n$ with $n = 1,2,...$ He estimated the
proportionality constant as $\alpha = 8\pi$ by assuming that the squared irreducible mass of the black hole is the analogue of
an adiabatic invariant action integral. Based on Bohr's correspondence principle, Hod \cite{hod} reached  a different
result for the proportionality constant, i.e., $\alpha = 4\ln 3$. This has been contested by other authors (see, for instance \cite{maggiore,ropotenko}), who confirmed the value $\alpha = 8\pi$ by independent analyses.

 It is interesting to verify if the present
description of a quantum black hole is consistent with such a discretization of the black hole area and to ascertain the expected value of
the constant $\alpha$. For a Kerr black hole, the horizon area is given by
\begin{equation}
\label{area}
A = 4\pi\left[r_+^2 + \left(\frac{J}{Mc}\right)^2\right]
\end{equation}
Inserting into the above equation the horizon radius given by eq.\ref{horizon3}, the quantization condition for the spin and the black
hole mass given by eq.\ref{mass}, one  finally obtains
\begin{equation}
\label{area2}
A = 4\pi\ell_P^2\left[\frac{2n^3}{(n^2+m^2)} + \frac{2m^2n}{(n^2+m^2)}\right] = 8\pi\ell_P^2n
\end{equation}
Thus, the quantization conditions imposed in the present description of a quantum rotating black hole are consistent with
the discrete area spectrum first proposed by Bekenstein. It worth mentioning that Medved \cite{medved}, adopting a perturbative
approach of the adiabatic invariant method, reached a similar result but included a fourth order spin correction term. Moreover,
for a Kerr black hole family of solutions, the Cauchy horizon is the future boundary of the domain of
dependence of the event horizon and the product of the areas of both horizons depends only on the squared angular momentum
\cite{marcus}. The area product $A_+A_-$ of both horizons can be simply written as
\begin{equation}
\label{product1}
A_+A_- = (8\pi)^2\left(\frac{GM}{c^2}\right)^2r_+r_-
\end{equation}
Inserting into this equation the expression for the mass and those for the inner and outer horizons, one obtains
\begin{equation}
\label{product2}
A_+A_- = (8\pi\ell^2)^2m^2 =\left(\frac{8\pi G}{c^3}\right)^2 J^2
\end{equation}
The last term on the right side of the above equation coincides with the relation given in \cite{marcus} when the electrical
charge is put equal to zero, being an additional test for the consistency of the particle-like model.

As we have seen previously, the "ground" state of the present quantum black hole model corresponds to $n = m = 1$, which also represents an extremal case. A further step is to examine the energy spectrum. For a Kerr black hole as well as for the Schwarzschild case, the
total energy inside the horizon depends only on the black hole mass \cite{chamorro}. In this case, using the preceding results,
\begin{equation}
\label{energy}
E_{n,m} = M_Pc^2\sqrt{\frac{(n^2+m^2)}{2n}}
\end{equation}
Hence the ground state corresponds to an energy $E_{1,1} = M_Pc^2$, the Planck energy. A larger black hole (with $n \neq m$)
will be in an excited state that can decay to a lower state by emitting an amount of energy $\Delta E = E_{n',m'}-E_{n,m}$. Such an energy quantum
$\Delta E$ does not correspond to a single particle (or photon) but should be imagined as a "bunch" of radiation/particles
with a well-defined total energy. If we appeal again to  Bohr's correspondence principle, one should expect that at large
quantum numbers, the equivalent transition frequency between two consecutive levels, i.e., $\omega = \Delta E/\hbar$ should
approach the ''classical'' frequency representing a quasi-normal mode \cite{hod,maggiore}. In fact, when $n > m >> 1$, the energy gap between two consecutive levels is
\begin{equation}
\label{quantumenergy}
\Delta E_{n,m} = (E_{n+1,m+1}-E_{n,m}) \approx \frac{M_Pc^2}{\sqrt{2}}\left[\frac{1}{\sqrt{2n}} + \frac{m}{n^{3/2}}\right]
\end{equation}
In order to interpret the terms in the above equation, we recall that when $n > m >> 1$, the black hole mass is given by $M \approx M_P\sqrt{n/2}$. Thus, the first term on the right side of eq.\ref{quantumenergy} can be written $\hbar c^3/4GM$ or, equivalently, $2\pi~kT_H$ where $T_H$
is the Hawking temperature. The second term requires a little  extra work with the expression defining
the angular rotational velocity of a Kerr black hole (in geometric units), that is
\begin{equation}
\label{omega}
\Omega = \frac{4\pi a}{(r_+^2 + a^2)}
\end{equation}
where $a$ is the spin parameter. Performing the quantization as before and recovering the physical constants, the angular
velocity can be recast as
\begin{equation}
\label{omega2}
\Omega_{n,m} = \frac{c}{2\ell_P}\frac{\sqrt{2}m}{\sqrt{n(n^2+m^2)}}
\end{equation}
For large values of the quantum numbers ($n > m$), one obtains  $\Omega_{n,m} \approx (c/\ell_P)(m/\sqrt{2n^3})$. Using these
results, eq.\ref{quantumenergy} can be rewritten as
\begin{equation}
\Delta E_{n,m} = \hbar\omega_{n,m} = 2\pi~kT_H + \hbar\Omega_{n,m}
\end{equation}
This expression is consistent with the limit at large quantum numbers for the real part of quasi-normal modes found by
Hod \cite{hod} in the case of rotating black holes. On the other hand, for a black hole in a given state $(n, m)$ there is a limit
for the energy that can be extracted from the ergosphere \cite{ruffini}, which corresponds to the total utilization of the
rotational energy leaving the black hole in a state $(n, 0)$ or with an energy
\begin{equation}
E_{n,0} = M_Pc^2\sqrt{\frac{n}{2}}
\end{equation}
For consistency, the above energy must coincide with that corresponding to the irreducible mass  given by
\begin{equation}
E_{ir} = \frac{c^4}{G}\sqrt{\frac{A_+}{16\pi}} = M_pc^2\sqrt{\frac{n}{2}}
\end{equation}
where eq.\ref{area2} was used to replace the area of the horizon. Despite this
 mathematical consistency, one should
expect that this result would be valid only for large black holes ($M > 10^{15}~g$) since for small ones
the state $(n, 0)$  does not correspond to the ground state and the stability against the Hawking process disappears. Hence,
for small black holes, one should expect that the extraction of rotational energy stops when $m =1$. In this case, the
irreducible mass becomes
\begin{equation}
M_{irr}^2 = \frac{A_+c^4}{16\pi G^2} + \frac{4\pi\hbar^2}{A_+c^2}
\end{equation}
In the equation above, the second term on the right represents a "quantum correction" to the irreducible mass that permits
the black hole to decay to the ground state. Notice that for large black holes, such a correction term is completely negligible.

\section{Accretion of PBHS by PBHs}
{
{Most discussions of PBH production argue that an extended mass function is fairly generic. Early production during an extended reheating phase boosts the PBH mass scale because of accretion by the rarer  more massive PBHs of the many smaller ones hovering near the ground state.  The generated mass function should peak near a mass limit determined by accretion over  the time available before reheating is completed.
Using the spherical thin  shell approximation to accretion in an Einstein-de Sitter universe
(Bertschinger 1985), we estimate that the PBH masses  increase by  a factor $\sim (1+z_{inf})/(1+z_{rh}),
$
where $z_{inf}$ is the inflationary scale, estimated to be $\sim 3.10^{16}$ GeV for a weakly interacting and slowly  rolling scalar field that yields the observed low level of scalar  density fluctuations. Also,
 $z_{rh}$ is the reheat scale at a temperature estimated to be anywhere from $\sim 10^{16} \rm GeV$ down to the QCD scale
 $\sim 100 \ \rm MeV$,    or even as low as  $\sim 10$ MeV  from BBN constraints.

 Detection of primordial  B-mode CMB polarization offers our best way of determining the reheat temperature, but is constrained by restriction to the experimentally feasible regime of
 $r \lower.5ex\hbox{$\; \buildrel > \over \sim \;$} 0.0001$,  while  $r\lower.2ex\hbox{$\; \buildrel \propto \over \sim \;$} T_{rh}^2 $
allows a much larger  but unmeasurable range.
Hence accretion allows PBHs produced during preheating to survive to the present epoch and  to populate the window accessible today by microlensing, provided they make up all or most of the dark matter of the universe.
There need be no adverse effects of early evaporation, eg during the epoch of nucleosynthesis, since the PBHs are mostly  evaporate later and are also very subdominant deep into the radiation era. Spectral $\mu$ distortions
of the CMB  produced via evaporation and energy injection into the thermal bath of the CMB  may provide a possible signature of their existence.
}}

\section{Concluding remarks}

Formation of PBHs in the very early universe, in particular during the short matter-dominated regime prevailing
during the oscillatory phase of the inflaton field, has recently been the subject of extensive investigations.
Small mass black holes can be
formed in a hybrid inflation scenario first proposed by Linde \cite{linde} (see also \cite{chen}) as a consequence of a resonant
instability affecting perturbations with wavelengths exiting the Hubble radius at the end of  inflation \cite{martin}. More
recently \cite{arya}, it was shown that PBHs with masses as small as $10^3~g$ can be formed in a warm inflation scenario since
the associated curvature power spectrum has a blue tilt that favors small-scale perturbations.

In these different scenarios, PBHs of small masses ($10^3~g$ of up to $10^{5}~g$) can be formed. Since these objects evaporate
in a short time-scale, the key question to be answered is whether or not a remnant will be left. Many authors have simply postulated the existence
of Planck-sized relics without any consideration about their nature or stability. If such relics are Planck-sized black holes,
their stability, at least from the thermodynamic point of view, is guaranteed if they are extremal black holes.
Regular or non-singular black holes described either by the Bardeen or the Hayward metrics are one possibility \cite{pacheco2}. However
these objects have a particular mass (or charge) distribution leading to a tenuous "outer" atmosphere, which is one reason to classify
them more appropriately as "Planck stars" \cite{rovelli}. Moreover, instabilities appear in the core during the evaporation process that
complicate the analysis of the last stages of the process leading to remnant formation \cite{visser}. Extremal singular charged
or rotating black holes are an alternative possibility. While the former include, for instance, difficulties concerning the discharge by
the Schwinger mechanism, the latter  option is favored by recent studies on the spin distribution of PBHs, suggesting that a substantial fraction
has spin parameters near the maximum value \cite{harada}. Moreover, other studies \cite{arbey} indicate that rotating PBHs are not constrained by the
so-called Thorne limit \cite{thorne} and might have spin parameters in excess of $a \sim 0.998.$

If remnants of the evaporation process are Planck-sized black holes, one would expect that quantum effects become important. In fact,
a possible quantum approach was first proposed by \cite{bekenstein}, who considered that the horizon surface of a charged, rotating black hole
behaves as a classical adiabatic invariant having a discrete spectrum. Another quantum picture was adopted by \cite{nicolini2} to describe
charged black holes. In such a particle-like formalism it is possible to find extremal configurations associated with the ground state. Here
the particle-like formalism is adopted to describe rotating black holes. The ground state of such a particle-like Kerr black hole
corresponds to a mass and a horizon having Planck sized values and a spin equal to $\hbar$. This state corresponds also to an
extremal case. Excited states
satisfying the condition $n = m$ are also extremal but are expected to decay following excitation by mass accretion
$n, m \rightarrow n', m$ with $n' > n$
and subsequent decay according to $n',m \rightarrow n'', m'$ with $n'', m' < n, m$. Similarly, the ground state population may oscillate due to these processes.

Using the derived expressions for the horizon radius and mass, the resulting area is also quantized and equally spaced according
to the original result first derived by Bekenstein \cite{bekenstein}. The energy spectrum is not equally spaced but satisfies  Bohr's correspondence principle, since the equivalent frequencies due to transitions between two consecutive levels at large quantum numbers
approach the frequencies of quasi-normal modes. Moreover the product between the inner and the outer horizon surfaces is proportional
to the squared angular momentum showing that the quantum picture reproduces the main properties expected for macroscopic black holes.

{
Accretion during  reheating by the largest surviving  but rare PBHs of the far more numerous but small, down to Planck mass,  relic PBHs allows a substantial boost in PBH mass especially if reheating is late. This means that there are potentially observable signatures. These include  late evaporation into the diffuse gamma-ray and x-ray backgrounds \cite{auffinger}, cosmic ray signature in the (very) local interstellar medium \cite{boudaud} gravitational microlensing  of deep stellar fields \cite{subaru}, and spectral distortions of the CMB,  from evaporation in the radiation era ($\mu$-type) and in the matter era ($y$-type), as well as the intermediate hybrid ($\mu$-$y$) variety \cite{chluba}.
}