I'm trying to prove that a sequence of functions $(k\_N)\_{N\in\mathbb{N}}$


$$k\_N(\vec{y}):=e^{-N^2r(\vec{y},Q\vec{y})}\sqrt{\frac{r}{\pi}}^kN^{k}$$


where $r>0$, $k>2$ and


**Edit**: I have forgot to say, that $k\in\mathbb{N}$ is the size of the matrix $Q$ i.e. $Q$ is $k\times k$. This means that $\det Q=0$ and $Q$ is positive semidefinte.


I have also changed $k$ from beeing $>0$ to $k>2$.


**End of the Edit**


$$Q:=\left(\begin{array}{cccccccc}
(1-\frac{1}{k})& -\frac{1}{k} & -\frac{1}{k} & \cdots & -\frac{1}{k} \\
-\frac{1}{k} & (1-\frac{1}{k}) & -\frac{1}{k} & \cdots & -\frac{1}{k} \\
\vdots & \ddots & & &\vdots \\
-\frac{1}{k} & \dots & &-\frac{1}{k} &(1-\frac{1}{k}) 
\end{array}\right)$$


goes to a delta function when $N\to\infty$. What I'm missing ist showing, that given $\varepsilon>0$ and $\eta>0$ there exists $n\_0$ such that $\forall n>n\_0$


$$\idotsint\_{|\vec{x}|>\eta}k\_n(\vec{x})d\vec{x}<\varepsilon$$.


I was looking for some expression for the error function of multivariate gaussians, but could find nothing of the sort. Is there a nice way of representing the error functions so that I don't have to diagonalize $Q$ and treat the function as several independent gaussians?


Thanks!



Error function of multivariate Gaussian
$\newcommand\R{\mathbb R}$Let us write $x$ for $\vec x$ and $t$ for $\eta$. Then
$$\int\_{x\in\R^k\colon|x|>t}k\_n(x)\,dx=\infty$$
for all real $t>0$.


Indeed, we can write
$$Q=I\_k-\frac1k\,1\_k1\_k^T,$$
where the matrix $I\_k$ is the $k\times k$ identity matrix and $1\_k$ is the $1\times k$ column matrix with unit entries, that is, $1\_K=e\_1+\dots+e\_k$, where $(e\_1,\dots,e\_k)$ is the standard basis of $\R^k$. So, $Q$ is the matrix of the orthogonal projector onto the orthogonal complement -- say $V$ -- of (the span of) the vector $1\_k$; in particular, $Q1\_k=0$.


Fix any orthonormal basis $(b\_1,\dots,b\_k)$ of $\R^d$ such that $b\_1=1\_k/\sqrt k$ (so that $(b\_2,\dots,b\_k)$ is an orthonormal basis of $V$).
Let $(x\_1,\dots,x\_k)$ denote the coordinates of an arbitrary vector $x\in\R^d$ in the orthonormal basis $(b\_1,\dots,b\_k)$. Then
$$(x,Qx)=x\_2^2+\dots+x\_k^2.$$
Therefore and because the Lebesgue measure is spherically invariant, for all real $t>0$ we have
$$
\begin{aligned}
&\int\_{x\in\R^k dx\,\colon|x|>t}dx\, k\_n(x) \\
&\ge\int\_{x\in\R^k\colon x\_1>t}dx\, k\_n(x) \\
&=\int\_t^\infty dx\_1\int\_\R dx\_2\cdots\,\int\_\R dx\_k 
\exp\{-n^2r(x\_2^2+\dots+x\_k^2)\}\Big(\sqrt{\frac{r}{\pi}}\Big)^k n^{2k}
=\infty,
\end{aligned}$$
because the inner $(k-1)$-fold integral is a strictly positive real number not depending on $x\_1$.


