I found this question: [Chernoff style concentration bound for ratio of variables](https://mathoverflow.net/questions/420837/chernoff-style-concentration-bound-for-ratio-of-variables).
I want to ask if we get similar thing for the ratio of the sum and the one Gaussian variable.


Given i.i.d. Gaussian random variables $X\_1,\dots, X\_k$ with $N(0, 1)$. Fix $\epsilon\in (0,1)$, can prove that for any $\delta>0$ there exists $1\le k=k(\epsilon, \delta)$ (some number) so that
$$ 
P\left(\frac{X\_1^2+X\_2^2+\dots+X\_k^2}{X\_1^2}>\frac{1}{\epsilon^2}\right)>1- \delta ?
$$


Can we find such $k$?




---


In [these 2020 slides by Andrew Nobel](https://nobel.web.unc.edu/wp-content/uploads/sites/13591/2020/10/Probability_Inequalities.pdf),
for $Y\sim \chi\_k^2$ where $(Y=\sum\_{I=1}^k X\_i^2)$, for $t\in (0,1)$
$$
P(Y\ge (1+\epsilon)k)\le \exp(-k(t^2-t^3)/4).
$$



Can we find such $k$ so that the following inequality holds?
$\newcommand\ep\epsilon $In the clever answer by Fedor Petrov, it was shown that
\begin{equation\*}
 Q:=P\Big(\frac{X\_1^2+\dots+X\_k^2}{X\_1^2}<C\Big)\le C/k, \tag{1}\label{1}
\end{equation\*}
where $C:=1/\ep^2>0$ and the $X\_i$'s are any iid random variables.


Let us show that for the standard normal $X\_i$'s as in the OP, the upper bound $C/k$ in \eqref{1} can be replaced by a bound decreasing exponentially in $k$.


For $C\le1$, $Q=0$. So, without loss of generality (wlog) $C>1$. Also, wlog $k\ge2$.
For
\begin{equation\*}
 c:=C-1>0, \tag{2}\label{2}
\end{equation\*}
any $x\in [0,\sqrt{(k-1)/c}\,]$, and
$$h:=\frac{k-1-cx^2}{4(k-1)},$$
we have
\begin{equation\*}
 Q=P\Big(\sum\_{i=2}^k X\_i^2<cX\_1^2\Big)\le Q\_1+Q\_2, \tag{3}\label{3}
\end{equation\*}
where
\begin{equation\*}
 Q\_1:=P(X\_1^2\ge x^2)\le e^{-x^2/2}=:R\_1 \tag{4}\label{4}
\end{equation\*}
and
\begin{equation\*}
\begin{aligned}
 Q\_2&:=P\Big(\sum\_{i=2}^k X\_i^2<cx^2\Big) \\ 
 &=P\Big(\sum\_{i=2}^k(1-X\_i^2)>k-1-cx^2\Big) \\ 
 &\le\exp\{-h(k-1-cx^2)+(k-1)\ln Ee^{h(1-X\_1^2)}\} \\ 
 &=\exp\{-h(k-1-cx^2)+(k-1)(h-\tfrac12\,\ln(1+2h))\} \\ 
 &\le\exp\{-h(k-1-cx^2)+2(k-1)h^2\} \\ 
 &=\exp-\frac{(k-1-cx^2)^2}{8(k-1)}=:R\_2. 
\end{aligned}
\tag{5}\label{5}
\end{equation\*}
Choosing now $x$ to be the positive root of the equation $R\_1=R\_2$, from \eqref{3}, \eqref{4}, and \eqref{5} we get
\begin{equation}
 Q\le 2\exp-\frac{k-1}{2(1+\sqrt C)^2}, \tag{6}\label{6}
\end{equation}
which is the promised bound, decreasing exponentially in $k$.


One may also note that typically $k$ and $C$ will be large, and then the exponent in the bound in \eqref{6} will be about $-k/(2C)$ -- compare this with the bound in \eqref{1}.


