A zero mean sub-Gaussian random variable Z satisfies $Eexp(tZ)≤exp(t^2k^2/2)$ for some constant k>0.
Let's say we have random variables: $X_1,X_2,...,X_n$ with $EX_i=\mu$ and $Var(X_i)=\sigma^2$. We know that $\hat{\sigma}$ is consistent. I am interested in obtaining tail bounds for the inverse of the estimator of the standard error: $1/\hat{\sigma}$, where $\hat{\sigma}=\sqrt{\frac{1}{n}\sum_{i=1}^n (X_i-\bar{X})^2}$ and $\bar{X}=\frac{1}{n}\sum_{i=1}^nX_i$.
I didn't find anything in the literature.
If the $X_i$ are standard Gaussian random variables, then $\hat{\sigma}^2$ is distributed according to a chi-squared distribution. Can we prove that if $X_i$ are sub-Gaussian, $\hat{\sigma}$ is sub-Gaussian or sub-exponential? And what about its inverse?
Or more generally, can we prove that under some assumptions on the $X_i$, it exists a $n$ such that $1/\hat{\sigma}$ is sub-Gaussian?
Thanks you in advance