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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

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