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Reposted from math stackexchange (https://math.stackexchange.com/questions/2779699/constant-concentration-for-chi-square-variables) as suggested by another user.

I would like to have a certain concentration inequality for chi square variables with $k$ degrees of freedom. As referenced in What are the sharpest known tail bounds for $\chi_k^2$ distributed variables?, Laurant and Massart give as a corollary that $$ \Pr\left(\left\lvert X-\mathbb E(X)\right\rvert\geq 2\sqrt{kx}+2x\right)\leq 2\exp(-x). $$ However, I only care about the case where the right hand side is constant, say $1/10$ (so $x\approx 3$), and I would like a weaker dependence on $k$, say $k^{1/4}$ instead of $k^{1/2}$. I only care about the asymptotic order of $k$, so anything $O(k^{1/2-\varepsilon})$ would be great. Could anyone suggest any results/approaches, or could it be impossible? Thank you!

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