What are the sharpest known tail bounds for χ2k\chi_k^2 distributed variables?

Let Xχ2k be a chi-squared distributed random variable with k degrees of freedom. What are the sharpest known bounds for the following probabilities

P[X>t]1δ1(t,k)

and

P[X<z]1δ2(z,k)

where δ1 and δ2 are some functions. Pointers to relevant papers would be appreciated.

Answer

The Sharpest bound I know is that of Massart and Laurent Lemma 1 p1325.

A corollary of their bound is:

P(Xk2kx+2x)exp(x)

P(kX2kx)exp(x)

Attribution
Source : Link , Question Author : mkolar , Answer Author : robin girard

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