Let X∼χ2k 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]≤1−δ1(t,k)

and

P[X<z]≤1−δ2(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(X−k≥2√kx+2x)≤exp(−x)

P(k−X≥2√kx)≤exp(−x)

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