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

Let $X \sim \chi^2_k$ be a chi-squared distributed random variable with $k$ degrees of freedom. What are the sharpest known bounds for the following probabilities

$$\mathbb{P}[X > t] \leq 1 - \delta_1(t, k)$$

and

$$\mathbb{P}[X < z] \leq 1 - \delta_2(z, k)$$

where $\delta_1$ and $\delta_2$ are some functions. Pointers to relevant papers would be appreciated.

• If you define the deltas to be complementary incomplete gamma functions, you obtain exact equalities. Obviously these are the sharpest possible bounds! I guess the point of this question is that your calculator doesn't compute incomplete gammas and you're looking for an approximation, but that still omits essential information: how can we answer this question until we know just what your calculator can compute? – whuber Nov 23 '10 at 16:04
• I am not interested in computing an upper bound, but obtaining something that I can control analytically. The answer that robin has provided is exactly what I was looking for. The question is, are there more precise bounds than those provided by Massart and Laurent? – mkolar Nov 23 '10 at 17:30
• Gamma integrals can be "controlled analytically," so what distinction are you making? – whuber Nov 23 '10 at 20:44

$$P(X-k\geq 2\sqrt{kx}+2x)\leq \exp (-x)$$
$$P(k-X\geq 2\sqrt{kx})\leq \exp (-x)$$