Let $X_n$ be a chi-squared random variable with $n$ degrees of freedom. What are the sharpest known lower bounds on the tails of its distribution? Specifically, I am looking for the lower bounds in the form:
$$P(X_n-n\geq f_1(x)\sqrt{n}+g_1(x))\geq \exp(-x)$$
$$P(n-X_n\geq f_2(x)\sqrt{n}+g_2(x))\geq \exp(-x)$$
for some $f_1(x)>0$, $f_2(x)>0$, and $g_1(x)\geq0$, $g_2(x)\geq0$.
Basically, I am wondering if the lower-bound "equivalent" of the following upper bounds from Massart and Laurent (see Lemma 1 on page 1325) exist:
$$P(X_n-n\geq 2\sqrt{xn}+2x)\leq\exp(-x)$$
$$P(n-X_n\geq 2\sqrt{xn})\leq\exp(-x)$$
I tried Cramer-Chernoff Theorem (i.e. Theorem 1 in here), but the lower bound isn't sharp enough…
Best Answer
This is not quite the form you requested. But if your goal was to control chi-squared left tails, I have found these lower bounds helpful.
For $X_1 \sim \chi^2_1$:
Let $Z \sim N(0,1)$, so that $Z/\sqrt{2} \sim N(0, 1/2)$. For $x>0$, we have $\text{erf}(x) = P\left(|Z/\sqrt{2}| \leq x\right) = P(X_1 \leq 2x)$.
Apply this bound or this refinement: $\text{erf}(x) \leq \sqrt{1-\exp(-4x^2/\pi)}$.
Then $P(X_1 \geq y) \geq 1- \sqrt{1-\exp(-2y/\pi)}$ for $y>0$.
For $X_2 \sim \chi^2_2$:
Directly from the chi-squared CDF for $n=2$, we see that $P(X_2 \geq x) = e^{-x/2}$.
Also, since the chi-squared distribution shifts right as the df increases, $P(X_n \geq x) \geq e^{-x/2}$ for $n\geq 2$.
For other $n$:
See this answer to another question.