I am wondering if there is a tight lower bound on the probability of a maximum of $n$ i.i.d. chi-square random variables, each with degree of freedom $d$ exceeding a value close to $d$. Formally, I need to lower bound the following expression:
$$P(\max_i X_i\geq d+\delta)$$
where each $X_i\sim\chi^2_d$ for $i=1,2,\ldots,n$ and $\delta$ is small. Ideally, I would like an expression involving elementary functions of $n$, $d$, and $\delta$. I am interested in the asymptotics and assume that $n$ and $d$ are large.
What I tried
We know that $P(\max_i X_i\geq d+\delta)=1-P(X<d+\delta)^n$ where $X\sim\chi^2_d$. Therefore, I tried to upper bound $P(X<d+\delta)$ using the CLT, which yields the normal approximation to chi-squared distribution, and the lower bounds on the Q-function. However, the resultant overall bound is not tight, and I am hoping something better exists.