Lower bound on the probability of maximum of $n$ i.i.d. chi-square random variables exceeding a value close to their number of degrees of freedom 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.

I'm not sure if this is helpful, but taking a look at what Wikipedia has to say about the cdf of the Chi-Square distribution, it appears that 
\begin{align*}
   P[X < d+\delta]^n
&= \left( \left( \frac{d+\delta}{2}\right)^{d/2} e^{-d/2} 
   \sum_{k=0}^\infty \frac{\left(\frac{d+\delta}{2}\right)^k}{\Gamma(d/2 + k + 1)}\right)^n \\
&= \left( e^{-d/2}  
   \sum_{y=d/2}^\infty \frac{\left(\frac{d+\delta}{2}\right)^y}{\Gamma(y + 1)}\right)^n \\
&= e^{n\delta/2}\left(e^{-(d+\delta)/2}   
   \sum_{y=d/2}^\infty \frac{\left(\frac{d+\delta}{2}\right)^y}{\Gamma(y + 1)}\right)^n,
\end{align*}
and so the term 
$$
 e^{-(d+\delta)/2}\sum_{y=d/2}^\infty \frac{\left(\frac{d+\delta}{2}\right)^y}{\Gamma(y + 1)}
$$
is bound above by $1$ (this is certainly clear for even $d$). It may be that the larger $d$ is, the further the term is from $1$. 
