Let $X_1,X_2,\ldots,X_n$ be a sequence of $n$ i.i.d. chi-squared random variables with $k$ degrees of freedom, and denote by $X_\max$ the maximum of this sequence.  Furthermore, let $k=\omega(1)$ increase, and $n=\omega(k)$ increasing faster than $k$ but not too much fast, i.e. $\log(n)=o(t)$.  For example, $n=k^2$.  This is the same scenario as in [my previous question][1].

Now, let 

$$S(n)=\frac{1}{n}\sum_{i=1}^n\exp\left[\frac{\sqrt{\log n}}{\sqrt{k}}(X_i-X_\max)\right]$$

I am wondering about the limiting behavior of $S(n)$ as $n\rightarrow\infty$.  My conjecture (based on the fact that the maximum isn't too far away from the rest of the sequence of random variables with exponential tail, as well as the [answer][2] and [comment][3] to the two [related][2] [questions][3]) is that $S(n)$ converges to unity in distribution, however, I am having hard time proving this formally, as I get bogged down in a nasty convolution when I try to do it (but, maybe I am approaching this wrong and perhaps I am incorrect).  Any suggestions?


  [1]: http://mathoverflow.net/questions/142834/asymptotic-behavior-of-max-of-chi-squared-distribution
  [2]: http://mathoverflow.net/questions/142834/asymptotic-behavior-of-max-of-chi-squared-distribution
  [3]: http://mathoverflow.net/questions/142772/how-far-away-is-the-maximum-of-n-i-i-d-chi-squared-random-variables-from-the