Let $X_1,X_2,\ldots,X_n$ be a sequence of i.i.d. chi-squared random variables with $t$ degrees of freedom, i.e. $X_i\sim\chi^2_t$. I am wondering what is known about the distribution of $S(n,t,a)=\sum_{i=1}^n a^{X_i}$ where $a>0$.
I don't think that the exact CDF and or PDF of $S(n,t,a)$ is known in the closed form. However, are there asymptotic results?
In my particular scenario, $t\gg 1$ and $n=t^{\delta}$ for $\delta>0$, and I am interested in the lower bound on $\mathbf{P}(S(n,t,a)\geq b)$ where $a=e^{\frac{v}{2(v+u)}}$ and $b=wn\left(1+\frac{v}{u}\right)^{t/2}$ with $u>0$, $v>0$, and $w>0$. That is, I am interested in lower-bounding: $$\mathbf{P}\left(\sum_{i=1}^n e^{\frac{v}{2(v+u)}X_i}\geq wn\left(1+\frac{v}{u}\right)^{t/2}\right)\tag{1}$$
This is from a problem in statistical signal processing that I working on, a generalization to multiple Gaussian signal pulses of this issue I posted on stats.SE.
What I tried
First, divide both sides inside the expression for probability in $(1)$ by $n$ and use Jensen's Inequality to get rid of the summation (by first moving it along with $1/n$ factor into the exponential and then using the summation property of independent chi-squared random variables). We thus obtain a lower bound on $\frac{1}{n}\sum_{i=1}^n e^{\frac{v}{2(v+u)}X_i}\geq e^{\frac{v}{2(v+u)}X}$ where $X\sim\chi^2_{t}$. Then, $$\mathbf{P}\left(\sum_{i=1}^n e^{\frac{v}{2(v+u)}X_i}\geq wn\left(1+\frac{v}{u}\right)^{t/2}\right)\geq\mathbf{P}\left(X\geq 2\left(1+\frac{u}{v}\right)\left(\frac{t}{2}\log\left(1+\frac{u}{v}\right)+\log w\right)\right)\tag{2}$$ and we can use the Central Limit Theorem to approximate the CCDF of $\frac{X-t}{\sqrt{2t}}$ by a standard normal distribution, which, in turn, has tight upper and lower bounds on CCDF. However, the bound in $(2)$ is not very tight and I wondering if something better exists.