Let $X$ and $Y$ be i.i.d. random variables with exponential distribution with mean $1$, and let $Z=(X-1)(Y-X)$. Let $Z\_1,...,Z\_n$ are i.i.d. copies of $Z$, and let $f(n)=P[\sum\_{i=1}^n Z\_i > 0]$. My question is to estimate $f(n)$. I am interested in asymptotic upper and lower bounds for large $n$, and also in efficient procedure to approximately compute, for example, $f(100)$.


The problem with asymptotic estimates is that the standard tool, Cramer's Theorem, is not applicable because the (logarithmic) moment generating function of $Z$ is not finite. So, the global question is how to estimate probabilities of large deviations for the sum of i.i.d copies of such $Z$.


The problem with computing $f(100)$ is that naive simulation (generate $100$ copies of $Z$ and compute the sum) returns negative sum all the time. The global question here is how to modify the naive experiment to be able to compute very small probabilities.



Estimating probability that a large sum of i.i.d variables is positive
The exact asymptotics of $f(n)$ for large $n$ follows by [Theorem 2.1, more specifically formula (2.4)](https://projecteuclid.org/euclid.aop/1176994938). 


However, to use that formula (2.4), you will have to compute lots of asymptotics regarding the distribution of the random variable $Z$, and also a few moments of it. Whereas these calculations are straightforward, it's lots of them! 


I think the crude, logarithmic asymptotics of $f(n)$ for large $n$ is given by $\log f(n)\sim-c\sqrt n$ for a certain real constant $c>0$. 


