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.

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