I am trying to find a reference (or, if it's false, a counterexample) for the following sort-of-intuitive fact: if $\tau$ is a stopping time with a subexponential probability distribution, and $(X\_n)\_{n\geq 1}$ are independent r.v.'s, also subexponential, then $\sum\_{n=1}^\tau X\_n$ aso has a subexponential distribution.


Specifically, I would like to know if the following statement is known:



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> Let $(X\_n)\_{n\geq 1}$ be independent random variables satisfying $\mathbb{E}[e^{X\_n}] \leq 1$ for all $n$, and $\tau$ be a stopping time. Suppose $\mathbb{E}[e^{\alpha \tau}] \leq e^\beta$ for some $\alpha >0$ and $\beta<\infty$. Then $\mathbb{E}[e^{\sum\_{n=1}^\tau X\_n}] \leq 1$.
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Distribution of a stopped random sum, with subexponential stopping time
The hypothesis implies that $M\_k=[e^{\sum\_{n=1}^k X\_n}]$ is a supermartingale, with $M\_0=1$. Then the optional stopping theorem for positive supermartingales implies the requested inequality. (see e.g. Williams' book "Probability with martingales"). 
Note that no moment conditions on the stopping time $\tau$ are needed,
just that it is an almost surely finite stopping time. Alternatively, look up "Wald's third identity"and apply it to the independent variables $e^{X\_n}/[\mathbb{E}e^{X\_n}]$ and the given stopping time. Note that the case of a general stopping time follows from the case of a bounded stopping time via Fatou's lemma.


