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As part of a homework exercise on a light-tailed Compound Poisson driven queue (i.e. a Compound Poisson process reflected at 0) I need to show that when $B$ denotes the job size of the Compound Poisson process, under the assumption that the mean of $B$ is finite we have that $$\lim_{y\rightarrow \infty}e^{y}y\mathbb{P}(B > y) = 0.$$ It seems to me that either the finiteness of the mean, or $X$ being light-tailed (i.e. $\mathbb{E}e^{\omega X_1} = 1$ for some $\omega >0$ and $\mathbb{E}(X_1e^{\omega X_1}) < \infty$) must imply that the probability $\mathbb{P}(B > y)$ must go to 0 superexponentially. However, I don't know how to show this.

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