= Exponential moment of a renewal count
{title2=$\mathbb E e^{\theta N(t)}$}
For nonnegative finite <independent and identically distributed random variables> as inter-arrivals with positive probability of exceeding $\varepsilon>0$, write $p=\mathbb P(X_1\geq\varepsilon)>0$. The count tail is bounded by $\mathbb P(\operatorname{Bin}(n,p)\leq\lfloor t/\varepsilon\rfloor)$, which has geometric decay times a polynomial in $n$. Hence $\mathbb E e^{\theta N(t)}<\infty$ for every fixed finite $t$ and sufficiently small $\theta>0$.
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