Exponential moment of a renewal count ($\mathbb{E}{e}^{\theta N(t)}$)

For nonnegative finite independent and identically distributed random variables as inter-arrivals with positive probability of exceeding $\epsilon >0$, write $p=\mathbb{P}(X_1\ge \epsilon )>0$. The count tail is bounded by $\mathbb{P}(\mathrm{Bin}(n,p)\le \lfloor t/\epsilon \rfloor )$, 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 $\theta >0$.