When can we compute the CDF of the Compound Poisson process

Question

Suppose we have a Compound Poisson Process with intensity $\lambda$ $$ S_t = \sum_{i=1}^{N_t} X_i. $$ We can compute the formula for the CDF as follows \begin{align*} F_{S_t}(x) = P(S_t \leq x) &= \sum_{k=0}^\infty P(S_t \leq x \mid N_t = k)P(N_t = k) \\ & = \sum_{k=0}^\infty P(\sum_{i=1}^k X_i \leq x)\frac{(\lambda t)^k}{k!}e^{-\lambda t} \\ & = \sum_{k=0}^\infty F_X^{*k}(x)\frac{(\lambda t)^k}{k!}e^{-\lambda t}. \\ \end{align*} My question is, Is there a known distribution for $X_i$ such that this formula can be explicitly computed? To me the exponential distribution seemed like a promising choice, but I wasn't able to get far with it.