Let $X_1,X_2,\ldots$ be a sequence of independent replicates of a random variable $X$ and let $\{N_t\}_{t\geq 0}$ be a non-negative integer valued random process and assume that $\{N_t\}_{t\geq 0}$ and $X$ are independent.
Then, under some conditions it is shown that the probability $P(\sum_{i=1}^{N_t} X_i\geq 0)$ decays exponentially fast as $t\to\infty$.
Moreover, we consider a testing problem in a compound Poisson process, and we study the exact slope of a test statistic based on the sum of random number of independent and exponentially distributed random variables.