A distribution [math] on [math] is said to belong to the class [math] for some [math] if [math] holds for all [math] and [math] exists and is finite. Let [math] and [math] be two independent random variables, where [math] has a distribution in the class [math] and [math] is non-negative with an endpoint [math] . We prove that the product [math] has a distribution in the class [math] . We further apply this result to investigate the tail probabilities of Poisson shot noise processes and certain stochastic equations with random coefficients.