A distribution on is said to belong to the class for some if holds for all and exists and is finite. Let and be two independent random variables, where has a distribution in the class and is non-negative with an endpoint . We prove that the product has a distribution in the class . We further apply this result to investigate the tail probabilities of Poisson shot noise processes and certain stochastic equations with random coefficients.
Qihe Tang. "On convolution equivalence with applications." Bernoulli 12 (3) 535 - 549, June 2006. https://doi.org/10.3150/bj/1151525135