On convolution equivalence with applications

Abstract

A distribution $F$ on $(-\mathrm{\infty },\mathrm{\infty })$ is said to belong to the class $S\left(\gamma \right)$ for some $\gamma \ge 0$ if ${lim}_{x\to \mathrm{\infty }}\overline{F}(x-u)/\overline{F}\left(x\right)=e^{\gamma u}$ holds for all $u$ and ${lim}_{x\to \mathrm{\infty }}\overline{F^{*2}}\left(x\right)\overline{F}\left(x\right)=2m_F$ exists and is finite. Let $X$ and $Y$ be two independent random variables, where $X$ has a distribution in the class $S\left(\gamma \right)$ and $Y$ is non-negative with an endpoint $\hat{y}=sup\{y:P(Y\le y\left)<1\right\}\in (0,\mathrm{\infty })$. We prove that the product $\mathrm{XY}$ has a distribution in the class $S(\gamma /\hat{y})$. We further apply this result to investigate the tail probabilities of Poisson shot noise processes and certain stochastic equations with random coefficients.