Distribution of the Superposition of Infinitely Divisible Processes

V. M. Zolotarev

Teoriâ veroâtnostej i ee primeneniâ, Tome 3 (1958) no. 2, pp. 197-200

Résumé

In this paper it is proved that for an arbitrary infinitely divisible process $\xi (t)$ and any non-negative infinitely divisible process $\eta(t)$ the distribution of their superposition $\xi(t)=\xi[\eta(t)]$ is also infinitely divisible. The corresponding spectral function $H(x)$ of that process (Levy function) is constructed. The second result is as follows: If in the sum $\zeta(t)=\xi_1+\cdots+\xi_{\eta(t)}$ all random variables are independent, process $\eta(t)$ has an infinitely divisible distribution, and the random variable $\xi_i$ satisfies condition $(V)$, then the distribution $\zeta(t)$ is infinitely divisible.