On the Compound Poisson Distribution

exist. We shall prove that under certain conditions we obtain (1) as a limit distribution of double sequences of independent and infinitesimal random variables and apply this theorem to stochastic processes with independent increments. Theorem 1 Let ξn1, ξn2, . . . , ξnkn (n = 1, 2, . . .) be a double sequence of random variables. Suppose that the random variables in each row are independent, they are infinitesimal, i.e. for every ε > 0 lim n→∞ max 1≤k≤kn P(|ξnk| > ε) = 0, finally, there exists a finite-valued, non-negative random variable η such that