A Refinement of the Limit Theorem for the Superposition of Independent Renewal Processes

We investigate a series scheme of random variables $\xi _{ni} \geqq 0,i = 1,2, \cdots n$, which are independent and equally distributed in every series. Every $\xi _{ni} $ is equal to the number of renewals of a renewal process $N_{ni} (t)$ in the interval $(u_0 ,u_0 + t]u_0 \geqq 0,t > 0$. If $H_n (t) = H_{ni} (t) = {\bf M}N_{ni} (t)$ is the renewal function of the process $N_{ni} (t)$, then we require that\[ nH_n (t) = H(t)\]for every n and t, where $H(t)$ is an arbitrary renewal function. Under these conditions we obtain the estimate (2) for the distribution of the sum $\zeta _n = \sum _{i = 1}^n \xi _{ni} $.