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

We investigate a series scheme of random variables ξ ₙᵢ 0,i = 1,2, ⋯ n, which are independent and equally distributed in every series. Every ξ ₙᵢ is equal to the number of renewals of a renewal process Nₙᵢ (t) in the interval (u₀ ,u₀ + t]u₀ 0,t > 0. If Hₙ (t) = Hₙᵢ (t) = MNₙᵢ (t) is the renewal function of the process Nₙᵢ (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 ζ ₙ = ∑ i = 1ⁿ ξ ₙᵢ.