A Theorem on Sums of Independent Positive Random Variables and Its Applications to Branching Random Processes

1964/01/01 by V. P. Chistyakov · 1 citation
Mathematics · #Stochastic processes and statistical mechanics

paper · doi:10.1137/1109088

crossref issued 1964/01/01 · crossref published 1964/01/01 · crossref published-print 1964/01/01 · openalex publication_date 1964/01/01 · crossref created 2005/03/07 · crossref deposited 2020/04/06 · openalex created_date 2025/10/10 · crossref indexed 2026/07/28 · openalex updated_date 2026/07/31

Abstract

Let ξ ₁ , ⋯ ,ξ _n , ⋯ be independent random positive variables and let \bf P \ ξ _k < t \ = G(t), k = 1, ⋯ ,n, ⋯ Let us denote \bf P\ ξ ₁ + ⋯ + ξ _n < t \ = G_n (t). Theorem.\mathop lim _{t → ∞} \frac1 - G_n (t)1 - G(t) = n, n = 1,2,3, ⋯ ,if and only if\mathop lim _{t → ∞} \frac1 - G₂ (t)1 - G(t) = 2. This theorem is useful in some investigations of age-dependent branching processes.

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