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

Let ξ ₁ , ⋯ ,ξ ₙ , ⋯ be independent random positive variables and let P \ ξ ₖ < t \ = G(t), k = 1, ⋯ ,n, ⋯ Let us denote \[ { P}\{ {ξ _1 + ⋯ + ξ _n < t} \} = G_n (t). \] Theorem.\[ {lim }t → ∞ {{1 - G_n (t)}}{{1 - G(t)}} = n, n = 1,2,3, ⋯ , \]if and only if\[ {lim }t → ∞ {{1 - G_2 (t)}}{{1 - G(t)}} = 2. \] This theorem is useful in some investigations of age-dependent branching processes.