We study conditions under which
P{Sτ > x} ∼ P{Mτ > x} ∼ EτP{ξ1 > x} as x → ∞,
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where Sτ is a sum ξ1 + ⋯ + ξτ of random size τ and Mτ is a maximum of partial sums Mτ = maxn≤τ Sn. Here, ξn, n = 1, 2, …, are independent identically distributed random variables whose common distribution is assumed to be subexponential. We mostly consider the case where τ is independent of the summands; also, in a particular situation, we deal with a stopping time.
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We also consider the case where Eξ > 0 and where the tail of τ is comparable with, or heavier than, that of ξ, and obtain the asymptotics
P{Sτ > x} ∼ EτP{ξ1 > x} + P{τ > x / Eξ} as x → ∞.
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This case is of primary interest in branching processes.
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In addition, we obtain new uniform (in all x and n) upper bounds for the ratio P{Sn > x} / P{ξ1 > x} which substantially improve Kesten’s bound in the subclass ${\mathcal{S}}^{*}$ of subexponential distributions.