Research output: Contribution to Journal/Magazine › Journal article › peer-review
Research output: Contribution to Journal/Magazine › Journal article › peer-review
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TY - JOUR
T1 - Asymptotics of randomly stopped sums in the presence of heavy tails
AU - Denisov, Denis
AU - Foss, Sergey
AU - Korshunov, Dmitry
PY - 2010
Y1 - 2010
N2 - We study conditions under which P{Sτ > x} ∼ P{Mτ > x} ∼ EτP{ξ1 > x} as x → ∞, 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. 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 → ∞. This case is of primary interest in branching processes. 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.
AB - We study conditions under which P{Sτ > x} ∼ P{Mτ > x} ∼ EτP{ξ1 > x} as x → ∞, 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. 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 → ∞. This case is of primary interest in branching processes. 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.
U2 - 10.3150/10-BEJ251
DO - 10.3150/10-BEJ251
M3 - Journal article
VL - 16
SP - 971
EP - 994
JO - Bernoulli
JF - Bernoulli
SN - 1350-7265
IS - 4
ER -