Advances in Applied Probability

Asymptotics for weighted random sums


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Let {Xi} be a sequence of independent, identically distributed random variables with an intermediate regularly varying right tail F̄. Let (N, C1, C2,...) be a nonnegative random vector independent of the {Xi} with N∈ℕ∪ {∞}. We study the weighted random sum SN =∑{i=1}N CiXi, and its maximum, MN=sup{1≤k N+1i=1k CiXi. This type of sum appears in the analysis of stochastic recursions, including weighted branching processes and autoregressive processes. In particular, we derive conditions under which P(MN > x)∼ P(SN > x)∼ E[∑i=1N F̄(x/Ci)] as x→∞. When E[X1]>0 and the distribution of ZN=∑ i=1NCi is also intermediate regularly varying, we obtain the asymptotics P(MN > x)∼ P(SN > x)∼ E[∑i=1N F̄}(x/Ci)] +P(ZN > x/E[X1]). For completeness, when the distribution of ZN is intermediate regularly varying and heavier than F̄, we also obtain conditions under which the asymptotic relations P(MN > x) ∼ P(SN > x)∼ P(ZN > x / E[X1] hold.

Article information

Adv. in Appl. Probab., Volume 44, Number 4 (2012), 1142-1172.

First available in Project Euclid: 5 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G50: Sums of independent random variables; random walks
Secondary: 60F10: Large deviations 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60G70: Extreme value theory; extremal processes

Randomly weighted sum randomly stopped sum heavy tail intermediate regular variation regular variation Breiman's theorem


OLVERA-CRAVIOTO, MARIANA. Asymptotics for weighted random sums. Adv. in Appl. Probab. 44 (2012), no. 4, 1142--1172. doi:10.1239/aap/1354716592.

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