Advances in Applied Probability

Marcinkiewicz law of large numbers for outer products of heavy-tailed, long-range dependent data

Hui He

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Given a supercritical Galton‒Watson process {Zn} and a positive sequence {εn}, we study the limiting behaviors of ℙ(SZn/Zn≥εn) with sums Sn of independent and identically distributed random variables Xi and m=𝔼[Z1]. We assume that we are in the Schröder case with 𝔼Z1 log Z1<∞ and X1 is in the domain of attraction of an α-stable law with 0<α<2. As a by-product, when Z1 is subexponentially distributed, we further obtain the convergence rate of Zn+1/Zn to m as n→∞.

Article information

Adv. in Appl. Probab., Volume 48, Number 3 (2016), 672-690.

First available in Project Euclid: 19 September 2016

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Mathematical Reviews number (MathSciNet)

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60F10: Large deviations

Galton‒Watson process domain of attraction stable distribution slowly varying function large deviation Lotka‒Nagaev estimator Schröder constant


He, Hui. Marcinkiewicz law of large numbers for outer products of heavy-tailed, long-range dependent data. Adv. in Appl. Probab. 48 (2016), no. 3, 672--690.

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