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

Abstract

Given a supercritical Galton‒Watson process {Z_n} and a positive sequence {ε_n}, we study the limiting behaviors of ℙ(S_Zn/Z_n≥ε_n) with sums S_n of independent and identically distributed random variables X_i 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 Z_n+1/Z_n to m as n→∞.