Convergence rates in the law of large numbers for random variables on partially ordered sets

Abstract

Let (A, ≤) be a partially ordered set, {X _α} a collection of i. i. d. random variables, indexed byA. Let$S_\alpha = \sum _{\beta \leqslant \alpha } X_\beta$, |α|=card {β∈A, β∈α}. We study the convergence rates ofS _α/|α|. We derive for a large class of partially ordered sets theorems, like the following one: For suitabler, t with 1/2< <r/t≤1:E|X| ^t M (|X| ^t/r)<∞ andEX=μ if and only if

$$S_\alpha = \sum _{\beta \leqslant \alpha } X_\beta $$

for all ε>0, where$M(x) = \sum _{j< x} d(j)$ withd(j)=card {α∈A, |α|=j}.