Laws of Large Numbers and Continuity of Processes

Abstract

Lai [7] has shown that for a sequence (X _k)_k≥₁ of independent copies of a realvalued, centered r.v. X, the strong law of large numbers (SLLN) holds if and only if that sequence is a.s. Abel convergent to 0. That surprising equivalence between Cesaro and Abel convergence remains true in other situations, for instance when the independent r.v. X _k. are symmetric, but not necessarily identically distributed (see Martikainen [8], Mikosch and Norvaisa [9]). Stated in a slightly different way, these two convergence results assert that for a sequence (X _k) of independent r.v. which are either centered and identically distributed or symmetric, the SLLN is equivalent to the a.s. paths-continuity of the following process (ζ(t), t ∈ [0,1]):

$$ \zeta (t)\left\{ {_{0a.s{\text{ if }}t = 1.}^{(1 - t)\sum\limits_{k > 1} {t^k X_k {\text{ }}\forall t \in [0,1['} } } \right. $$