 
Summary: A strong law of large numbers for martingale arrays
Yves F. Atchad´e
(March 2009)
Abstract: We prove a martingale triangular array generalization of the ChowBirnbaum
Marshall's inequality. The result is used to derive a strong law of large numbers for martingale
triangular arrays whose rows are asymptotically stable in a certain sense. To illustrate, we
derive a simple proof, based on martingale arguments, of the consistency of kernel regression
with dependent data. Another application can be found in [1] where the inequality is used
to prove a strong law of large numbers for adaptive Markov Chain Monte Carlo methods.
AMS 2000 subject classifications: Primary 60J27, 60J35, 65C40.
Keywords and phrases: Martingales and Martingale arrays, Strong law of large numbers,
Kernel regression.
1. Strong law of large numbers for martingale arrays
Let (, F, P) be a probability space and E the expectation operator with respect to P. Let
{Dn,i, Fn,i, 1 i n}, n 1 be a martingaledifference array. That is for each n 1, {Fn,i, 1
i n} is a nondecreasing sequence of subsigmaalgebra of F, for any 1 i n, E (Dn,i) <
and E (Dn,iFn,i1) = 0. We assume throughout the paper that Fn,0 = {, } for all n 0. We
introduce the partial sums
Mn,k :=
k
