Convergence rates in the law of large numbers for arrays of martingale differences

We study the convergence rates in the law of large numbers for arrays of martingale differences. For n⩾1n⩾1, let Xn1,Xn2,…Xn1,Xn2,… be a sequence of real valued martingale differences with respect to a filtration {∅,Ω}=Fn0⊂Fn1⊂Fn2⊂⋯{∅,Ω}=Fn0⊂Fn1⊂Fn2⊂⋯, and set Snn=Xn1+⋯+XnnSnn=Xn1+⋯+Xnn. Under a simple moment condition on ∑j=1nE[|Xnj|γ|Fn,j−1] for some γ∈(1,2]γ∈(1,2], we show necessary and sufficient conditions for the convergence of the series ∑n=1∞ϕ(n)P{|Snn|>εnα}, where α  , ε>0ε>0 and ϕ   is a positive function; we also give a criterion for ϕ(n)P{|Snn|>εnα}→0ϕ(n)P{|Snn|>εnα}→0. The most interesting case where ϕ   is a regularly varying function is considered with attention. In the special case where (Xnj)j⩾1(Xnj)j⩾1 is the same sequence (Xj)j⩾1(Xj)j⩾1 of independent and identically distributed random variables, our result on the series ∑n=1∞ϕ(n)P{|Snn|>εnα} corresponds to the theorems of Hsu, Robbins and Erdös (1947, 1949) if α=1α=1 and ϕ(n)=1ϕ(n)=1, of Spitzer (1956) if α=1α=1 and ϕ(n)=1/nϕ(n)=1/n, and of Baum and Katz (1965) if α>1/2α>1/2 and ϕ(n)=nb−1ϕ(n)=nb−1 with b⩾0b⩾0. In the single martingale case (where Xnj=XjXnj=Xj for all n and j  ), it generalizes the results of Alsmeyer (1990). The consideration of martingale arrays (rather than a single martingale) makes the results very adapted in the study of weighted sums of identically distributed random variables, for which we prove new theorems about the rates of convergence in the law of large numbers. The results are established in a more general setting for sums of infinitely many martingale differences, say Sn,∞=∑j=1∞Xnj instead of SnnSnn. The obtained results improve and extend those of Ghosal and Chandra (1998). The one-sided cases and the supermartingale case are also considered.