Convergence rates in the law of large numbers for martingales

AbstractIn this paper we extend well-known results by Baum and Katz (1965) and others on the rate of convergence in the law of large numbers for sums of i.i.d. random variables to general zero-mean martingales SN. For 12 < α ⩽ 1, p>1α and f(x) = |x| (two-sided case) or = x+ or x− (one-sided case), it is e.g. shown that if, for some γ ϵ (1α, 2] and q>(pα − 1)(γα − 1), supn⩾1n−1∑j=1nE(|Xj|γ|S0,…Sn)q<∞ and an additional mixing condition holds in the one-sided case, then ∑n⩾1npα−2P(f(Sn)>εnα<∞for allε> 0 holds iff ∑n⩾1∑j⩾njpα−2P(f(Xn)>εjα)<∞for all ε>0, X1, X2, … being the increments of SN. The latter condition reduces to the well-known moment condition Ef(X1)p<∞, if X1, X2, … are i.i.d. Our results also extend recent ones by Irle (1985, 1987)