On strong laws of large numbers with rates

Guy Cohen, Roger Jones, Michael Lin

Research output: Contribution to journalArticlepeer-review

Abstract

Let {fn} � Lp(µ), 1 < p < 1, be a sequence of functions with supn ||fn||p < 1. We prove that if for some 0 < � � 1 we have sup n 1 n1 � n X k=1 fk p < 1, then for � < p 1 pthe sequence { 1 n1 � n X k=1 fk} has a.e. bounded p-variation, hence converges, and the p-variation norm func- tion is in Lp(µ). If we replace supn ||fn||p < 1 by supn ||fn||1 < 1, then the a.e. convergence holds for � < p p+1�. Furthermore, in each case we also have a.e. convergence of the series 1 X k=1 fk k1 � for the corresponding values of �, and in the first case we even have that the sequence of partial sums has bounded p-variation. Some applications are given. In particular, we show that if {gn} are centered independent (not necessarily identically distributed) random variables with supn ||gn||q < 1 for some q � 2, then almost every realization an = gn(y) has the property that for every Dunford-Schwartz operator T on a probability space (,µ) and f 2 Lp(µ), p > q
Original languageEnglish
Pages (from-to)101-126
JournalContemporary Mathematics
Volume356
DOIs
StatePublished - 1 Jan 2004

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