Unknown Book Proceedings Series Volume 00, XXXX On strong laws of large numbers with rates Guy Cohen, Roger L. Jones, and Michael Lin Abstract. Let {fn}⊂ Lp(µ), 1 <p< ∞, be a sequence of functions with sup n ||fn||p < ∞. We prove that if for some 0 <β ≤ 1 we have sup n    1 n 1−β n  k=1 f k    p < ∞, then for δ< p - 1 p β the sequence { 1 n 1−δ n  k=1 f k } has a.e. bounded p-variation, hence converges, and the p-variation norm func- tion is in Lp(µ). If we replace sup n ||fn||p < ∞ by sup n ||fn||∞ < ∞, then the a.e. convergence holds for δ< p p+1 β. Furthermore, in each case we also have a.e. convergence of the series ∞  k=1 f k k 1−δ for the corresponding values of δ, 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 sup n ||gn||q < ∞ 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 (Ω,µ) and f ∈ Lp(µ),p> q q−1 the series ∞  k=1 a k T k f k converges a.e. The same result holds for 1 <q< 2 if in addition the random varaibles {gn} are all symmetric. When the {gn} are i.i.d. the symmetry is not needed, and a.e. convergence of the above series holds also for f ∈ L q q−1 (µ). 1. Introduction It is known that there is no general speed of convergence in the pointwise ergodic theorem for ergodic measure preserving transformations; Krengel [Kr1] has shown that for every measure preserving transformation θ of the unit circle with Lebesgue measure and for every sequence {a n } of positive numbers converging to 0 there exists a continuous function f with integral 0 such that lim sup n | 1 n ∑ n k=1 f ◦ θ k |/a n = ∞ a.e. For further discussion see pp. 14-15 of [Kr2]. 1991 Mathematics Subject Classification. Primary 47A35, 28D05; Secondary 42A16, 60F15. Key words and phrases. strong laws of large numbers, ergodic theorems, speed of conver- gence, random Fourier series. Roger Jones was partially supported by a research leave granted by the Research Council of DePaul University. c XXXX American Mathematical Society 1