Geodesic
Obtaining Prescribed Rates of Convergence for the Ergodic Theorem
Canadian journal of mathematics, Tome 35 (1983) no. 6, pp. 1129-1146

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Let {Yn, n ∊ Z} be an ergodic strictly stationary sequence of random variables with mean zero, where Z denotes the set of integers. For n ∊ N = {1, 2, ...}, let Sn = Y 1 + Y 2 + ... + Yn . The ergodic theorem, alias the strong law of large numbers, says that n –l Sn → 0 as n → ∞ a.s. If the Yn 's are independent and have variance one, the law of the iterated logarithm tells us that this convergence takes place at the rate in the sense that 1 It is our purpose here to investigate what other rates of convergence are possible for the ergodic theorem, that is to say, what sequences {bn, n ≧ 1} have the property that 2 for some ergodic stationary sequence {Yn, n ∊ Z}. 
O'Brien, G. L. Obtaining Prescribed Rates of Convergence for the Ergodic Theorem. Canadian journal of mathematics, Tome 35 (1983) no. 6, pp. 1129-1146. doi: 10.4153/CJM-1983-062-3
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