Abstract
We find conditions on a sequence of random variables to satisfy the strong law of large numbers (SLLN) under a rearrangement. It turns out that these conditions are necessary and sufficient for the permutational SLLN (PSLLN). By PSLLN we mean that the SLLN holds under almost all simple permutations within blocks the lengths of which grow exponentially (Prokhorov blocks). In the case of orthogonal random variables it is shown that Kolmogorov's condition, that is known not to be sufficient for SLLN, is actually sufficient for PSLLN. It is also shown that PSLLN holds for sequences that are strictly stationary with finite first moments. In the case of weakly stationary sequences a Gaposhkin result implies that SLLN and PSLLN are equivalent. Finally we consider the case of general norming and generalization of the Nikishin theorem. The methods of proof uses on the one hand the idea of Prokhorov blocks and Garsia's construction of product measure on the space of simple permutations, and on the other hand, a maximal inequality for permutations.
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Chobanyan, S., Levental, S. & Mandrekar, V. Prokhorov Blocks and Strong Law of Large Numbers Under Rearrangements. Journal of Theoretical Probability 17, 647–672 (2004). https://doi.org/10.1023/B:JOTP.0000040292.52298.fd
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DOI: https://doi.org/10.1023/B:JOTP.0000040292.52298.fd