Summary
Let {Y i} be iid with EY 1=0, EY 21 =1. Let {Xi} be iid normal mean zero, variance one random variables. According to Strassen's first almost sure invariance principle {X i} and {Y i} can be reconstructed on a new probability space without changing the distribution of each sequence such that \(\sum\limits_{i = 1}^n {Y_i - \sum\limits_{i = 1}^n {X_i } = o(n{\text{ }}\log _2 n)^{1/2} } \) a.s., thus improving on the trivial bound obtainable from the law of the iterated logarithm: \(\sum\limits_{i = 1}^n {Y_i - \sum\limits_{i = 1}^n {X_i } = O(n{\text{ }}\log _2 n)^{1/2} } \) a.s. In this work we establish analogous improvements for symmetric {Y i} in the domain of normal attraction to a symmetric stable law with index 0<α<2. (We make this assumption of symmetry in order to avoid messy details concerning centering constants.) Let {X i} be iid symmetric stable random variables with index 0<α<2. Then, for example, hypotheses are stated which imply for a given γ satisfying 2>γ≧α that \(\sum\limits_{i = 1}^n {Y_i - \sum\limits_{i = 1}^n {X_i } = o(n^{1/\gamma } )} \) a.s., thus improving on the trivial bound: \(\sum\limits_{i = 1}^n {Y_i - \sum\limits_{i = 1}^n {X_i } = o(n^{1/\alpha + \varepsilon } )} \) a.s., ɛ>0.
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This research was supported in part by a National Science Foundation grant, USA
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Stout, W. Almost sure invariance principles when EX 21 =∞. Z. Wahrscheinlichkeitstheorie verw Gebiete 49, 23–32 (1979). https://doi.org/10.1007/BF00534337
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DOI: https://doi.org/10.1007/BF00534337