Summary
A generalization of the classical Law of the Iterated Logarithm (LIL) is obtained for the weighted i.i.d. case consisting of sequences {σ n Y n } where the weights {σ n } are nonzero constants and {Y n} are i.i.d. random variables. If Y is symmetric but not necessarily square integrable and if the weights satisfy a certain growth rate, conditions are given which guarantee that {σ n Y n} obey a Generalized Law of the Iterated Logarithm (GLIL) in the sense that \(\mathop {\lim \sup }\limits_{n \to \infty } \sum\limits_1^n {\sigma _j } Y_j /a_n = 1\) almost certainly for some positive conslants a n . Teicher has shown that such weights entail the classical LIL when EY 2<∞ and Feller has treated the GLIL when σ n =1 and EY 2=∞. The main finding here asserts that if {qn} satisfies q 2 n =nG(qn)loglogq n where G is a specified slowly varying function, asymptotically equivalent to the truncated second moment of Y, and if a certain series converges, then the GLIL obtains with \(a_n = (2/n)^{\tfrac{1}{2}} s_n q_n \) where \({\text{s}}_{\text{n}}^{\text{2}} = \sum\limits_1^n {\sigma _j^2 } \).
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Rosalsky, A. A generalization of the Iterated Logarithm Law for weighted sums with infinite variance. Z. Wahrscheinlichkeitstheorie verw Gebiete 58, 351–372 (1981). https://doi.org/10.1007/BF00542641
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DOI: https://doi.org/10.1007/BF00542641