## 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 {q_{n}} satisfies *q*
^{2}_{
n
}
=*nG*(q_{n})loglog*q*
_{
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