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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

A law of the iterated logarithm for stable summands


Authors: R. P. Pakshirajan and R. Vasudeva
Journal: Trans. Amer. Math. Soc. 232 (1977), 33-42
MSC: Primary 60F15
MathSciNet review: 0455093
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Abstract: Let $ {X_1},{X_2}, \ldots $ be a sequence of independent indentically distributed stable random variables with parameters $ \alpha \;(0 < \alpha < 2)$ and $ \beta (\vert\beta \vert \leqslant 1)$. Let $ {S_n} = \sum\nolimits_{i = 1}^n {{X_i}} $. Suppose that $ ({S_{1,n}})$ and $ ({S_{2,n}})$ are independent copies of the sequence $ ({S_n})$. In this paper we obtain the set of all limit points in the plane of the sequence

$\displaystyle \left\{ {\vert{n^{ - 1/\alpha }}({S_{1,n}} - {a_n}){\vert^{1/(\lo... ...},\vert{n^{ - 1/\alpha }}({S_{2,n}} - {a_n}){\vert^{1/(\log \log n)}}} \right\}$

where $ ({a_n})$ is zero if $ \alpha \ne 1$ and is $ (2\beta n\log n)/\pi $ if $ \alpha = 1$.

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DOI: https://doi.org/10.1090/S0002-9947-1977-0455093-4
Article copyright: © Copyright 1977 American Mathematical Society