A law of the iterated logarithm for stable summands

Abstract:

Let ${X_1},{X_2}, \ldots$ be a sequence of independent indentically distributed stable random variables with parameters $\alpha \;(0 < \alpha < 2)$ and $\beta (|\beta | \leqslant 1)$. 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 \[ \left \{ {|{n^{ - 1/\alpha }}({S_{1,n}} - {a_n}){|^{1/(\log \log n)}},|{n^{ - 1/\alpha }}({S_{2,n}} - {a_n}){|^{1/(\log \log n)}}} \right \}\] where $({a_n})$ is zero if $\alpha \ne 1$ and is $(2\beta n\log n)/\pi$ if $\alpha = 1$.