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A characteristic function 9(t) (or the corresponding distribution F) is said to be semi-stable, if for some constants T>I and c>1, 9(t)=9~(c-lt),for all * holds. Every semi-stable characteristic function is infinitely divisible. A complex valued function 9(t) defined on the real line is a non-normal semi-stable characteristic function which satisfies (1) converges (or contains a subsequence which converges) weakly to F, where asterisk denotes the convolution operator. Every semi-stable (or stable) distribution belongs to the domain of partial attraction (or the domain of attraction) of itself. It is known 245

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