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Convolutions of Distributions Attracted to Stable Laws

Howard G. Tucker
The Annals of Mathematical Statistics
Vol. 39, No. 5 (Oct., 1968), pp. 1381-1390
Stable URL: http://www.jstor.org/stable/2239395
Page Count: 10
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Convolutions of Distributions Attracted to Stable Laws
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Abstract

This paper deals with the domains of attraction of the stable distributions and the normalizing coefficients associated with distributions in those domains of attraction. Using the notation F ε D(α) and F ε DN(α) to mean that the distribution function F is in the domain of attraction and the domain of normal attraction respectively of a stable law of characteristic exponent α, the following result is obtained: if F ε D(α) and G ε D(β), where $0 < \alpha \leqq \beta \leqq 2$, and if {Bn} and {Cn} are normalizing coefficients respectively of F and G, then F * G ε D(α) and its normalizing coefficients are {(Bα n + Cα n)1/α}. Two more specialized results are obtained on convolutions of distribution functions in D(2), namely: (i) if F ε DN(2) and $G \varepsilon \mathscr{D}(2)\backslash\mathscr{D}_\mathscr{N}(2)$, then $F \ast G\varepsilon \mathscr{D}(2)\backslash\mathscr{D}_\mathscr{N}(2)$, and (ii) if F and G are distribution functions, and if the four tail probabilities vary regularly with exponent -2 and involve possibly four different slowly varying functions, then F, G and F* G are in D(2). These latter two results hold only for D(2) and not for D(α) for $0 < \alpha < 2$, thus adding two exceptional properties to the normal law within the family of stable laws.

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