Open Access
October, 1968 Convolutions of Distributions Attracted to Stable Laws
Howard G. Tucker
Ann. Math. Statist. 39(5): 1381-1390 (October, 1968). DOI: 10.1214/aoms/1177698119


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 $F \varepsilon \mathscr{D}(\alpha)$ and $F \varepsilon \mathscr{D}_\mathscr{N}(\alpha)$ 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 $\alpha$, the following result is obtained: if $F \varepsilon \mathscr{D}(\alpha)$ and $G \varepsilon \mathscr{D}(\beta)$, where $0 < \alpha \leqq \beta \leqq 2$, and if $\{B_n\}$ and $\{C_n\}$ are normalizing coefficients respectively of $F$ and $G$, then $F \ast G \varepsilon \mathscr{D}(\alpha)$ and its normalizing coefficients are $\{(B^\alpha_n + C^\alpha_n)^{1/\alpha}\}$. Two more specialized results are obtained on convolutions of distribution functions in $\mathscr{D}(2)$, namely: (i) if $F \varepsilon \mathscr{D}_\mathscr{N}(2)$ 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 $F\ast G$ are in $\mathscr{D}(2)$. These latter two results hold only for $\mathscr{D}(2)$ and not for $\mathscr{D}(\alpha)$ for $0 < \alpha < 2$, thus adding two exceptional properties to the normal law within the family of stable laws.


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Howard G. Tucker. "Convolutions of Distributions Attracted to Stable Laws." Ann. Math. Statist. 39 (5) 1381 - 1390, October, 1968.


Published: October, 1968
First available in Project Euclid: 27 April 2007

zbMATH: 0165.20102
MathSciNet: MR230358
Digital Object Identifier: 10.1214/aoms/1177698119

Rights: Copyright © 1968 Institute of Mathematical Statistics

Vol.39 • No. 5 • October, 1968
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