Convolutions of Distributions Attracted to Stable Laws

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 $F \varepsilon \mathscr{D}(\alpha)$ 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.