Open Access
October, 1980 Joint Limit Laws of Sample Moments of a Symmetric Distribution
Howard G. Tucker
Ann. Probab. 8(5): 991-998 (October, 1980). DOI: 10.1214/aop/1176994627


Let $\{X_n\}$ be a sequence of i.i.d. random variables with a common symmetric distribution $F$. Let $\mathfrak{L}(Z)$ denote the distribution of a random variable $Z$, and let $\mathfrak{D}(\alpha)$ denote the domain of attraction of a stable law of characteristic exponent $\alpha$. It is assumed that $\mathfrak{L}(X^k_1) \in \mathfrak{D}(\alpha)$ for some integer $k \geqslant 2$ and $\alpha \in (0, 2].$ Let $\mathbf{S}_n$ denote the $k$-dimensional random vector whose $j$th coordinate is $\Sigma^n_{i=1} X^j_i$, and let $m = \max\{j: k\alpha/j \geqslant 2\}$. Then there exist a sequence of $k \times k$ matrices $\{A_n\}$ and a sequence of vectors $\{\mathbf{b}_n\}$ in $\mathbb{R}^k$ such that $\mathbf{A}_n\mathbf{S}_n + \mathbf{b}_n$ converges in law to a random vector $\mathbf{S}$. The first $m$ coordinates of $\mathbf{S}$ are jointly normal and are independent of the remaining $k - m$ coordinates. No pair of these remaining $k - m$ coordinates are independent, but their joint distribution is operator-stable with two orbits.


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Howard G. Tucker. "Joint Limit Laws of Sample Moments of a Symmetric Distribution." Ann. Probab. 8 (5) 991 - 998, October, 1980.


Published: October, 1980
First available in Project Euclid: 19 April 2007

zbMATH: 0441.60017
MathSciNet: MR586782
Digital Object Identifier: 10.1214/aop/1176994627

Primary: 60F05

Keywords: domain of attraction of a stable distribution , multivariate stable distribution , operator-stable distribution , Sample moments

Rights: Copyright © 1980 Institute of Mathematical Statistics

Vol.8 • No. 5 • October, 1980
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