Joint Limit Laws of Sample Moments of a Symmetric Distribution

Abstract

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.