Limit Distributions of Self-normalized Sums

Abstract

If $X_i$ are i.i.d. and have zero mean and arbitrary finite variance the limiting probability distribution of $S_n(2) = (\sum^n_{i=1} X_i)/(\sum^n_{j=1} X_j^2)^{\frac{1}{2}}$ as $n \rightarrow \infty$ has density $f(t) = (2\pi)^{- \frac{1}{2}}\exp (-t^2/2)$ by the central limit theorem and the law of large numbers. If the tails of $X_i$ are sufficiently smooth and satisfy $P(X_i > t) \sim rt^{-\alpha}$ and $P(X_i < -t) \sim lt^{-\alpha}$ as $t\rightarrow\infty$, where $0 < \alpha < 2, r > 0, l > 0, S_n(2)$ still has a limiting distribution $F$ even though $X_i$ has infinite variance. The density $f$ of $F$ depends on $\alpha$ as well as on $r/l$. We also study the limiting distribution of the more general $S_n(p) = (\sum^n_{i=1} X_i)/(\sum^n_{j=1} |X_j|^p)^{1/p}$ where $X_i$ are i.i.d. and in the domain of a stable law $G$ with tails as above. In the cases $p = 2$ (see (4.21)) and $p = 1$ (see (3.7)) we obtain exact, computable formulas for $f(t) = f(t,\alpha, r/l)$, and give graphs of $f$ for a number of values of $\alpha$ and $r/l$. For $p = 2$, we find that $f$ is always symmetric about zero on $(-1, 1)$, even though $f$ is symmetric on $(-\infty, \infty)$ only when $r = l$.