Limit Distributions of Self-normalized Sums

If Xᵢ are i.i.d. and have zero mean and arbitrary finite variance the limiting probability distribution of Sₙ(2) = (∑ⁿᵢ₌₁ Xᵢ)/(∑ⁿⱼ₌₁ Xⱼ²)1/2 as n → ∞ has density f(t) = (2π)- 1/2exp (-t²/2) by the central limit theorem and the law of large numbers. If the tails of Xᵢ are sufficiently smooth and satisfy P(Xᵢ > t) ~ rt-α and P(Xᵢ < -t) ~ lt-α as t→∞, where 0 < α < 2, r > 0, l > 0, Sₙ(2) still has a limiting distribution F even though Xᵢ has infinite variance. The density f of F depends on α as well as on $r/l$. We also study the limiting distribution of the more general Sₙ(p) = (∑ⁿᵢ₌₁ Xᵢ)/(∑ⁿⱼ₌₁ |Xⱼ|ᵖ)1/p where Xᵢ 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,α, r/l), and give graphs of f for a number of values of α and $r/l$. For $p = 2$, we find that f is always symmetric about zero on $(-1, 1)$, even though f is symmetric on (-∞, ∞) only when $r = l$.