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) = (ⁿ₈=₁ Xᵢ) / (ⁿ₉=₁ Xⱼ²) ^1{2} as n has density f (t) = (2) ^- 1{2} (-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^- and P (Xᵢ 0, l > 0, Sₙ (2) still has a limiting distribution F even though Xᵢ has infinite variance. The density f of F depends on as well as on r/l. We also study the limiting distribution of the more general Sₙ (p) = (ⁿ₈=₁ Xᵢ) / (ⁿ₉=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, , r/l), and give graphs of f for a number of values of 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.