A central limit theorem for self-normalized products of random variables
Conditions on the distribution function \(F\) of independent and identically distributed random variables \(X_1,X_2,\dots\) are given under which the self-normalized product \[ Z_n=\frac{n\prod_{i=1}^n X_i} {\sqrt{\sum^* X_{i_1}^2 X_{i_2}^2\cdots X_{i_{n-1}}^2}} \] is asymptotically normally distributed as \(n\to\infty\); \(\sum^*\) denotes the sum over all \((n-1)\)-long sequences of integers \(1\leq i_1<i_2<\cdots<i_{n-1}\leq n\). The main condition is that for any \(k>0\), \[ \frac{F(1/\sqrt x)- F(-1/\sqrt x)}{F(1/\sqrt{kx})-F(-1/\sqrt{kx})}\to\sqrt k\quad\text{as } x\to\infty. \] The possible limiting behaviour of the product when this condition fails is also investigated. The limiting behaviour of the product \(\prod_{i=1}^n X_i\) with deterministic norming is also discussed. An appendix presents relevant facts from the theory of stable laws.
scientific article; zbMATH DE number 1310364