Relatively stable walks

B. A. Rogosin

Teoriâ veroâtnostej i ee primeneniâ, Tome 21 (1976) no. 2, pp. 383-387

Résumé

Let $\xi_1,\xi_2,\dots$ he a sequence of i.i.d.r.v.'s, $\displaystyle S_n=\sum_{k=1}^n\xi_k$, $n=1,\dots$. The following statements are equivalent: 1) $S_n/a_n\to 1$ in probability for some sequence of positive numbers $a_1,a_2,\dots$; 2) $\displaystyle\nu(x)=\int_{\{|\xi_1|<x\}}\xi_1d\mathbf P>0$0$ for sufficiently large $x>0$, $\displaystyle\qquad\lim_{x\to\infty}\nu(xy)/\nu(x)=1$ for all $y>0\quad$ and $\displaystyle\quad\lim_{x\to\infty}x\mathbf P(|\xi_1|\ge x)/\nu(x)=0$.