The recurrency of oscillating random walks

B. A. Rogozin ; S. G. Foss

Teoriâ veroâtnostej i ee primeneniâ, Tome 23 (1978) no. 1, pp. 161-169

Résumé

Let $Y=\{y_n\}_{n=0}^{\infty}$ be an oscillating random walk ([1]): $$ y_0=0,\qquad y_{n+1}-y_n= \begin{cases} \xi'_{n+1},\le 0,\\ \xi''_{n+1},>0, \end{cases} \qquad(n=1,2,\dots), $$ $\{\xi'_n\}_{n=1}^{\infty}$ and $\{\xi''_n\}_{n=1}^{\infty}$ be two sequences of independent identically distributed, in each sequence, random variables with values in the set $\{0,\pm 1,\pm 2,\dots\}$, \begin{gather*} S'_0=S''_0=0,\\ S'_n=\sum_{k=1}^n\xi'_k,\qquad S''_n=\sum_{k=1}^n\xi''_k,\qquad n=1,2,\dots \end{gather*} The random walks $S'_n=\{S'_n\}_{n=0}^{\infty}$ and $S''_n=\{S''_n\}_{n=0}^{\infty}$ are aperiodic. It is shown that $Y$ can be transient in the case $\mathbf M\xi'_1=\mathbf M\xi''_1=0$. A recurrency condition for $Y$ is obtained when $S'$ and $S''$ are stable random walks.