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12 papers
Short Communications
The recurrency of oscillating random walks
B. A. Rogozin,
S. G. Foss Novosibirsk
Abstract:
Let
$Y=\{y_n\}_{n=0}^{\infty}$ be an oscillating random walk ([1]):
$$
y_0=0,\qquad y_{n+1}-y_n=
\begin{cases}
\xi'_{n+1},&y_n\le 0,\\
\xi''_{n+1},&y_n>0,
\end{cases}
\qquad(n=1,2,\dots),
$$
$\{\xi'_n\}_{n=1}^{\infty}$ 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.
Received: 09.06.1976