Oscillating random walk with a moving boundary
\(S_{n + 1}\), an oscillating random walk with moving boundary of parameter \(\gamma\), is defined by the recursive relation: \(S_ n + U_ n\) (if \(S_ n < n\gamma\)); \(S_ n + V_ n\) (if \(S_ n > n\gamma\)); \(S_ n + (B_ n U_ n + (1 - B_ n)V_ n\)) (if \(S_ n = n\gamma\)), where \(\{B_ n\}\), \(\{U_ n\}\), \(\{V_ n\}\) are i.i.d. sequences, \(P(B_ n = 1) = 1 - P(B_ n = 0)\). It is shown that if \(\{U_ n\}\), \(\{V_ n\}\) are integer-valued, then interesting difference appears depending on \(\gamma\) being rational or not. Let \(EU_ n = \mu\), \(EV_ n = \nu\). The main results are: if \(\mu < \nu\), then on \((\mu,\nu)\), \(P(\liminf_{n \to \infty} \{S_ n < n\gamma\}\)) is continuous only at every irrational \(\gamma\); and if \(\mu > \nu\), then for an irrational \(\gamma \in (\nu,\mu)\), there is a unique invariant probability measure [this fills in a gap of a theorem of \textit{A. A. Borovkov}, Theory Probab. Appl. 25, 649-651 (1980)], while for rational \(\gamma\), there exist an uncountable number of mutually singular discrete invariant probability measures.
scientific article; zbMATH DE number 720065
scientific article