The oscillating random walk

{Y_n;n=0, 1, …} denotes a stationary Markov chain taking values in R^d. As long as the process stays on the same side of a fixed hyperplane E⁰, it behaves as an ordinary random walk with jump measure μ or ν, respectively. Thus ordinary random walk would be the special case μ = ν. Also the process Y′_n = |Y′_n?1?Z_n| (with the Z_n as i.i.d. real random varia bles) may be regarded as a special case. The general process is studied by a Wiener–Hopf type method. Exact formulae are obtained for many quantities of interest. For the special case that the Y_n are integral-valued, renewal type conditions are established which are necessary and sufficient for recurrence.