Clipped Gaussian processes are never M-step Markov (Q1121593)

Given real-valued random variables \(Z_ t\), \(t=0,\pm 1,...\), the process \(X_ t=I(Z_ t\geq c)\), \(t=0,\pm 1,..\). is referred to as the clipped process. Here I(\(\cdot)\) denotes the indicator function and c is any given real number. The author shows that if \(Z_ t\), \(t=0,\pm 1,..\). is a stationary but not independent Gaussian process, then its clipped process can never be an m-th order Markov sequence \((m>1)\), although its correlation-sequence can agree exactly with that of a Markov sequence. This result shows the incompatibility of an assumption implicit in \textit{B. Kedem}, Biometrika 65, 207-210 (1978; Zbl 0371.62133).