Celebratio Mathematica — Rosenblatt

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G. F. Newell and M. Rosenblatt: “Zero crossing probabilities for Gaussian stationary processes,” Ann. Math. Stat. 33 : 4 (1962), pp. 1306–1313. MR 141153 Zbl 0113.33401 article

Let \( X(t) \) be a separable stationary Gaussian process with mean zero, \( EX(t) \equiv 0 \), and continuous covariance function \[ \rho(t) = EX(\tau)\,X(\tau + t) \] normalized so that \( \rho(0) = 1 \). Questions relating to the probability \[ H_X(T) = P[X(t) > 0], \quad 0 \leq t\leq T\ \] that \( X(t) \) does not cross zero during some time interval \( T \) arise in various applications [Rice 1944–1945; Slepian 1962].

Many of the difficulties that arise in treating such questions are due to the fact that most of the interesting stationary Gaussian processes are not Markovian. Here we shall obtain bounds on the behavior of \( H_X(T) \) particularly for large \( T \) using an interesting inequality of D. Slepian [1962] and estimates of \( H_X(T) \) for some simple processes.