Study of statistical properties for level crossings in discrete time series

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In the paper by Tanaka and Shimizu (2001), the expectation formula of level crossings for discrete time series is studied in the case where the underlying process is a strictly stationary ellipsoidal process with zero-mean, unit-variance and autocorrelation function ρ(h), which is actually governed by a scale mixture of normal distributions. The present study assumes the same process and aims to get expectation and variance formulae of the numbers of one-step and two-step level-crossings for an observed discrete sample Y_1,...,Y_N. Transformation into trinary series 1, 0, -1 is used to get the results. Here the number of one-step level-crossings is defined by the number t (2≦t≦N) satisfying that [Y_t≧u and v≦Y_<t-1><u], or [v≦Y_t<u and Y_<t-1>≧u], or [v≦Y_t<u and Y_<t-1><v], or finally [Y_t<v and v≦Y_<t-1><u] for u, v(u>v). Similarly the number of two-step level-crossings is defined by the number t (2≦t≦N) satisfying that [Y_t≧u and Y_<t-1><u] or [ Y_t<v and Y_<t-1>≧u]. The expectation formulae are obtained in the paper by Shimizu and Tanaka (2003) and the paper also provides applications to the expectation formula for the number of two-step level-crossings after taking the first- and second-difference time series in a stationary Gaussian zero-mean, unit-variance and autocorrelation function ρ(h). The numbers generalize the number of extremes (local minima and maxima) and that of inflection points. Asymptotic behavior of the length of excursions is studied in Tanaka and Shimizu (2004). A paper by Shikama and Shimizu which studies variance formulae is submitted to a journal for possible publication and is now under review.