Study of statistical properties for level crossings in discrete time series

在Tanaka和Shimizu（2001）的论文中，研究了离散时间序列的水平交叉点的期望公式，其中基础过程是具有零均值、单位方差和自相关函数ρ（h）的严格平稳椭球过程，实际上是由正态分布的尺度混合所支配。本文假设相同的过程，对观测的离散样本Y_1，…，给出了一步和两步水平交叉数的期望和方差公式，Y_N。转换成三进制序列1，0，-1是用来得到结果。这里，一步电平交叉的数量由数量t（2 &lt;$t &lt;$N）定义，满足[Y_t &lt;$u和v &lt;$Y_<t-1>&lt;u]，或[v &lt;$Y_t &lt;u和Y_<t-1>&lt;$u]，或[v &lt;$Y_t&lt;u和Y_<t-1>&lt;v]，或最后对于u，v（u&gt;v）满足[Y_t&lt;v和v &lt;$Y_<t-1>&lt;u]。类似地，两步电平交叉的数量由满足[Y_t &lt;v和Y_<t-1>&lt;u]或[ Y_t&lt;v和Y_&lt;u<t-1>]的数量t（2 &lt;$t &lt;$N）定义。Shimizu和Tanaka（2003）在论文中得到了期望公式，论文还提供了在平稳高斯零均值、单位方差和自相关函数ρ（h）中取第一和第二差分时间序列后两步水平交叉次数的期望公式的应用。这些数字概括了极值（局部最小值和最大值）和拐点的数量。Tanaka and Shimizu（2004）研究了漂移长度的渐近行为。Shikama和Shimizu的一篇研究方差公式的论文已提交给一家杂志，供可能发表，目前正在审查中。

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.