符号变更数量：AR的分段（1）
Number of Sign Changes: Segment of AR(1)
摘要：
设$X_{t}$表示平稳的一阶自回归过程。考虑序列的$n$连续观测（在时间$t$上）（例如，$X_{1}, ..., X_{n}$）。设其均值为零，滞后一序列相关为$ρ$，满足$|ρ| < 1$。Rice（1945）证明了$(n-1) \arccos(ρ)/π$是符号变化的期望数。Nyberg，Lizana & Ambjörnsson（2018）基于独立区间近似提出了高阶矩的相应公式。我们只关注小$n$的方差，并看到理论和模型之间有希望的拟合。
英文摘要：
Let $X_{t}$ denote a stationary first-order autoregressive process. Consider $n$ contiguous observations (in time $t$) of the series (e.g., $X_{1}, ..., X_{n}$). Let its mean be zero and its lag-one serial correlation be $ρ$, which satisfies $|ρ| < 1$. Rice (1945) proved that $(n-1) \arccos(ρ)/π$ is the expected number of sign changes. A corresponding formula for higher-order moments was proposed by Nyberg, Lizana & Ambjörnsson (2018), based on an independent interval approximation. We focus on the variance only, for small $n$, and see a promising fit between theory and model.
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