关于一般滑动窗口过程的一个注记
A note on general sliding window processes
摘要：
令$f:\mathbb{R}^k\to \mathbb{R}$是可测函数，令$\{U_i\}_{i\in\mathbb{N}}$是i.i.d.随机变量。考虑随机过程$Z_i=f(U_{i},...,U_{i+k-1})$。我们证明，对于所有$\ell$，$Z_1,...,Z_\ell$单调的正概率在$f$中是一致的。我们给出了这个概率的上下界，并得出了有限值域的$k$ -块因子过程的推论。证明是基于应用组合结果从拉姆齐理论的境界连续概率。
英文摘要：
Let $f:\mathbb{R}^k\to \mathbb{R}$ be a measurable function, and let $\{U_i\}_{i\in\mathbb{N}}$ be a sequence of i.i.d. random variables. Consider the random process $Z_i=f(U_{i},...,U_{i+k-1})$. We show that for all $\ell$, there is a positive probability, uniform in $f$, for $Z_1,...,Z_\ell$ to be monotone. We give upper and lower bounds for this probability, and draw corollaries for $k$-block factor processes with a finite range. The proof is based on an application of combinatorial results from Ramsey theory to the realm of continuous probability.
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