A note on general sliding window processes

作者：Alon Noga^*; Feldheim Ohad N

来源：Electronic Communications in Probability, 2014, 19(0): 1-7.

DOI：10.1214/ECP.v19-3341

• k-factor
• d-dependent
• de Bruijn
• Ramsey

Let f : R-k -> [ r] = {1, 2, . . . , r} be a measurable function, and let {U-i}(i is an element of N) be a sequence of i.i.d. random variables. Consider the random process {Z(i)}(i is an element of N) defined by Z(i) = f (U-i , . . . ,Ui+ k-1). We show that for all q, there is a positive probability, uniform in f, that Z(1) - Z(2) - . . . - Z(q). A continuous counterpart is that if f : R-k -> R, and U-i and Z(i) are as before, then there is a positive probability, uniform in f, for Z(1) , . . . , Z(q) to be monotone. We prove these theorems, give upper and lower bounds for this probability, and generalize to variables indexed on other lattices.
The proof is based on an application of combinatorial results from Ramsey theory to the realm of continuous probability.

• 出版日期2014-9-22