TY - JOUR

T1 - A Note on General Sliding Window Processes

AU - Alon, Noga

AU - Feldheim, Ohad N.

PY - 2014/9/22

Y1 - 2014/9/22

N2 - Let f: Rk→[r] = {1, 2,…,r} be a measurable function, and let {Ui}i∈N be a sequence of i.i.d. random variables. Consider the random process {Zi}i∈N defined by Zi = f(Ui,…,Ui+k-1). We show that for all q, there is a positive probability, uniform in f, that Z1 = Z2 = … = Zq. A continuous counterpart is that if f: Rk → R, and Ui and Zi are as before, then there is a positive probability, uniform in f, for Z1,…,Zq 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.

AB - Let f: Rk→[r] = {1, 2,…,r} be a measurable function, and let {Ui}i∈N be a sequence of i.i.d. random variables. Consider the random process {Zi}i∈N defined by Zi = f(Ui,…,Ui+k-1). We show that for all q, there is a positive probability, uniform in f, that Z1 = Z2 = … = Zq. A continuous counterpart is that if f: Rk → R, and Ui and Zi are as before, then there is a positive probability, uniform in f, for Z1,…,Zq 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.

KW - D-dependent

KW - De Bruijn

KW - K-factor

KW - Ramsey

UR - http://www.scopus.com/inward/record.url?scp=84907710904&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84907710904&partnerID=8YFLogxK

U2 - 10.1214/ECP.v19-3341

DO - 10.1214/ECP.v19-3341

M3 - Article

AN - SCOPUS:84907710904

VL - 19

JO - Electronic Communications in Probability

JF - Electronic Communications in Probability

SN - 1083-589X

ER -