TY - JOUR

T1 - Nonconventional random matrix products

AU - Kifer, Yuri

AU - Sodin, Sasha

N1 - Publisher Copyright:
© 2018, University of Washington. All rights reserved.

PY - 2018

Y1 - 2018

N2 - Let ξ1, ξ2, … be independent identically distributed random variables and F: ℝl → SLd(ℝ) be a Borel measurable matrix-valued function. Set Xn = F(ξq1(n), ξq2(n), …, ξql(n)) where 0 ≤ q1 < q2 < … < ql are increasing functions taking on integer values on integers. We study the asymptotic behavior as N → ∞ 1 of the singular values of the random matrix product ΠN = XN … X2X1 and show, in particular, that (under certain conditions) [math] converges with probability one as N → ∞. We also obtain similar results for such products when ξi form a Markov chain. The essential difference from the usual setting appears since the sequence (Xn, n ≥ 1) is long-range dependent and nonstationary.

AB - Let ξ1, ξ2, … be independent identically distributed random variables and F: ℝl → SLd(ℝ) be a Borel measurable matrix-valued function. Set Xn = F(ξq1(n), ξq2(n), …, ξql(n)) where 0 ≤ q1 < q2 < … < ql are increasing functions taking on integer values on integers. We study the asymptotic behavior as N → ∞ 1 of the singular values of the random matrix product ΠN = XN … X2X1 and show, in particular, that (under certain conditions) [math] converges with probability one as N → ∞. We also obtain similar results for such products when ξi form a Markov chain. The essential difference from the usual setting appears since the sequence (Xn, n ≥ 1) is long-range dependent and nonstationary.

KW - Avalanche principle

KW - Large deviations

KW - Nonconventional limit theorems

KW - Random matrix products

UR - http://www.scopus.com/inward/record.url?scp=85050943926&partnerID=8YFLogxK

U2 - 10.1214/18-ECP140

DO - 10.1214/18-ECP140

M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???

AN - SCOPUS:85050943926

SN - 1083-589X

VL - 23

JO - Electronic Communications in Probability

JF - Electronic Communications in Probability

ER -