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2018 Nonconventional random matrix products
Yuri Kifer, Sasha Sodin
Electron. Commun. Probab. 23: 1-12 (2018). DOI: 10.1214/18-ECP140

Abstract

Let $\xi _1,\xi _2,...$ be independent identically distributed random variables and $F:{\mathbb R}^\ell \to SL_d({\mathbb R})$ be a Borel measurable matrix-valued function. Set $X_n=F(\xi _{q_1(n)},\xi _{q_2(n)},...,\xi _{q_\ell (n)})$ where $0\leq q_1<q_2<...<q_\ell $ are increasing functions taking on integer values on integers. We study the asymptotic behavior as $N\to \infty $ of the singular values of the random matrix product $\Pi _N=X_N\cdots X_2X_1$ and show, in particular, that (under certain conditions) $\frac 1N\log \|\Pi _N\|$ converges with probability one as $N\to \infty $. We also obtain similar results for such products when $\xi _i$ form a Markov chain. The essential difference from the usual setting appears since the sequence $(X_n,\, n\geq 1)$ is long-range dependent and nonstationary.

Citation

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Yuri Kifer. Sasha Sodin. "Nonconventional random matrix products." Electron. Commun. Probab. 23 1 - 12, 2018. https://doi.org/10.1214/18-ECP140

Information

Received: 25 March 2018; Accepted: 29 May 2018; Published: 2018
First available in Project Euclid: 9 June 2018

zbMATH: 1394.60005
MathSciNet: MR3820127
Digital Object Identifier: 10.1214/18-ECP140

Subjects:
Primary: 60B20, 60F10, 60F15, 82B44

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