Electronic Communications in Probability

Nonconventional random matrix products

Yuri Kifer and Sasha Sodin

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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.

Article information

Source
Electron. Commun. Probab., Volume 23 (2018), paper no. 37, 12 pp.

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

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1528509621

Digital Object Identifier
doi:10.1214/18-ECP140

Mathematical Reviews number (MathSciNet)
MR3820127

Zentralblatt MATH identifier
1394.60005

Subjects
Primary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52) 60F15: Strong theorems 60F10: Large deviations 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.)

Keywords
random matrix products large deviations avalanche principle nonconventional limit theorems

Rights
Creative Commons Attribution 4.0 International License.

Citation

Kifer, Yuri; Sodin, Sasha. Nonconventional random matrix products. Electron. Commun. Probab. 23 (2018), paper no. 37, 12 pp. doi:10.1214/18-ECP140. https://projecteuclid.org/euclid.ecp/1528509621


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