Electronic Communications in Probability

Nonconventional random matrix products

Yuri Kifer and Sasha Sodin

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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

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

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

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Zentralblatt MATH identifier

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.)

random matrix products large deviations avalanche principle nonconventional limit theorems

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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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  • [1] P. Bougerol, Comparison des exposants de Lyapunov des processus markoviens multiplicatifs, Ann. Inst. H. Poincaré (sec. B) 24 (1988), 439–489.
  • [2] P. Bougerol, Théorémes limite pour les systémes linéares á coefficients markovien, Probab. Theor. Rel. Fields 78 (1988), 193–221.
  • [3] Y. Benoist and J.-F. Quint, Random Walks on Reductive Groups, Springer, Heidelberg, 2016.
  • [4] P. Bougerol and J. Lacroix, Products of Random Matrices and Applications to Schrödinger Operators, Birkhäuser, Boston, 1985.
  • [5] R.C. Bradley, Introduction to Strong Mixing Conditions, Kendrick Press, Heber City, 2007.
  • [6] P. Duarte and S. Klein, Lyapunov Exponents of Linear Cocycles, Continuity via Large Deviations, Atlantis Press, 2016.
  • [7] H. Furstenberg and H. Kesten, Products of random matrices, Ann. Math. Stat. 31 (1960), 457–469.
  • [8] M. Goldstein and W. Schlag, Hölder continuity of the integrated density of states for quasi-periodic Schrödinger equation and averages of shifts of subharmonic functions, Ann. Math. 154 (2001), 155–203.
  • [9] Y. Hafouta, Stein’s method for nonconventional sums, arXiv 1704.01094
  • [10] Y. Hafouta and Y. Kifer, Nonconventional Limit Theorems and Random Dynamics, World Scientific, Singapore, 2018.
  • [11] I.A. Ibragimov and Y.V. Linnik, Independent and Stationary Sequences of Random Variables, Wolters-Noordhoff, Groningen, 1971.
  • [12] A.D. Virtser, On the simplicity of the spectrum of the Lyapunov characteristic indices of a product of random matrices, Theor. Probab. Appl. 28 (1984), 122–135.