Asymptotics of Products of Nonnegative Random Matrices

V. Yu. Protasov

Funkcionalʹnyj analiz i ego priloženiâ, Tome 47 (2013) no. 2, pp. 68-79.

Résumé

Asymptotic properties of products of random matrices $\xi_k=X_k\cdots X_1$ as $k\to\infty$ are analyzed. All product terms $X_i$ are independent and identically distributed on a finite set of nonnegative matrices $\mathcal{A}=\{A_1,\dots, A_m\}$. We prove that if $\mathcal{A}$ is irreducible, then all nonzero entries of the matrix $\xi_k$ almost surely have the same asymptotic growth exponent as $k\to\infty$, which is equal to the largest Lyapunov exponent $\lambda(\mathcal{A})$. This generalizes previously known results on products of nonnegative random matrices. In particular, this removes all additional “nonsparsity” assumptions on matrices imposed in the literature. We also extend this result to reducible families. As a corollary, we prove that Cohen's conjecture (on the asymptotics of the spectral radius of products of random matrices) is true in case of nonnegative matrices.

Détail

Mots-clés : random matrix, nonnegative matrix
Keywords: Lyapunov exponent, asymptotics, sparsity, irreducibility.

Bibliographie

[1] H. Furstenberg, H. Kesten, “Products of random matrices”, Ann. Math. Statist., 31 (1960), 457–469 | DOI | MR | Zbl

[2] V. I. Oseledets, “Multiplikativnaya ergodicheskaya teorema. Kharakteristicheskie pokazateli Lyapunova dinamicheskikh sistem”, Trudy MMO, 19, Izd-vo Mosk. un-ta, M., 1968, 179–210

[3] V. N. Tutubalin, “O predelnykh teoremakh dlya proizvedeniya sluchainykh matrits”, TVP, 10:1 (1965), 19–32 | MR | Zbl

[4] E. Le Page, “Théorème limites pour les produits de matrices aléatories”, Lecture Notes in Math., 928, 1982, 258–303 | DOI | MR | Zbl

[5] A. V. Lëtchikov, “Uslovnaya predelnaya teorema dlya proizvedenii sluchainykh matrits”, Matem. sb., 186:3 (1995), 65–84 | MR | Zbl

[6] I. Ya. Goldsheid, G. A. Margulis, “Pokazateli Lyapunova proizvedeniya sluchainykh matrits”, UMN, 44(269):5 (1989), 13–60 | MR

[7] W. C. Watkins, “Limit theorems for products of random matrices: a comparison of two points of view”, Contemp. Math., 50, Amer. Math. Sci., Providence, RI, 1986, 5–29 | DOI | MR

[8] H. Ishitani, “A central limit theorem for the subadditive process and its application to products of random matrices”, Publ. Res. Inst. Math. Sci., 12:3 (1976/77), 565–575 | DOI | MR

[9] E. S. Key, “Lower bounds for the maximal Lyapunov exponent”, J. Theoret. Probab., 3:3 (1990), 477–488 | DOI | MR | Zbl

[10] H. Hennion, “Limit theorems for products of positive random matrices”, Ann. Probab., 25:4 (1997), 1545–1587 | DOI | MR | Zbl

[11] D. Hong, Lyapunov exponents: when the top joins the bottom, Technical Report RR-4198, INRIA, 2001 http://hal.inria.fr/docs/00/07/24/24/pdf/rr-4198.pdf

[12] J. E. Cohen, “Convexity properties of products of random nonnegative matrices”, Proc. Nati. Acad. Sci. USA, 77:7, part 1 (1980), 3749–3752 | DOI | MR | Zbl

[13] R. Jungers, V. Yu. Protasov, V. Blondel, “Overlap-free words and spectra of matrices”, Theor. Comput. Sc., 410:38–40 (2009), 3670–3684 | DOI | MR | Zbl

[14] V. Yu. Protasov, “Invariantnye funktsionaly sluchainykh matrits”, Funkts. analiz i ego pril., 44:3 (2010), 84–88 | DOI | MR | Zbl

[15] V. Yu. Protasov, “Invariantnye funktsii dlya pokazatelei Lyapunova sluchainykh matrits”, Matem. sb., 202:1 (2011), 105–132 | DOI | MR | Zbl

[16] Random Matrices and Their Applications, Contemp. Math., 50, eds. J. E. Cohen, H. Kesten, C. M. Newman, Amer. Math. Soc., Providence, RI, 1986 | MR | Zbl

[17] J. N. Tsitsiklis, V. D. Blondel, “The Lyapunov exponent and joint spectral radius of pairs of matrices are hard—when not impossible—to compute and to approximate”, Math. Control Signals Systems, 10:1 (1997), 31–40 | DOI | MR | Zbl

[18] L. Gerencser, G. Michaletzky, Z. Orlovits, “Stability of block-triangular stationary random matrices”, Systems Control Lett., 57:8 (2008), 620–625 | DOI | MR | Zbl