Limit theorems for products of positive random matrices. (English) Zbl 0903.60027

Summary: Let \(S\) be the set of \(q\times q\) matrices with positive entries, such that each column and each row contains a strictly positive element, and denote by \(S^0\) the subset of these matrices, all entries of which are strictly positive. Consider a random ergodic sequence \((X_n)_{n\geq 1}\) in \(S\). The aim of this paper is to describe the asymptotic behavior of the random products \(X^{(n)}= X_n\cdots X_1\), \(n\geq 1\), under the main hypothesis \(P\left(\bigcup_{n\geq 1}[X^{(n)}\in S^0]\right)> 0\). We first study the behavior “in direction” of row and column vectors of \(X^{(n)}\). Then, adding a moment condition, we prove a law of large numbers for the entries and lengths of these vectors and also for the spectral radius of \(X^{(n)}\). Under the mixing hypotheses that are usual in the case of sums of real random variables, we get a central limit theorem for the previous quantities. The variance of the Gaussian limit law is strictly positive except when \((X^{(n)})_{n\geq 1}\) is tight. This tightness property is fully studied when the \(X_n\), \(n\geq 1\), are independent.


60F99 Limit theorems in probability theory
60F05 Central limit and other weak theorems
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[1] Arnold, L., Demetrius, L. and Gundlach, V. M. (1994). Evolutionary formalism for products of positive random matrices. Ann. Appl. Probab. 4 859-901. · Zbl 0818.15015
[2] Bougerol, P. (1987). Tightness of products of random matrices and stability of linear stochastic systems. Ann. Probab. 15 40-74. · Zbl 0614.60008
[3] Bougerol, P. and Lacroix, J. (1985). Products of Random Matrices with Applications to Schrödinger Operators. Birkhäuser, Boston. · Zbl 0572.60001
[4] Bushell, P. J. (1973). Hilbert’s metric and positive contraction mappings in a Banach space. Arch. Rational Mech. Anal. 52 330-338. · Zbl 0275.46006
[5] Cohen, J. (1986). Products of random matrices and related topics in mathematics and science-a bibliography. Contemp. Math. 50 337-358. · Zbl 0584.60020
[6] Cohn, H., Nerman, O. and Peligrad, M. (1993). Weak ergodicity and products of random matrices. J. Theoret. Probab. 6 389-405. · Zbl 0770.60005
[7] Cornfeld, I. P., Fomin, S. V. and Sinai, Ya. (1982). Ergodic Theory. Springer, New York. · Zbl 0493.28007
[8] Doukhan, P. (1994). Mixing: Properties and Examples. Lecture Notes in Statist. 85. Springer, New York. · Zbl 0790.60037
[9] D ürr, D. and Goldstein, S. (1984). Remarks on the CLT for weakly dependent random variables. Lecture Notes in Math. 1158. Springer, New York. · Zbl 0582.60036
[10] Durrett, R. (1991). Probability: Theory and Examples. Wadsworth and Brooks/Cole, Monterey, CA. · Zbl 0709.60002
[11] Esseen, C. G. and Janson, S. (1985). On moment conditions for sums of independent variables and martingale differences. Stochastic Process. Appl. 19 173-182. · Zbl 0554.60050
[12] Evstigneev, C. G. I.V. (1974). Positive matrix-valued cocycles over dynamical systems. Uspehi Mat. Nauk 29 219-226. · Zbl 0314.28015
[13] Ferrero, P. and Schmitt, B. (1988). Produits aléatoires d’opérateurs matrices de transfert. Probab. Theory Related Fields 79 227-248. · Zbl 0633.60014
[14] Furstenberg, H. (1963). Non-commuting random products. Trans. Amer. Math. Soc. 108 377-428. JSTOR: · Zbl 0203.19102
[15] Furstenberg, H. and Kesten, H. (1960). Products of random matrices. Ann. Math. Statist. 31 457-469. · Zbl 0137.35501
[16] Gordin, M. I. (1969). The central limit theorem for stationary processes. Soviet Math. Dokl 10 1174-1176. · Zbl 0212.50005
[17] Guivarc’h, Y. and Raugi, A. (1986). Products of random matrices: convergence theorems. Contemp. Math. 50 31-54. · Zbl 0592.60015
[18] Hall, P. and Heyde, C. C. (1980). Martingale Limit Theory and Its Application. Academic Press, New York. · Zbl 0462.60045
[19] Hennion, H. (1991). Dérivabilité du plus grand exposant caractéristique des produits de matrices aléatoires indépendantes a coefficients positifs. Ann. Inst. H. Poincaré Probab. Statist. 27 27-59. · Zbl 0724.60009
[20] Kesten, H. and Spitzer, F. (1984). Convergence in distribution of products of random matrices.Wahrsch. Verw. Gebiete 67 363-386. · Zbl 0535.60016
[21] Kingman, J. F. C. (1973). Subadditive ergodic theory. Ann. Probab. 1 883-899. · Zbl 0311.60018
[22] Kranosel’skij, M. A., Lifshits, Je. A. and Sobolev, A. V. (1980). Positive Linear SystemsThe Method of Positive Operators. Sigma Ser. Appl. Math. Heldermann, Berlin.
[23] Ledrappier, F. (1982). Quelques propriétés des exposants caractéristiques. Ecole d’été de Saint-Flour 12. Lecture Notes in Math. 1097. Springer, Berlin. · Zbl 0578.60029
[24] Le Page, E. (1982). Théor emes limites pour les produits de matrices aléatoires. Probability Measures on Groups. Lecture Notes in Math. 928 258-303. Springer, Berlin. · Zbl 0506.60019
[25] Liverani, C. (1996). Central limit theorem for deterministic systems. In International Conference on Dynamical Systems, Montevideo 1995: a Tribute to Ricardo Mané. · Zbl 0871.58055
[26] Mukherjea, A. (1987). Convergence in distribution of products of random matrices: a semigroup approach. Trans. Amer. Math. Soc. 303 395-411. JSTOR: · Zbl 0638.60007
[27] Pellaumail, J. (1990). Graphes et algorithme de calcul de probabilités stationnaires d’un processus markovien discret. Ann. Inst. H. Poincaré Probab. Statist. 26 121-143. · Zbl 0701.60064
[28] Peres, Y. (1992). Domain of analytic continuation for the top Lyapunov exponent. Ann. Inst. H. Poincaré Probab. Statist. 28 131-148. · Zbl 0794.58023
[29] Raugi, A. (1983). Une démonstration du théor eme de Choquet-Deny par les martingales. Ann. Inst. H. Poincaré Statist. Probab. 19 101-109.
[30] Seneta, E. (1981). Nonnegative Matrices and Markov Chains. Springer, New York. · Zbl 0471.60001
[31] Volny, D. (1993). Approximating martingales and the central limit theorem for strictly stationary processes. Stochastic Process. Appl. 44 41-74. · Zbl 0765.60025
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