Convergence in distribution of products of random matrices: a semigroup approach

Mukherjea, Arunava

doi:10.1090/S0002-9947-1987-0896029-8

Convergence in distribution of products of random matrices: a semigroup approach
HTML articles powered by AMS MathViewer

by Arunava Mukherjea PDF

Trans. Amer. Math. Soc. 303 (1987), 395-411 Request permission

Abstract:

The problem of weak convergence of the sequence of convolution powers of a probability measure has been considered in this paper in the general context of a noncompact semigroup and in particular, in the semigroup of nonnegative and real matrices. Semigroup methods have been used to give simple proofs of some recent results of Kesten and Spitzer in nonnegative matrices. It has been also shown that these methods often lead to similar results in the more general context of real matrices.

References

Philippe Bougerol, Tightness of products of random matrices and stability of linear stochastic systems, Ann. Probab. 15 (1987), no. 1, 40–74. MR 877590
W. Edwin Clark, Remarks on the kernel of a matrix semigroup, Czechoslovak Math. J. 15(90) (1965), 305–310 (English, with Russian summary). MR 177047
Göran Högnäs and Arunava Mukherjea, Recurrent random walks and invariant measures on semigroups of $n\times n$ matrices, Math. Z. 173 (1980), no. 1, 69–94. MR 584350, DOI 10.1007/BF01215525
Harry Kesten and Frank Spitzer, Convergence in distribution of products of random matrices, Z. Wahrsch. Verw. Gebiete 67 (1984), no. 4, 363–386. MR 761563, DOI 10.1007/BF00532045
Arunava Mukherjea, Limit theorems: stochastic matrices, ergodic Markov chains, and measures on semigroups, Probabilistic analysis and related topics, Vol. 2, Academic Press, New York-London, 1979, pp. 143–203. MR 556681
Arunava Mukherjea, A new result on the convergence of nonhomogeneous stochastic chains, Trans. Amer. Math. Soc. 262 (1980), no. 2, 505–520. MR 586731, DOI 10.1090/S0002-9947-1980-0586731-3
Arunava Mukherjea and Nicolas A. Tserpes, Measures on topological semigroups: convolution products and random walks, Lecture Notes in Mathematics, Vol. 547, Springer-Verlag, Berlin-New York, 1976. MR 0467871