Central limit theorem for products of random matrices

Berger, Marc A.

doi:10.1090/S0002-9947-1984-0752503-3

Central limit theorem for products of random matrices
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Trans. Amer. Math. Soc. 285 (1984), 777-803 Request permission

Abstract:

Using the semigroup product formula of $\text {P}$. Chernoff, a central limit theorem is derived for products of random matrices. Applications are presented for representations of solutions to linear systems of stochastic differential equations, and to the corresponding partial differential evolution equations. Included is a discussion of stochastic semigroups, and a stochastic version of the Lie-Trotter product formula.

References

Richard Bellman, Limit theorems for non-commutative operations. I, Duke Math. J. 21 (1954), 491–500. MR 62368
Marc A. Berger, A remark on stochastic fundamental matrices, Ann. Probab. 11 (1983), no. 1, 215–216. MR 682810
Marc A. Berger and Shmuel Friedland, The generalized Radon-Hurwitz numbers, Compositio Math. 59 (1986), no. 1, 113–146. MR 850125
Marc A. Berger and Victor J. Mizel, Theorems of Fubini type for iterated stochastic integrals, Trans. Amer. Math. Soc. 252 (1979), 249–274. MR 534121, DOI 10.1090/S0002-9947-1979-0534121-3

Semigroup approach to the radical differential calculus

Leo Breiman, Probability, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1968. MR 0229267
G. P. Bucan, The mixed product of nonhomogeneous stochastic semigroups, Teor. Veroyatnost. i Primenen. 24 (1979), no. 1, 168–175 (Russian, with English summary). MR 522249
Paul R. Chernoff, Note on product formulas for operator semigroups, J. Functional Analysis 2 (1968), 238–242. MR 0231238, DOI 10.1016/0022-1236(68)90020-7
Paul R. Chernoff, Product formulas, nonlinear semigroups, and addition of unbounded operators, Memoirs of the American Mathematical Society, No. 140, American Mathematical Society, Providence, R.I., 1974. MR 0417851
John D. Dollard and Charles N. Friedman, Product integration with applications to differential equations, Encyclopedia of Mathematics and its Applications, vol. 10, Addison-Wesley Publishing Co., Reading, Mass., 1979. With a foreword by Felix E. Browder; With an appendix by P. R. Masani. MR 552941
J. L. Doob, Stochastic processes, John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1953. MR 0058896
Harry Furstenberg, Noncommuting random products, Trans. Amer. Math. Soc. 108 (1963), 377–428. MR 163345, DOI 10.1090/S0002-9947-1963-0163345-0
Harry Furstenberg, A Poisson formula for semi-simple Lie groups, Ann. of Math. (2) 77 (1963), 335–386. MR 146298, DOI 10.2307/1970220
H. Furstenberg and H. Kesten, Products of random matrices, Ann. Math. Statist. 31 (1960), 457–469. MR 121828, DOI 10.1214/aoms/1177705909
Ĭ. Ī. Gīhman and A. V. Skorohod, Stochastic differential equations, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 72, Springer-Verlag, New York-Heidelberg, 1972. Translated from the Russian by Kenneth Wickwire. MR 0346904
Jerome A. Goldstein, Semigroup-theoretic proofs of the central limit theorem and other theorems of analysis, Semigroup Forum 12 (1976), no. 3, 189–206. MR 415408, DOI 10.1007/BF02195925
Ulf Grenander, Probabilities on algebraic structures, John Wiley & Sons, Inc., New York-London; Almqvist & Wiksell, Stockholm-Göteborg-Uppsala, 1963. MR 0206994
Y. Guivarc’h, Quelques propriétés asymptotiques des produits de matrices aléatoires, Eighth Saint Flour Probability Summer School—1978 (Saint Flour, 1978), Lecture Notes in Math., vol. 774, Springer, Berlin, 1980, pp. 177–250 (French). MR 590627
Yves Guivarc’h, Michael Keane, and Bernard Roynette, Marches aléatoires sur les groupes de Lie, Lecture Notes in Mathematics, Vol. 624, Springer-Verlag, Berlin-New York, 1977 (French). MR 0517359
G. A. Hunt, Semi-groups of measures on Lie groups, Trans. Amer. Math. Soc. 81 (1956), 264–293. MR 79232, DOI 10.1090/S0002-9947-1956-0079232-9

Necessary and sufficient conditions for the asymptotic stability of linear stochastic systems

12

E. J. McShane, Stochastic calculus and stochastic models, Probability and Mathematical Statistics, Vol. 25, Academic Press, New York-London, 1974. MR 0443084
A. Pazy, A Trotter-Lie formula for compactly generated semigroups of nonlinear operators, J. Functional Analysis 23 (1976), no. 4, 353–361. MR 0425706, DOI 10.1016/0022-1236(76)90061-6

Semi-groups of linear operators and applications to partial differential equations

Mark A. Pinsky, Multiplicative operator functionals and their asymptotic properties, Advances in probability and related topics, Vol. 3, Dekker, New York, 1974, pp. 1–100. MR 0368182
Mark A. Pinsky, Isolated singularities of degenerate elliptic equations in $R^{2}$, J. Differential Equations 24 (1977), no. 2, 274–281. MR 430517, DOI 10.1016/0022-0396(77)90151-6

On limit theorems for the product of random matrices

10

Joseph C. Watkins, A central limit problem in random evolutions, Ann. Probab. 12 (1984), no. 2, 480–513. MR 735850
Eugene Wong and Moshe Zakai, On the convergence of ordinary integrals to stochastic integrals, Ann. Math. Statist. 36 (1965), 1560–1564. MR 195142, DOI 10.1214/aoms/1177699916