Strong Laws of Large Numbers for Products of Random Matrices

Abstract

ABSTRACT. This work, on products of random matrices, is inspired by papers of Furstenberg and Resten (Ann. Math. Statist. 31 (1960), 457-469) and Furstenberg (Trans. Amer. Math. Soc. 108 (1963), 377-428). In particular, a formula was known for almost sure limits for normalized products of random matrices in terms of a stationary measure. However, no explicit computational techniques were known for these limits, and little was known about the stationary measures.We prove two main theorems. The first assumes that the random matrices are upper triangular and computes the almost sure limits in question. For the second, we assume the random matrices are 2x2 and Bernoulli, i.e., random matrices whose support is two points.Then the second theorem gives an asymptotic result for the almost sure limits, with rates of convergence in some cases.

Lyapunov Exponents for the Random Product of Two Shears

“…In figure 1a the bounds are all so close to the true value of λ that the details of the graph are difficult to resolve. It is clear however, that the standard bound given by (5) (plotted in cyan) is a worse upper bound than all others in the figure, despite being calculated from the expected value of matrix products of length 2 12 , and decreases in accuracy for this fixed k for increasing α.…”

“…In [10] the bound is described as "easy, if not efficient", since the number of matrix product calculations required increases exponentially with k. Further progress in this direction has tended to be either for specific simple cases, or algorithmic procedures leading to (sometimes very accurate) approximations. For example, [11] and [12] discuss cases where the Lyapunov exponent can be computed exactly, in particular when matrices can be grouped in commuting blocks. Chassain et al [13] establish the distribution for the matrix product, in terms of a continued fraction, in the case that the matrices are 2 × 2 shear matrices, but observe that even for these simple matrices, the Lyapunov exponent is still unobtainable.…”

A Matrix-Based Regularity Measure for Symbolic Sequences

“…They take the projection (ax + b)/(cx + d) of the matrix product, a b c d , and link the distribution to continued fractions on [0, 1], which follows a Denjoy-Minkowski measure. Other results in the general 2 by 2 case include [55] which computes (analytically) an exact Lyapunov exponent for a simple 2 by 2 matrix pair associated with a random Fibonacci sequence; [2] which numerically computes Lyapunov exponents for several simple 2 by 2 matrix pairs; [30] which looks at properties of products of 2-dimensional matrices in context of random continued fractions of a certain type; as well as [4,41,28], all of which consider Lyapunov exponents primarily or in part. The literature indicates the difficulties of analytical methods even in the simple 2 by 2 case, and for results on the distribution of the M i products in the general case, we'll turn to classical probability analogs in random matrix theory.…”

Strong law of large numbers on graphs and groups

“…In 1963, Furstenberg solved this problem for normalized products of random matrices in terms of stationary measures ( [13]). Computational techniques that would allow to compute these measures are investigated in [28]. A concise account of a number of illuminating results in the direction of the generalization of the SLLN to groups can be found in [20], where the authors prove the theorem about the directional distribution of the product g 1 g 2 .…”

“…It is not hard to see that if we find C 1 and C 2 satisfying ( 27) and ( 28) respectively, then (26) holds for C = C 1 + C 2 and the theorem is proved. First we argue (28). Choose any v 0 ∈ V (Γ) such that µ(v 0 ) > 0 and r ∈ N such that the inequality (25) holds.…”