Products of Random Matrices, Non-Commutative Branching Random Walks and Multitype Branching Random Walks in Random Environment
Final Report Abstract
Consider a random variable A, taking values in the set M (d × d, R) of d × d-matrices with real valued entries. Given a sequence of independent, identically distributed (i.i.d.) copies A1 , A2 , . . . of A, we define the product of random matrices Πn := An · · · A1 . It arises as a fundamental object in various models and is of importance in its own right, for it can be seen as the archetypical model of a multiplicative random walk on a noncommutative (semi-)group. Although products of random matrices have been studied for a long time, starting with a strong law of large numbers for log ∥Πn ∥ by Furstenberg and Kesten (1960), this is a very active field of research. The aim of this project was twofold, the fundamental concept being mutual enrichment of basic research and applications. Firstly, we have used our experience with products of random matrices to study models from applied probability; in particular models with a branching mechanism, where the study of extremal particles has attracted a great deal of attention in the last few years. We have considered a branching process with particles moving in Rd , where the displacement of new particles is given by the action of random matrices on their parent’s position. We have studied the minimal and maximal distance of particles to the origin, and established convergence results in particular for the corresponding so-called derivative martingale; which provides the third-order term in the asymptotics of positions of extremal particles. Secondly, studying these and further models like multivariate financial time series or stochastic gradient descent in deep learning gave rise to challenging new problems in the theory of products of random matrices. In particular, we have proved refined limit theorems like precise large deviations and Berry-Esseen estimates for the entries Πi,jn. For products of positive matrices, we were able to obtain an approximate duality result that allows for the first time to study renewal theory for centered processes (where 1/n log ∥Πn ∥ → 0 a.s.).
Publications
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Berry-Esseen bounds and moderate deviations for the norm, entries and spectral radius of products of positive random matrices. Science China Mathematics, 67(3), 627-646.
Xiao, Hui; Grama, Ion & Liu, Quansheng
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Conditioned limit theorems for hyperbolic dynamical systems. Ergodic Theory and Dynamical Systems, 44(1), 50-117.
GRAMA, ION; QUINT, JEAN-FRANÇOIS & XIAO, HUI
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Large deviation expansions for the coefficients of random walks on the general linear group. The Annals of Probability, 51(4).
Xiao, Hui; Grama, Ion & Liu, Quansheng
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Limit theorems for first passage times of multivariate perpetuity sequences
Mentemeier S. & Xiao H.
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Analysing heavy-tail properties of Stochastic Gradient Descent by means of Stochastic Recurrence Equations.
Damek, E. & Mentemeier, S.
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The extremal position of a branching random walk on the general linear group. Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, 61(4).
Grama, Ion; Mentemeier, Sebastian & Xiao, Hui
