Products of Random Matrices

Furstenberg, Harry; Kesten, Harry

doi:10.1214/aoms/1177705909

Cited by 931 publications

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“…(1. 90) In addition, we also define for β ∈ R ∪ {∞} H 0 +,n0 = H +,n0+1 , H β +,n0 = H +,n0 − a(n 0 )β −1 δ n0+1 , . δ n0+1 , β = 0, H ∞ −,n0 = H −,n0 , H β −,n0 = H −,n0+1 − a(n 0 )β δ n0 , .…”

Section: Jacobi Operatorsmentioning

confidence: 99%

Jacobi Operators and Completely Integrable Nonlinear Lattices

Teschl

1999

Mathematical Surveys and Monographs

323

349

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Abstract. This book is intended to serve both as an introduction and a reference to spectral and inverse spectral theory of Jacobi operators (i.e., second order symmetric difference operators) and applications of these theories to the Toda and Kac-van Moerbeke hierarchy.Starting from second order difference equations we move on to self-adjoint operators and develop discrete Weyl-Titchmarsh-Kodaira theory, covering all classical aspects like Weyl m-functions, spectral functions, the moment problem, inverse spectral theory, and uniqueness results. Next, we investigate some more advanced topics like locating the essential, absolutely continuous, and discrete spectrum, subordinacy, oscillation theory, trace formulas, random operators, almost periodic operators, (quasi-)periodic operators, scattering theory, and spectral deformations.Then, the Lax approach is used to introduce the Toda hierarchy and its modified counterpart, the Kac-van Moerbeke hierarchy. Uniqueness and existence theorems for the initial value problem, solutions in terms of Riemann theta functions, the inverse scattering transform, Bäcklund transformations, and soliton solutions are discussed.Corrections and complements to this book are available from:

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Section: Jacobi Operatorsmentioning

confidence: 99%

Jacobi Operators and Completely Integrable Nonlinear Lattices

Teschl

1999

Mathematical Surveys and Monographs

323

349

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“…1 Products of random matrices and their asymptotic behavior were originally studied by Bellman (1954). One of the decisive steps was made by Furstenberg and Kesten (1960), who investigated a matrix-valued stationary stochastic process X 1 , ... , X n , ... , and proved that the limit of n −1 E (log X 1 ...X n ) exists (but might equal ±∞) and that under certain assumptions n −1 log X 1 ...X n converges to this limit almost surely. Essentially, the only facts that are used in the proof of this result are the ergodic theorem, the norm inequality X 1 X 2 ≤ X 1 X 2 and the fact that the unit sphere is compact in finite-dimensional spaces.…”

Section: Introductionmentioning

confidence: 99%

The norm of products of free random variables

Kargin

2006

Probab. Theory Relat. Fields

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Let Xi denote free identically-distributed random variables. This paper investigates how the norm of products Πn = X1X2...Xn behaves as n approaches infinity. In addition, for positive Xi it studies the asymptotic behavior of the norm of Yn = X1 • X2 • ...• Xn, where • denotes the symmetric product of two positive operators:It is proved that if the expectation of Xi is 1, then the norm of the symmetric product Yn is between c1n 1/2 and c2n for certain constant c1 and c2. That is, the growth in the norm is at most linear.For the norm of the usual product P in, it is proved that the limit of n −1 log N orm(P in) exists and equals log p E (X * i Xi). In other words, the growth in the norm of the product is exponential and the rate equals the logarithm of the Hilbert-Schmidt norm of operator X.Finally, if π is a cyclic representation of the algebra generated by Xi, and if ξ is a cyclic vector, then n −1 log N orm(π (Πn) ξ) = log p E (X * i Xi) for all n. In other words, the growth in the length of the cyclic vector is exponential and the rate coincides with the rate in the growth of the norm of the product.These results are significantly different from analogous results for commuting random variables and generalize results for random matrices derived by Kesten and Furstenberg. IntroductionSuppose X 1 , X 2 , ... , X n are identically-distributed free random variables. These variables are infinite-dimensional linear operators but the reader may find it convenient to think of them as very large random matrices. The first question we will address in this paper is how the norm of Π n = X 1 X 2 ...X n behaves. If X i are all positive, then it is natural to look also at the symmetric product operation • defined as follows:The benefit is that unlike the usual operator product, this operation maps the set of positive variables to itself. For this operation we can ask how the norm of symmetric products Y n = X 1 • X 2 • ... • X n behaves. 1 * Courant Institute of Mathematical Sciences; 109-20 71st Road, Apt. 4A, Forest Hills NY 11375; kargin@cims.nyu.edu1 The operation • is neither commutative, nor associative. By convention we multiply starting on the right, so, for example, X 1 • X 2 • X 3 • X 4 = X 1 • (X 2 • (X 3 • X 4 )) . However,

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“…We need to consider how such products behave on average. Such products have been studied for several decades (Furstenberg and Kesten 1960) and have found applications in physics, biology, and economics. Based on our discussion of criticality in Sect.…”

Section: Stochastic Stabilitymentioning

confidence: 99%

Stochastic stability of particle swarm optimisation

2017

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Particle swarm optimisation (PSO) is a metaheuristic algorithm used to find good solutions in a wide range of optimisation problems. The success of metaheuristic approaches is often dependent on the tuning of the control parameters. As the algorithm includes stochastic elements that effect the behaviour of the system, it may be studied using the framework of random dynamical systems (RDS). In PSO, the swarm dynamics are quasi-linear, which enables an analytical treatment of their stability. Our analysis shows that the region of stability extends beyond those predicted by earlier approximate approaches. Simulations provide empirical backing for our analysis and show that the best performance is achieved in the asymptotic case where the parameters are selected near the margin of instability predicted by the RDS approach.Keywords Particle swarm optimisation · Criticality · Random dynamical systems · Random matrix products · Parameter selection Particle swarm optimisationParticle swarm optimisation (PSO) (Kennedy and Eberhart 1995) is a metaheuristic algorithm which is widely used in search and optimisation tasks. It aims at locating solutions to problems that may be characterised by high dimensionality, heterogeneity, the presence of many suboptimal solutions, and the absence of gradient information. An optimal solution is a global minimum of a given cost function (or, depending on problem, a global maximum) the domain of which is explored by a swarm of particles. In many problems, where PSO is applied, also solutions with near-optimal costs can be considered as good.The number of particles N is quite low in most applications, usually amounting to a few dozens. Each particle represents a potential solution and shares knowledge about the currently B J. Michael Herrmann

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Products of Random Matrices

Cited by 931 publications

References 0 publications

Jacobi Operators and Completely Integrable Nonlinear Lattices

Jacobi Operators and Completely Integrable Nonlinear Lattices

The norm of products of free random variables

Stochastic stability of particle swarm optimisation

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