A Vector-valued Random Ergodic Theorem

Laws of large numbers and Birkhoff’s ergodic theorem

In preparation for the next post on the central limit theorem, it's worth recalling the fundamental results on convergence of the average of a sequence of random variables: the law of large numbers (both weak and strong), and its strengthening to non-IID sequences, the Birkhoff ergodic theorem. 1 Convergence of random variables First we need to recall the different ways in which a sequence of r...

Finer filtration for matrix-valued cocycle based on Oseledec's multiplicative ergodic theorem

We consider a measurable matrix-valued cocycle A : Z+ × X → Rd×d, driven by a measurepreserving transformation T of a probability space (X,F , μ), with the integrability condition log ‖A(1, ·)‖ ∈ L1(μ). We show that for μ-a.e. x ∈ X, if limn→∞ 1 n log ‖A(n, x)v‖ = 0 for all v ∈ Rd \ {0}, then the trajectory {A(n, x)v}n=0 is far away from 0 (i.e. lim supn→∞ ‖A(n, x)v‖ > 0) and there is some nonz...

A Mean Ergodic Theorem For Asymptotically Quasi-Nonexpansive Affine Mappings in Banach Spaces Satisfying Opial's Condition

Individual ergodic theorem for intuitionistic fuzzy observables using intuitionistic fuzzy state

The classical ergodic theory hasbeen built on σ-algebras. Later the Individual ergodictheorem was studied on more general structures like MV-algebrasand quantum structures. The aim of this paper is to formulate theIndividual ergodic theorem for intuitionistic fuzzy observablesusing  m-almost everywhere convergence, where  m...

Discrete random electromagnetic Laplacians

We consider discrete random magnetic Laplacians in the plane and discrete random electromagnetic Laplacians in higher dimensions. The existence of these objects relies on a theorem of Feldman-Moore which was generalized by Lind to the nonabelian case. For example, it allows to realize ergodic Schrr odinger operators with stationary independent magnetic elds on discrete two dimensional lattices ...