Ergodic theorems for lower probabilities

We first establish a general version of the Birkhoff Ergodic Theorem for quasi-integrable extended real-valued random variables without assuming ergodicity. The key argument involves the Poincaré Recurrence Theorem. Our extension of the Birkhoff Ergodic Theorem is also shown to hold for asymptotic mean stationary sequences. This is formulated in terms of necessary and sufficient conditions. In particular, we examine the case where the probability space is endowed with a metric and we discuss the validity of the Birkhoff Ergodic Theorem for continuous random variables. The interest of our results is illustrated by an application to the convergence of statistical transforms, such as the moment generating function or the characteristic function, to their theoretical counterparts.

Duke Mathematical Journal, 1948

An instability property of the Birkhoff's ergodic theorem and related asymptotic laws with respect to small violations of algorithmic randomness is studied. The Shannon--McMillan--Breiman theorem and all universal compression schemes are also among them.

Ergodic Theory and Dynamical Systems, 2012

We introduce methods that allow us to derive continuous-time versions of various discrete-time ergodic theorems. We then illustrate these methods by giving simple proofs and refinements of some known results as well as establishing new results of interest.

arXiv: Probability, 2017

In this paper we obtain some possibilistic variants of the probabilistic laws of large numbers, different from those obtained by other authors, but very natural extensions of the corresponding ones in probability theory. Our results are based on the possibility measure and on the "maxitive" definitions for possibility expectation and possibility variance. Also, we show that in this frame, the weak form of the law of large numbers, implies the strong law of large numbers.

Israel Journal of Mathematics, 2005

We prove extensions of Menchoff's inequality and the Menchoff-Rademacher theorem for sequences {f n } ⊂ L p , based on the size of the norms of sums of sub-blocks of the first n functions. The results are applied to the study of a.e. convergence of series n a n T n g n α when T is an L 2-contraction, g ∈ L 2 , and {a n } is an appropriate sequence. Given a sequence {f n } ⊂ L p (Ω, µ), 1 < p ≤ 2, of independent centered random variables, we study conditions for the existence of a set of x of µ-probability 1, such that for every contraction T on L 2 (Y, π) and g ∈ L 2 (π), the random power series n f n (x)T n g converges π-a.e. The conditions are used to show that for {f n } centered i.i.d. with f 1 ∈ L log + L, there exists a set of x of full measure such that for every contraction T on L 2 (Y, π) and g ∈ L 2 (π), the random series n fn(x)T n g n converges π-a.e.

Proceedings of the American Mathematical Society

In this article we prove a strong law of large numbers for Borel measurable nonseparably valued random elements in the case of generalized random sets.

Fuzzy Sets and Systems, 1993

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2009

k=n Ak. If E occurs, then infinitely many of Ak:s occur. Sometimes we write this as E = {An i.o.} where i.o. is to be read as ”infinitely often”, i.e., infinitely many times. E is sometimes denoted with lim sup Ak. We need a couple of auxiliary results (the lemmas below) of probability calculus (found, e.g., on page 3 of [1]) that are basic in the sense that they are derived directly from the Kolmogorov axioms.

Journal of Theoretical Probability, 2011

Let {X n ; n ≥ 1} be a sequence of independent copies of a real-valued random variable X and set S n = X 1 + · · · + X n , n ≥ 1. This paper is devoted to a refinement of the classical Kolmogorov-Marcinkiewicz-Zygmund strong law of large numbers. We show that for 0 < p < 2,