Birkhoff's Pointwise Ergodic Theorem Research Articles

On Stability of Probability Laws with Respect to Small Violations of Algorithmic Randomness

We study a stability property of probability laws with respect to small violations of algorithmic randomness. Some sufficient condition of stability is presented in terms of Schnorr tests of algorithmic randomness. Most probability laws, like the strong law of large numbers, the law of iterated logarithm, and even Birkhoff's pointwise ergodic theorem for ergodic transformations, are stable in this sense. Nevertheless, the phenomenon of instability occurs in ergodic theory. Firstly, the stability property of Birkhoff's ergodic theorem is non-uniform. Moreover, a computable non-ergodic measure-preserving transformation can be constructed such that the ergodic theorem is non-stable.

Every flat surface is Birkhoff and Oseledets generic in almost every direction

We prove that the Birkhoff pointwise ergodic theorem and the Oseledets multiplicative ergodic theorem hold for every flat surface in almost every direction. The proofs rely on the strong law of large numbers, and on recent rigidity results for the action of the upper triangular subgroup of $SL(2,\mathbb R)$ on the moduli space of flat surfaces. Most of the results also use a theorem about continuity of splittings of the Kontsevich-Zorich cocycle recently proved by S. Filip.

Statistical properties of controlled fluid flows with applications to control of mixing

In this paper, we study statistical properties of fluid flows that are actively controlled. Statistical properties such as Lagrangian and Eulerian time-averages are important flow quantities in fluid flows, particularly during mixing processes. Due to the assumption of incompressibility, the transformations in the state space can be described by a sequence of measure preserving transformations on a measure space. The classical Birkhoff's pointwise ergodic theorem does not necessarily apply in the context of sequences of transformations. We call B-regular a sequence for which this theorem holds. Motivated by mixing control concepts, we define three notions of asymptotic equivalence for sequences of transformations. We show an example in which Birkhoff's pointwise ergodic theorem does not hold even when a ‘strong’ asymptotic equivalence to a B-regular sequence is assumed. Under a ‘very strong’ asymptotic equivalence condition, we prove B-regularity. In the context of optimize-then-stabilize strategy for mixing control, we also prove that very strong asymptotic equivalence to a mixing sequence implies mixing. The mean ergodic theorem and the Poincare’ recurrence theorem are also proven for sequences of transformations under suitable asymptotic equivalence assumptions.

Discretization of Continuous Markov Chains and Markov Chain Monte Carlo Convergence Assessment

We show that continuous state-space Markov chains can be rigorously discretized into finite Markov chains. The idea is to subsample the continuous chain at renewal times related to small sets that control the discretization. Once a finite Markov chain is derived from the Markov chain Monte Carlo output, general convergence properties on finite state spaces can be exploited for convergence assessment in several directions. Our choice is based on a divergence criterion derived from Kemeny and Snell, which is first evaluated on parallel chains with a stopping time and then implemented, more efficiently, on two parallel chains only, using Birkhoff's pointwise ergodic theorem for stopping rules. The performance of this criterion is illustrated on three standard examples.

Rate of convergence for computing expectations of stopping functionals of an α-mixing process

Theshiftmethod consists in computing the expectation of an integrable functionalFdefined on the probability space ((ℝd)N,B(ℝd)⊗N, μ⊗N) (μ is a probability measure on ℝd) using Birkhoff's Pointwise Ergodic Theorem, i.e.asn→ ∞, where θ denotes the canonical shift operator. WhenFlies inL2(FT, μ⊗N) for some integrable enough stopping timeT, several weak (CLT) or strong (Gàl-Koksma Theorem or LIL) converging rates hold. The method successfully competes with Monte Carlo. The aim of this paper is to extend these results to more general probability distributionsPon ((ℝd)N,B(ℝd)⊗N), namely when the canonical process (Xn)n∊NisP-stationary, α-mixing and fulfils Ibragimov's assumptionfor some δ > 0. One application is the computation of the expectation of functionals of an α-mixing Markov Chain, under its stationary distributionPν. It may both provide a better accuracy and save the random number generator compared to the usual Monte Carlo or shift methods on independent innovations.

Rate of convergence for computing expectations of stopping functionals of an α-mixing process

The shift method consists in computing the expectation of an integrable functional F defined on the probability space ((ℝ d ) N , B(ℝ d )⊗N , μ⊗N ) (μ is a probability measure on ℝ d ) using Birkhoff's Pointwise Ergodic Theorem, i.e. as n → ∞, where θ denotes the canonical shift operator. When F lies in L 2( F T , μ⊗N ) for some integrable enough stopping time T, several weak (CLT) or strong (Gàl-Koksma Theorem or LIL) converging rates hold. The method successfully competes with Monte Carlo. The aim of this paper is to extend these results to more general probability distributions P on ((ℝ d ) N , B(ℝ d )⊗N ), namely when the canonical process (X n ) n∊N is P-stationary, α-mixing and fulfils Ibragimov's assumption for some δ > 0. One application is the computation of the expectation of functionals of an α-mixing Markov Chain, under its stationary distribution P ν. It may both provide a better accuracy and save the random number generator compared to the usual Monte Carlo or shift methods on independent innovations.

A generalization of Birkhoff's pointwise ergodic theorem

A generalization of Birkhoff's pointwise ergodic theorem

On the simulation of expectations of random variables depending on a stopping time

Birkhoff's pointwise ergodic theorem with the shift operator over [0,1]11yields a new practical method to compute expectations of functionals of stochastic process. Indeed converges to as Nconverges toward infinity, almost surely. By numerical simulations we will explain the efficiency of this method especially when compared to the classic Monte-Carlo one. It will furthermore be proven that under suitable assumptions a. central limit theorem holds. These assumptions are satisfied in most encountered practical problems. It will precisely be fulfilled when with a stopping time T having a moment of order p, p > 2.Moreover, under this assumptions a “weak” law of iterated logarithm applies. Such that: Numerical simulations were processed.