A perturbed ergodic theorem

Few, if any, of the properties enjoyed by ergodic measure-preserving transformations defined on a finite measure space generalize in a natural way to those defined on an infinite measure space. Concrete examples of ergodic transformations which preserve a finite measure and ones which preserve an infinite measure exist in the literature, see [1]. It is not difficult to see that ergodic transformations never admit wandering sets of positive measure. In [2 ] it was shown that a basic difference exists between ergodic transformations which preserve a finite measure and those which preserve an infinite measure; namely, an ergodic measure-preserving transformation defined on an infinite measure space always admits weakly wandering sets of positive measure (Theorem 2 of [2]). Unlike wandering sets, it is not true in general that the union of two weakly wandering sets is again a weakly wandering set even if we require that a class of mutually disjoint images of one weakly wandering set does not intersect a class of mutually disjoint images of the other. One may ask then if there are any ergodic measure-preserving transformations defined on an infinite measure space which admit only weakly wandering sets of finite measure. In this paper we show that this is not the case. We prove that there always exist weakly wandering sets of infinite measure for any ergodic measure-preserving transformation defined on an infinite measure space (Theorem 3). In [2] the existence of a weakly wandering set of positive measure for an ergodic measure-preserving transformation defined on an infinite measure space was discovered while studying the necessary and sufficient conditions for the existence of a finite, invariant, and equivalent measure for a given measurable and nonsingular transformation. In this paper we construct the weakly wandering sets in a different way. Using the pointwise ergodic theorem we prove a simple and yet a useful fact about ergodic measure-preserving transformations defined on an infinite measure space (Theorem 2). It states that given two sets A and B both of finite measure, it is possible to find an image of A under some power of the transformation T which has small intersection with the set B. This fact is basic in proving Lemma 1 which shows the existence of a

In the online prediction framework, we use generalized entropy of to study the loss rate of predictors when outcomes are drawn according to stationary ergodic distributions over the binary alphabet. We show that the notion of generalized entropy of a regular game \cite{KVV04} is well-defined for stationary ergodic distributions. In proving this, we obtain new game-theoretic proofs of some classical information theoretic inequalities. Using Birkhoff's ergodic theorem and convergence properties of conditional distributions, we prove that a classical Shannon-McMillan-Breiman theorem holds for a restricted class of regular games, when no computational constraints are imposed on the prediction strategies. If a game is mixable, then there is an optimal aggregating strategy which loses at most an additive constant when compared to any other lower semicomputable strategy. The loss incurred by this algorithm on an infinite sequence of outcomes is called its predictive complexity. We use our version of Shannon-McMillan-Breiman theorem to prove that when a restriced regular game has a predictive complexity, the predictive complexity converges to the generalized entropy of the game almost everywhere with respect to the stationary ergodic distribution.

Motivated by computing medians and means, two proximal splitting methods, backward-backward scheme introduced by Passty [Ergodic convergence to a zero of the sum of monotone operators in Hilbert spaces. J Math Anal Appl. 1979;72:383–390.] and barycentric method introduced by Lehdili and Lemaire [Metric spaces of non-positive curvature. Berlin: Springer; 1999.] are investigated in Hadamard space setting. A weak ergodic convergence theorem and some strong convergene results for both schemes are studied.

We initiate the study of effective pointwise ergodic theorems in resource-bounded settings. Classically, the convergence of the ergodic averages for integrable functions can be arbitrarily slow [Ulrich Krengel, 1978]. In contrast, we show that for a class of PSPACE L¹ functions, and a class of PSPACE computable measure-preserving ergodic transformations, the ergodic average exists and is equal to the space average on every EXP random. We establish a partial converse that PSPACE non-randomness can be characterized as non-convergence of ergodic averages. Further, we prove that there is a class of resource-bounded randoms, viz. SUBEXP-space randoms, on which the corresponding ergodic theorem has an exact converse - a point x is SUBEXP-space random if and only if the corresponding effective ergodic theorem holds for x.

Inertial primal-dual projection neurodynamic approaches for constrained convex optimization problems and application to sparse recovery.

We prove that return time sequences for dynamical systems which are abelian extensions of translations, are universaly good for the pointwise ergodic theorem. This can be used to prove the pointwise ergodic theorem along Morse sequence. This last result can also be proved by means of estimations of trigonometric sums.

Let C be a bounded closed convex subset of a uniformly convex multi-Banach space X and let $${\mathfrak {I}}_{j} = \{T_j(t) : t\in G\}$$ be a commutative semigroup of asymptotically nonexpansive in the intermediate mapping from C into itself. In this paper, we prove the strong mean ergodic convergence theorem for the almost-orbit of $$\mathfrak {I}$$ . Our results extend and unify many previously known results especially (Dong et al. On the strong ergodic theorem for commutative semigroup of non-Lipschitzian mappings in Banach space, preprint).

In this paper on the weighted pointwise ergodic theorem we bring together a substantial amount of previous work by a number of mathematicians and integrate it into a systematic consistent scheme; we also bring some original contributions to the subject which extend its boundaries and suggest further avenues of research. The paper is divided into six sections. The most significant new results are contained in $\S \S 5$ and $6$.

Let ( X , μ) be a probability measure space and T 1 , . . . , T n be a family of commuting, measure preserving invertible transformations on X . Let Q ( m 1 , . . . , m n ) be a homogeneous, positive polynomial with integer coefficients, and let S λ = { m ∈ Z n : Q ( m ) = λ} denote the set of integer solutions m = ( m 1 , . . . m n ) of the diophantine equation Q ( m ) = λ. We prove that under a certain nondegeneracy condition on the polynomial Q ( m ) and an ergodic condition on the family of transformations T = ( T 1 , . . . , T n ) the images of the solution sets: Ω x ,λ = {( T m1 1 T m2 2 … T mn n x ): m ∈ S λ} become uniformly distributed on X w.r.t. μ for a.e. x ∈ X as λ → ∞. That is the pointwise ergodic theorem holds when the standard averages are replaced by the ones, where the exponents satisfy a diophantine equation. The proof uses a variant of the Hardy-Littlewood method of exponential sums developed by Birch and Davenport and techniques from harmonic analysis. A key point is the corresponding maximal theorem, which is a discrete analogue of a maximal theorem on R n corresponding to the level surfaces of the polynomial Q ( x ).

At the core of any MANET simulation is a mobility model. To help ensure reliable simulation results, it is of interest to know if the mobility model is stable: Will time-averaged measurements of "mobility model events" converge? For example, does the time-averaged distance between a pair of nodes converge as simulation time increases? In this paper, we study the stability of a class of discrete Random Waypolnt Mobility Models (RWMMs). This class includes the classic Random Waypolnt Mobility Model. We show that each mobility model in this class satisfies a pointwise ergodic theorem (a generalized strong law of large numbers); thus, all bounded time-averaged measurements of mobility model events converge with probability one. A corollary of this ergodic theorem shows that each mobility model in this class also possesses a time-stationary regime.

We study the almost sure asymptotic behavior of the ergodic sums of $L^1$--functions, for the infinite measure preserving system given by the horocycle flow on the unit tangent bundle of a $\Z^d$--cover of a hyperbolic surface of finite area, equipped with the volume measure. We prove rational ergodicity, identify the return sequence, and describe the fluctuations of the ergodic sums normalized by the return sequence. One application is a 'second order ergodic theorem': almost sure convergence of properly normalized ergodic sums, subject to a certain summability method (the ordinary pointwise ergodic theorem fails for infinite measure preserving systems).

In this paper we introduce the notion of stochastic convergence of &#964- measurable operators and prove a noncommutative extension of pointwise ergodic theorem of G. D. Birkhoff by means of it by using the techniques developed by Petz in [12] Journal of the Nigerian Association of Mathematical Physics Vol. 9 2005: pp. 37-42

Adaptive estimated maximum-entropy distribution model

Soit T un operateur de Dunford-Schwartz sur un espace de probabilite (Ω, μ). Pour f ∈ L P (μ), p > 1, nous obtenons des theoremes ergodiques du type 1/n 1/p ∑ n k=1 T k f → 0 μ-p.s. sous des conditions portant sur la croissance de ∥∑ n k=1 T k f∥ρ. Lorsque T est induit par une transformation preservant la mesure et que p = 2, nous obtenons de meilleurs resultats. Ces derniers sont alors utilises pour obtenir le theoreme central limite quenched pour les sommes partielles associees aux fonctionnelles de chaines de Markov stationnaires et ergodiques. Nous ameliorons ainsi des resultats anterieurs de Derriennic-Lin et Wu-Woodroofe.

Under a general hypothesis an expanding map T of a Riemannian manifold M is known to preserve a measure equivalent to the Liouville measure on that manifold. As a consequence of this and Birkhoff’s pointwise ergodic theorem, the orbits of almost all points on the manifold are asymptotically distributed with regard to this Liouville measure. Let T be Lipschitz of class τ for some τ in (0,1], let Ω(x) denote the forward orbit closure of x and for a positive real number δ and let E(x0, δ) denote the set of points x in M such that the distance from x0 to Ω is at least δ. Let dim A denote the Hausdorff dimension of the set A. In this paper we prove a result which implies that there is a constant C(T) > 0 such that \(\dim E(x_0,\delta) \ge \dim M - \frac{C(T)}{\vert\!\log \delta \vert} \) if τ = 1 and \(\dim E(x_0,\delta) \ge \dim M - \frac{C(T)}{\log \vert \log \delta \vert}\) if τ < 1. This gives a quantitative converse to the above asymptotic distribution phenomenon. The result we prove is of sufficient generality that a similar result for expanding hyperbolic rational maps of degree not less than two follows as a special case.