Uniform Ergodicity Research Articles

Ergodicity for a family of operators

The aim of this paper is to introduce the notions of power boundedness, Cesàro boundedness, mean ergodicity, and uniform ergodicity for a family of bounded linear operators on a Banach space. The authors present some elementary results in this setting and show that some main results about power bounded, Cesàro bounded, mean ergodic, and the uniform ergodic operator can be extended from the case of a linear bounded operator to the case of a family of bounded linear operators acting on a Banach space. Also, we show that the Yosida theorem can be extended from the case of a bounded linear operator to the case of a family of bounded linear operators acting on a Banach space.

Nonlinear Stochastic Operators and Associated Inhomogeneous Entangled Quantum Markov Chains

In the present paper, we introduce a class of F-stochastic operators on a finite-dimensional simplex, each of which is regular, ascertaining that the species distribution in the succeeding generation corresponds to the species distribution in the previous one in the long run. It is proposed a new scheme to define non-homogeneous Markov chains contingent on the F-stochastic operators and given initial data. By means of the uniform ergodicity of the non-homogeneous Markov chain, we define a non-homogeneous (quantum) entangled Markov chain. Furthermore, it is established that the non-homogeneous entangled Markov chain enables ψ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\psi$$\\end{document}-mixing property.

The small mass limit for long time statistics of a stochastic nonlinear damped wave equation

We study the long time statistics of a class of semi–linear damped wave equations with polynomial nonlinearities and perturbed by additive Gaussian noise in dimensions 2 and 3. We find that if sufficiently many directions in the phase space are stochastically forced, the system is exponentially attractive toward its unique invariant measure with a convergent rate that is uniform with respect to the mass. Then, in the small mass limit, we prove the convergence of the first marginal of the invariant measures in a suitable Wasserstein distance toward the unique invariant measure of a stochastic reaction–diffusion equation. Together with the uniform geometric ergodicity, we obtain the validity of the small mass limit for the solutions on the infinite time horizon [0,∞). This extends previously known results established for the damped wave equations under Lipschitz nonlinearities.

On the uniform ergodicity rate of a fractional Ehrenfest urn model

On the uniform ergodicity rate of a fractional Ehrenfest urn model

Hoeffding’s inequality for non-irreducible Markov models

In this article, we establish Hoeffding’s inequality for bounded Lipschitz functions of a class of not necessarily irreducible Markov models. The result complements the existing literature on this topic where Hoeffding’s inequality for bounded measurable functions of a class of irreducible Markov models has been considered. Our approach is based on the assumption of uniform ergodicity of the underlying Markov model in L1-Wasserstein space.

Joint ergodicity of piecewise monotone interval maps

For , let µ i be a Borel probability measure on which is equivalent to the Lebesgue measure λ and let be µ i -preserving ergodic transformations. We say that transformations are uniformly jointly ergodic with respect to if for any , limN−M→∞1N−M∑n=MN−1f0(T0 nx)⋅f1(T1 nx)⋯fk(Tk nx)=∏i=0k∫fidμi in L2(λ). We establish convenient criteria for uniform joint ergodicity and obtain numerous applications, most of which deal with interval maps. Here is a description of one such application. Let T G denote the Gauss map, , and, for β > 1, let T β denote the β-transformation defined by . Let T 0 be an ergodic interval exchange transformation. Let be distinct real numbers with and assume that for all . Then for any , limN−M→∞1N−M∑n=MN−1f0(T0nx)⋅f1(Tβ1nx)⋯fk(Tβknx)⋅fk+1(TGnx)=∫f0dλ⋅∏i=1k∫fidμβi⋅∫fk+1dμGin L2(λ). We also study the phenomenon of joint mixing. Among other things we establish joint mixing for skew tent maps and for restrictions of finite Blaschke products to the unit circle.

Concentration inequality for U-statistics of order two for uniformly ergodic Markov chains

We prove a new concentration inequality for U-statistics of order two for uniformly ergodic Markov chains. Working with bounded and π-canonical kernels, we show that we can recover the convergence rate of Arcones and Giné who proved a concentration result for U-statistics of independent random variables and canonical kernels. Our result allows for a dependence of the kernels hi,j with the indexes in the sums, which prevents the use of standard blocking tools. Our proof relies on an inductive analysis where we use martingale techniques, uniform ergodicity, Nummelin splitting and Bernstein’s type inequality. Assuming further that the Markov chain starts from its invariant distribution, we prove a Bernstein-type concentration inequality that provides sharper convergence rate for small variance terms.

Hold-out estimates of prediction models for Markov processes

We consider the selection of prediction models for Markovian time series. For this purpose, we study the theoretical properties of the hold-out method. In the econometrics literature, the hold-out method is called ‘out-of-sample’ and is the main method to select a suitable time series model. This method consists of estimating models on a learning set and picking up the model with minimal empirical error on a validation set of future observations. Hold-out estimates are well studied in the independent case, but, as far as we know, this is not the case when the validation set is not independent of the learning set. In this paper, assuming uniform ergodicity of the Markov chain, we state generalization bounds and oracle inequalities for such method; in particular, we show that the ‘out-of-sample’ selection method is adaptive to noise condition.

High-Performance Computing Analysis and Location Selection of Logistics Distribution Center Space Based on Whale Optimization Algorithm.

As a meta-heuristic algorithm based on swarm intelligence, the WOA algorithm has few control parameters and searches for the optimal solution by encircling the prey, searching for the prey, and attacking the bubble net. During the whole process, only two internal parameters A and C are utilized for the control of the exploration and development process. BWOA is simple to implement. In the process of algorithm execution, the initial population, global exploration, and local development stages have shortcomings. Therefore, it is necessary to optimize the WOA algorithm. Based on WOA, this study conducts a high-performance computing analysis and location selection of logistics distribution center space. It is concluded that: (1) by using the combination of direct logistics distribution and hierarchical logistics distribution, the WOA algorithm optimizes the cross selection strategy, the population fitness S-LO is improved, the quality of LA is guaranteed, and the chaotic S-LO mapping eliminates inferior individuals in the population. Direct distribution is carried out for bulky goods and important distribution customers, and hierarchical logistics distribution is used for customers in intensive logistics distribution destinations. (2) WOA uses the second reverse learning, chaotic mapping, and logistic chaotic mapping to improve the location update mode. The direct distribution method is mostly used for the logistics business with short journeys, fixed distribution points, and more goods delivered at one time, and logistics enterprises do not need to store and distribute goods. The uniform ergodicity of the Tent chaotic map and logistic chaotic map is improved. Ka adaptive inertia weights are a good complement to optimize the limitations of the Ao whale algorithm. (3) The inertia weight of the levy flight behavior can play a powerful role in balancing the global exploration ability and optimization performance of the intelligent algorithm. The long-term short-distance search of HED and the long-distance jump of KVAR are combined. Variant individuals undergo vector synthesis. It reduces the construction and operation costs of logistics sites and is suitable for logistics distribution under specific conditions.

Robustness to incorrect models and data-driven learning in average-cost optimal stochastic control

We study continuity and robustness properties of infinite-horizon average expected cost problems with respect to (controlled) transition kernels, and applications of these results to the problem of robustness of control policies designed for approximate models applied to actual systems. We show that sufficient conditions presented in the literature for discounted-cost problems are in general not sufficient to ensure robustness for average-cost problems. However, we show that the average optimal cost is continuous in the convergences of controlled transition kernel models where convergence of models entails (i) continuous weak convergence in state and actions, and (ii) continuous setwise convergence in the actions for every fixed state variable, in addition to either uniform ergodicity or some regularity conditions. We establish that the mismatch error due to the application of a control policy designed for an incorrectly estimated model to the true model decreases to zero as the incorrect model approaches the true model under the stated convergence criteria. Our findings significantly relax related studies in the literature which have primarily considered the more restrictive total variation convergence criteria. Applications to robustness to models estimated through empirical data (where almost sure weak convergence criterion typically holds, but stronger criteria do not) are studied and conditions for asymptotic robustness to data-driven learning are established.

Convergence of likelihood ratios and estimators for selection in nonneutral Wright–Fisher diffusions

AbstractA number of discrete time, finite population size models in genetics describing the dynamics of allele frequencies are known to converge (subject to suitable scaling) to a diffusion process termed the Wright–Fisher diffusion. This diffusion evolves on a bounded interval, such that many standard results in diffusion theory, assuming evolution on the real line, no longer apply. In this article we derive conditions to establish ‐uniform ergodicity for diffusions on bounded intervals, and use them to prove that the Wright–Fisher diffusion is uniformly in the selection and mutation parameters ergodic, and that the measures induced by the solution to the stochastic differential equation are uniformly locally asymptotically normal. We subsequently use these results to show that the maximum likelihood and Bayesian estimators for the selection parameter are uniformly over compact sets consistent, asymptotically normal, display moment convergence, and are asymptotically efficient for a suitable class of loss functions.

Convergence Rates in Uniform Ergodicity by Hitting Times and $$L^2$$-Exponential Convergence Rates

Generally, the convergence rate in \(L^2\)-exponential ergodicity \(\lambda \) is an upper bound for the convergence rate \(\kappa \) in uniform ergodicity for a Markov process, that is, \(\lambda \geqslant \kappa \). In this paper, we prove that \(\kappa \geqslant \inf \left\{ \lambda ,1/M_H\right\} \), where \(M_H\) is a uniform bound on the moment of the hitting time to a “compact” set H. In the case where \(M_H\) can be made arbitrarily small for H large enough. we obtain that \(\lambda =\kappa \). The general results are applied to Markov chains, diffusion processes and solutions to stochastic differential equations driven by symmetric stable processes.

Moderate deviation principles for bifurcating Markov chains: case of functions dependent of one variable

The main purpose of this article is to establish moderate deviation principles for additive functionals of bifurcating Markov chains. Bifurcating Markov chains are a class of processes which are indexed by a regular binary tree. They can be seen as the models which represent the evolution of a trait along a population where each individual has two offsprings. Unlike the previous results of Bitseki, Djellout & Guillin (2014), we consider here the case of functions which depend only on one variable. So, mainly inspired by the recent works of Bitseki & Delmas (2020) about the central limit theorem for general additive functionals of bifurcating Markov chains, we give here a moderate deviation principle for additive functionals of bifurcating Markov chains when the functions depend on one variable. This work is done under the uniform geometric ergodicity and the uniform ergodic property based on the second spectral gap assumptions. The proofs of our results are based on martingale decomposition recently developed by Bitseki & Delmas (2020) and on results of Dembo (1996), Djellout (2001) and Puhalski (1997).

Spectrums and Uniform Mean Ergodicity of Weighted Composition Operators on Fock Spaces

For holomorphic pairs of symbols (u, psi ), we study various structures of the weighted composition operator W_{(u,psi )} f= u cdot f(psi ) defined on the Fock spaces mathcal {F}_p. We have identified operators W_{(u,psi )} that have power-bounded and uniformly mean ergodic properties on the spaces. These properties are described in terms of easy to apply conditions relying on the values |u(0)| and |u(frac{b}{1-a})|, where a and b are coefficients from linear expansion of the symbol psi . The spectrum of the operators is also determined and applied further to prove results about uniform mean ergodicity.

Constructing Keyed Strong S-Box Using an Enhanced Quadratic Map

As the only nonlinear component for symmetric cryptography, S-Box plays an important role. An S-Box may be vulnerable because of the existence of fixed point, reverse fixed point or short iteration cycles. To construct a keyed strong S-Box, first, a 2D enhanced quadratic map (EQM) was constructed, and its dynamic behaviors were analyzed through phase diagram, Lyapunov exponent, Kolmogorov entropy, bifurcation diagram and randomness testing. The results demonstrated that the state points of EQM have uniform distribution, ergodicity and better randomness. Then a keyed strong S-Box construction algorithm was designed based on EQM, and the fixed point, reverse fixed point, and short cycles were eliminated. Experimental results verified the algorithm’s feasibility and effectiveness.

Averaging principle for stochastic differential equations in the random periodic regime

We present the validity of stochastic averaging principle for non-autonomous slow–fast stochastic differential equations (SDEs) whose fast motions admit random periodic solutions. Our investigation is motivated by some problems arising from multi-scale stochastic dynamical systems, where configurations are time dependent due to nonlinearity of the underlying vector fields and the onset of time dependent random invariant sets. Averaging principle with respect to uniform ergodicity of the fast motion is no longer available in this scenario. The ergodicity of time periodic measures of the fast motion on certain minimal Poincaré section is used to identify the averaging limit.

Regeneration-enriched Markov processes with application to Monte Carlo

We study a class of Markov processes that combine local dynamics, arising from a fixed Markov process, with regenerations arising at a state-dependent rate. We give conditions under which such processes possess a given target distribution as their invariant measures, thus making them amenable for use within Monte Carlo methodologies. Since the regeneration mechanism can compensate the choice of local dynamics, while retaining the same invariant distribution, great flexibility can be achieved in selecting local dynamics, and the mathematical analysis is simplified. We give straightforward conditions for the process to possess a central limit theorem, and additional conditions for uniform ergodicity and for a coupling from the past construction to hold, enabling exact sampling from the invariant distribution. We further consider and analyse a natural approximation of the process which may arise in the practical simulation of some classes of continuous-time dynamics.

Control problem for the impulse process under stochastic optimization procedure and Levy conditions

A stochastic approximation procedure and a limit generator of the original problem are constructed for a system of stochastic differential equations with Markov switching and impulse perturbation under Levy approximation conditions with control, which is determined by the condition for the extremum of the quality criterion function.The control problem using the stochastic optimization procedure is a generalization of the control problem with the stochastic approximation procedure, which was studied in previous works of the authors. This generalization is not simple and requires non-trivial approaches to solving the problem. In particular we discuss how the behavior of the boundary process depends on the prelimiting stochastic evolutionary system in the ergodic Markov environment. The main assumption is the condition for uniform ergodicity of the Markov switching process, that is, the existence of a stationary distribution for the switching process over large time intervals. This allows one to construct explicit algorithms for the analysis of the asymptotic behavior of a controlled process. An important property of the generator of the Markov switching process is that the space in which it is defined splits into the direct sum of its zero-subspace and a subspace of values, followed by the introduction of a projector that acts on the subspace of zeros.For the first time, a model of the control problem for the diffusion transfer process using the stochastic optimization procedure for control problem is proposed. A singular expansion in the small parameter of the generator of the three-component Markov process is obtained, and the problem of a singular perturbation with the representation of the limiting generator of this process is solved.

On Uniform Exponential Ergodicity of Markovian Multiclass Many-Server Queues in the Halfin–Whitt Regime

We study ergodic properties of Markovian multiclass many-server queues that are uniform over scheduling policies and the size of the system. The system is heavily loaded in the Halfin–Whitt regime, and the scheduling policies are work conserving and preemptive. We provide a unified approach via a Lyapunov function method that establishes Foster–Lyapunov equations for both the limiting diffusion and the prelimit diffusion-scaled queuing processes simultaneously. We first study the limiting controlled diffusion and show that if the spare capacity (safety staffing) parameter is positive, the diffusion is exponentially ergodic uniformly over all stationary Markov controls, and the invariant probability measures have uniform exponential tails. This result is sharp because when there is no abandonment and the spare capacity parameter is negative, the controlled diffusion is transient under any Markov control. In addition, we show that if all the abandonment rates are positive, the invariant probability measures have sub-Gaussian tails regardless whether the spare capacity parameter is positive or negative. Using these results, we proceed to establish the corresponding ergodic properties for the diffusion-scaled queuing processes. In addition to providing a simpler proof of previous results in Gamarnik and Stolyar [Gamarnik D, Stolyar AL (2012) Multiclass multiserver queueing system in the Halfin-Whitt heavy traffic regime: asymptotics of the stationary distribution. Queueing Systems 71(1–2):25–51], we extend these results to multiclass models with renewal arrival processes, albeit under the assumption that the mean residual life functions are bounded. For the Markovian model with Poisson arrivals, we obtain stronger results and show that the convergence to the stationary distribution is at an exponential rate uniformly over all work-conserving stationary Markov scheduling policies.

On the uniform ergodic for α−times integrated semigroups

Let $A$ be a generator of an $\alpha-$times integrated semigroup$(S(t))_{t\geq 0}$. We study the uniform ergodicity of $(S(t))_{t\geq 0}$ and we show that the range of $A$ is closed if and only if $\lambda R(\lambda,A)$ is uniformly ergodic.Moreover, we obtain that $(S(t))_{t\geq 0}$ is uniformly ergodic if and only if $\alpha=0$. Finally, we get that $\frac{1}{t^{\alpha+1}}\int_{0}^{t}S(s)ds$ converge uniformly for all $\alpha\geq 0$.