Ergodic Process Research Articles

Asymptotically efficient estimation of Ergodic rough fractional Ornstein-Uhlenbeck process under continuous observations

Asymptotically efficient estimation of Ergodic rough fractional Ornstein-Uhlenbeck process under continuous observations

Development of a method for calculating statistical characteristics of the input material flow of a transport conveyor

The object of this study is the material flow incoming the conveyor. The actual problem of calculating the stochastic characteristics of the input material flow of a transport system, based on the typification of the input material flow, is being solved. When constructing a model of the input material flow, methods of similarity theory were used. A criterion has developed for dividing the realization of the input material flow into a deterministic and stochastic component, which makes it possible to represent the stochastic component of the input flow in the form of an realization of a centered ergodic process. A method is presented for calculating amplitude and phase frequency spectra for the components of the input material flow, based on specified types of theoretical correlation functions. The calculating accuracy of the normalized correlation function values is e~0.05. Distinctive features of the obtained results are that the typification method of the input material flow is based on the use of the amplitude spectrum for the input material flow. A special feature of the results obtained is that a single realization of the input material flow was used to model the input material flow. The scope of application of the obtained results is the mining industry. The developed methodology for calculating the statistical characteristics of the input material flow allow to improve the accuracy of algorithms for optimal control of the flow parameters of the transport system for a mining enterprise. The condition for the practical application of the obtained results is the presence in the sections of the transport conveyor of measuring sensors that determine the speed of the belt and the amount of material in the bunker

Sequential Offloading for Distributed DNN Computation in Multiuser MEC Systems

This paper studies a sequential task offloading problem for a multiuser mobile edge computing (MEC) system. While most of the existing works consider static one-shot offloading optimization with fixed wireless channel conditions and fixed computational tasks, we consider a dynamic optimization approach, which embraces wireless channel fluctuations and random deep neural network (DNN) task arrivals over an infinite horizon. Specifically, we introduce a local CPU workload queue (WD-QSI) and an MEC server workload queue (MEC-QSI) to model the dynamic workload of DNN tasks at each WD and the MEC server, respectively. The transmit power and the partitioning of the local DNN task at each WD are dynamically determined based on the instantaneous channel conditions (to capture the transmission opportunities) and the instantaneous WD-QSI and MEC-QSI (to capture the dynamic urgency of the tasks) to minimize the average latency of the DNN tasks. The joint optimization can be formulated as an ergodic Markov decision process (MDP), in which the optimality condition is characterized by a centralized Bellman equation. However, the brute force solution of the MDP is not viable due to the curse of dimensionality as well as the requirement for knowledge of the global state information. To overcome these issues, we first decompose the MDP into multiple lower dimensional sub-MDPs, each of which can be associated with a WD or the MEC server. Next, we further develop a parametric online Q-learning algorithm, so that each sub-MDP is solved locally at its associated WD or the MEC server. The proposed solution is completely decentralized in the sense that the transmit power for sequential offloading and the DNN task partitioning can be determined based on the local channel state information (CSI) and the local WD-QSI at the WD only. Additionally, no prior knowledge of the distribution of the DNN task arrivals or the channel statistics will be needed for the MEC server. The proposed solution can achieve the superb performance over various state-of-the-art baselines.

Square-mean pseudo almost automorphic solutions of infinite class under the light of measure theory

The aim of this work is to present new concept of square-mean pseudo almost automorphic of infinite class using the measure theory. We use the (µ,ν)-ergodic process to define the spaces of (µ,ν)-pseudo almost automorphic processes of infinite class in the square-mean sense. We present many interesting results on those spaces like completeness and composition theorems and we study the existence and the uniqueness of the square-mean (µ,ν)-pseudo almost automorphic solutions of infinite class for of the stochastic evolution equation. We provide an example to illustrate ours results.

Anomalous diffusion, nonergodicity, non-Gaussianity, and aging of fractional Brownian motion with nonlinear clocks.

How do nonlinear clocks in time and/or space affect the fundamental properties of a stochastic process? Specifically, how precisely may ergodic processes such as fractional Brownian motion (FBM) acquire predictable nonergodic and aging features being subjected to such conditions? We address these questions in the current study. To describe different types of non-Brownian motion of particles-including power-law anomalous, ultraslow or logarithmic, as well as superfast or exponential diffusion-we here develop and analyze a generalized stochastic process of scaled-fractional Brownian motion (SFBM). The time- and space-SFBM processes are, respectively, constructed based on FBM running with nonlinear time and space clocks. The fundamental statistical characteristics such as non-Gaussianity of particle displacements, nonergodicity, as well as aging are quantified for time- and space-SFBM by selecting different clocks. The latter parametrize power-law anomalous, ultraslow, and superfast diffusion. The results of our computer simulations are fully consistent with the analytical predictions for several functional forms of clocks. We thoroughly examine the behaviors of the probability-density function, the mean-squared displacement, the time-averaged mean-squared displacement, as well as the aging factor. Our results are applicable for rationalizing the impact of nonlinear time and space properties superimposed onto the FBM-type dynamics. SFBM offers a general framework for a universal and more precise model-based description of anomalous, nonergodic, non-Gaussian, and aging diffusion in single-molecule-tracking observations.

Light impurities: equilibrium, transport and density profiles in tokamak and stellarator plasmas

The density profiles of light impurities observed in recent decades in a relatively wide variety of experiments on tokamaks and stellarators are analyzed and modeled based on the concept of self-consistent equilibrium and transport of impurity charge states (Shurygin 2020 Nuclear Fusion 60 046001). The emphasis is on the formation of charge-radial equilibrium and its invariant properties affecting transport and density profile. The impurity invariant equilibrium arises due to random compatible charge-radial variations, linking the impurity motion into a general ergodic (Markovian) process governed by a matrix of final probabilities. More than 60 profiles of helium, boron and carbon with central accumulation, flat and hollow were accurately reproduced by invariant dimensionless transport functions. It is shown that the leading processes determining the formation of impurity invariant density profiles in stationary plasma are the processes of ionization-recombination, and not the transport of particles. Successive variations in the hollow density profiles of helium, boron, and carbon observed in experiments can be described as successive invariant profile changes in the recombination rate profiles. The invariant particle diffusive-convective profiles can be calculated directly from the impurity density profiles. A general empirical formula for equilibrium constant of light impurities was proposed.

From graph theory and geometric probabilities to a representative width for three-dimensional detonation cells

We present a model for predicting a representative width for the three-dimensional irregular patterns observed on the front views of cellular detonation fronts in reactive gases. Its physical premise is that the cellular combustion process produces the same burnt mass per unit of time as the average planar steady Zel’dovich-von Neuman-Döring (ZND) process. The transverse waves of irregular cells are described as a stochastic system subject to a stationary ergodic process, considering that the distributions of the patterns should have identical temporal and statistical average properties. Graph theory then defines an ideal cell whose grouping is equivalent to the actual 3D cellular front, geometric probabilities determine the mean burned fraction that parameterizes the model, and ZND calculations close the problem with the time-position relationship of a fluid element in the ZND reaction zone. The model is limited to detonation reaction zones whose only ignition mechanism is adiabatic shock compression, such as those of the mixtures with H2, C3H8 or C2H4 as fuels considered in this work. The comparison of their measured and calculated widths shows an agreement better than or within the accepted experimental uncertainties, depending on the quality of the chemical kinetic scheme used for the ZND calculations. However, the comparison for CH4:O2 mixtures shows high overestimates, indirectly confirming that the detonation reaction zones in these mixtures certainly include other ignition mechanisms contributing to the combustion process, such as turbulent diffusion. In these situations, the cell widths on longitudinal soot recordings have a very large dispersion, so a mean width may thus not be a relevant detonation characteristic length. The model is easily implementable as a post-process of ZND profiles and provides rapid estimates of the cell width, length and reaction time.

Law of large numbers and central limit theorem for ergodic quantum processes

A discrete quantum process is represented by a sequence of quantum operations, which are completely positive maps that are not necessarily trace preserving. We consider quantum processes that are obtained by repeated iterations of a quantum operation with noise. Such ergodic quantum processes generalize independent quantum processes. An ergodic theorem describing convergence to equilibrium for a general class of such processes has been recently obtained by Movassagh and Schenker. Under irreducibility and mixing conditions, we obtain a central limit type theorem describing fluctuations around the ergodic limit.

Local asymptotic normality for ergodic jump-diffusion processes via transition density approximation

We show local asymptotic normality (LAN) for a statistical model of discretely observed ergodic jump-diffusion processes, where the drift coefficient, diffusion coefficient, and jump structure are parametrized. Under the LAN property, we can discuss the asymptotic efficiency of regular estimators, and the quasi-maximum-likelihood and Bayes-type estimators proposed in Shimizu and Yoshida (Stat. Inference Stoch. Process. 9 (2006) 227–277) and Ogihara and Yoshida (Stat. Inference Stoch. Process. 14 (2011) 189–229) are shown to be asymptotically efficient in this model. Moreover, we can construct asymptotically uniformly most powerful tests for the parameters. Unlike with a model for diffusion processes, Aronson-type estimates of the transition density functions do not hold, which makes it difficult to prove LAN. Therefore, instead of Aronson-type estimates, we employ the idea of Theorem 1 in Jeganathan (Sankhyā Ser. A 44 (1982) 173–212) and use the L2 regularity condition. Moreover, we show that local asymptotic mixed normality of a statistical model is implied from that for a model generated by approximated transition density functions under suitable conditions. Together with density approximation by means of thresholding techniques, the LAN property for the jump-diffusion processes is proved.

Neural Estimation and Optimization of Directed Information Over Continuous Spaces

This work develops a new method for estimating and optimizing the directed information rate between two jointly stationary and ergodic stochastic processes. Building upon recent advances in machine learning, we propose a recurrent neural network (RNN)-based estimator which is optimized via gradient ascent over the RNN parameters. The estimator does not require prior knowledge of the underlying joint/marginal distributions and can be easily optimized over continuous input processes realized by a deep generative model. We prove consistency of the proposed estimation and optimization methods and combine them to obtain end-to-end performance guarantees. Applications for channel capacity estimation of continuous channels with memory are explored, and empirical results demonstrating the scalability and accuracy of our method are provided. When the channel is memoryless, we investigate the mapping learned by the optimized input generator.

Large deviations for the Pearson family of ergodic diffusion processes involving a quadratic diffusion coefficient and a linear force

The Pearson family of ergodic diffusions with a quadratic diffusion coefficient and a linear force is characterized by explicit dynamics of their integer moments and by explicit relaxation of spectral properties towards their steady state. Besides the Ornstein–Uhlenbeck process with a Gaussian steady state, other representative examples of the Pearson family are the square root or the Cox–Ingersoll–Ross process converging towards the gamma distribution, the Jacobi process converging towards the beta distribution, the reciprocal gamma process (corresponding to an exponential functional of the Brownian motion) that converges towards the inverse-gamma distribution, the Fisher–Snedecor process and the Student process. The last three steady states display heavy tails. The goal of the present paper is to analyze the large deviation properties of these various diffusion processes in a unified framework. We first consider level 1 concerning time-averaged observables over a large time window T. We write the first rescaled cumulants for generic observables and identify specific observables whose large deviations can be explicitly computed from the dominant eigenvalue of the appropriate deformed generator. The explicit large deviations at level 2 concerning the time-averaged density are then used to analyze the statistical inference of model parameters from data on a very long stochastic trajectory in order to obtain the explicit rate function for the two inferred parameters of the Pearson linear force.

On the Kemeny time for continuous-time reversible and irreversible Markov processes with applications to stochastic resetting and to conditioning towards forever-survival

For continuous-time ergodic Markov processes, the Kemeny time is the characteristic time needed to converge towards the steady state : in real-space, the Kemeny time corresponds to the average of the mean-first-passage-time over the final configuration x drawn with the steady state , which turns out to be independent of the initial configuration x 0; in the spectral domain, the Kemeny time corresponds to the sum of the inverses of all the non-vanishing eigenvalues of the opposite generator. We describe many illustrative examples involving jumps and/or diffusion in one dimension, where the Kemeny time can be explicitly computed as a function of the system-size, via its real-space definition and/or via its spectral definition: we consider both reversible processes satisfying detailed-balance where the eigenvalues are real, and irreversible processes characterized by non-vanishing steady currents where the eigenvalues can be complex. In particular, we study the specific properties of the Kemeny times for Markov processes with stochastic resetting, and for absorbing Markov processes conditioned to survive forever.

Inferring the Mixing Properties of a Stationary Ergodic Process From a Single Sample-Path

We propose strongly consistent estimators of the ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> norm of the sequence of α-mixing (respectively β-mixing) coefficients of a stationary ergodic process. We further provide strongly consistent estimators of individual α-mixing (respectively β-mixing) coefficients for a subclass of stationary α-mixing (respectively β-mixing) processes with summable sequences of mixing coefficients. The estimators are in turn used to develop strongly consistent goodness-of-fit hypothesis tests. In particular, we develop hypothesis tests to determine whether, under the same summability assumption, the α-mixing (respectively β-mixing) coefficients of a process are upper bounded by a given rate function. Moreover, given a sample generated by a (not necessarily mixing) stationary ergodic process, we provide a consistent test to discern the null hypothesis that the ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> norm of the sequence α of α-mixing coefficients of the process is bounded by a given threshold γ ∈ [0,∞) from the alternative hypothesis that ∥α∥ > γ. An analogous goodness-of-fit test is proposed for the ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> norm of the sequence of β-mixing coefficients of a stationary ergodic process. Moreover, the procedure gives rise to an asymptotically consistent test for independence.

Best Arm Identification in Restless Markov Multi-Armed Bandits

We study the problem of identifying the best arm in a multi-armed bandit environment when each arm is a time-homogeneous and ergodic discrete-time Markov process on a common, finite state space. The state evolution on each arm is governed by the arm’s transition probability matrix (TPM). A decision entity that knows the set of arm TPMs but not the exact mapping of the TPMs to the arms, wishes to find the index of the best arm as quickly as possible, subject to an upper bound on the error probability. The decision entity selects one arm at a time sequentially, and all the unselected arms continue to undergo state evolution ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">restless</i> arms). For this problem, we derive the first-known problem instance-dependent asymptotic lower bound on the growth rate of the expected time required to find the index of the best arm, where the asymptotics is as the error probability vanishes. Further, we propose a sequential policy that, for an input parameter <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</i> , forcibly selects an arm that has not been selected for <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</i> consecutive time instants. We show that this policy achieves an upper bound that depends on <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</i> and is monotonically non-increasing as <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</i> → ∞. The question of whether, in general, the limiting value of the upper bound as <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</i> → ∞ matches with the lower bound, remains open. We identify a special case in which the upper and the lower bounds match. Prior works on best arm identification have dealt with (a) independent and identically distributed observations from the arms, and (b) rested Markov arms, whereas our work deals with the more difficult setting of restless Markov arms.

Stochastic Ordering of Stationary Distributions of Linear Recurrences: Further Results and Economic Applications

We investigate pairwise stochastic comparisons of stationary solutions to the linear recurrence Xt+1=AtXt+Bt, where At and Bt are non-negative random variables. We establish novel order-preserving properties, which enable us to obtain comparison theorems about well-known measures of conditional size, tail variability and skewness across probability distributions. While useful in studies of ergodic wealth accumulation processes and the persistence of inequality, our results can fruitfully be exploited to conduct comparative statics exercises in structural models entailing Kesten-type reduced-form representations. An application of our analysis to a dynamic asset accumulation model uncovers the qualitatively similar effects of capital income and earnings taxation on expected wealth concentration over higher quantiles as well as on conditional upper tail dispersion of wealth holdings, qualifying previous results that solely rely on the determination of Pareto exponents.

Robustness of iterated function systems of Lipschitz maps

AbstractLet $\{X_n\}_{n\in{\mathbb{N}}}$ be an ${\mathbb{X}}$ -valued iterated function system (IFS) of Lipschitz maps defined as $X_0 \in {\mathbb{X}}$ and for $n\geq 1$ , $X_n\;:\!=\;F(X_{n-1},\vartheta_n)$ , where $\{\vartheta_n\}_{n \ge 1}$ are independent and identically distributed random variables with common probability distribution $\mathfrak{p}$ , $F(\cdot,\cdot)$ is Lipschitz continuous in the first variable, and $X_0$ is independent of $\{\vartheta_n\}_{n \ge 1}$ . Under parametric perturbation of both F and $\mathfrak{p}$ , we are interested in the robustness of the V-geometrical ergodicity property of $\{X_n\}_{n\in{\mathbb{N}}}$ , of its invariant probability measure, and finally of the probability distribution of $X_n$ . Specifically, we propose a pattern of assumptions for studying such robustness properties for an IFS. This pattern is implemented for the autoregressive processes with autoregressive conditional heteroscedastic errors, and for IFS under roundoff error or under thresholding/truncation. Moreover, we provide a general set of assumptions covering the classical Feller-type hypotheses for an IFS to be a V-geometrical ergodic process. An accurate bound for the rate of convergence is also provided.

Non-ergodic fragmentation upon collision-induced activation of cysteine-water cluster cations.

Cysteine-water cluster cations Cys(H2O)3,6+ and Cys(H2O)3,6H+ are assembled in He droplets and probed by tandem mass spectrometry with collision-induced activation. Benchmark experimental data for this biologically important system are complemented with theory to elucidate the details of the collision-induced activation process. Experimental energy thresholds for successive release of water are compared to water dissociation energies from DFT calculations showing that clusters do not only fragment exclusively by sequential emission of single water molecules but also by the release of small water clusters. Release of clustered water is observed also in the ADMP (atom centered density matrix propagation) molecular dynamics model of small Cys(H2O)3+ and Cys(H2O)3H+ clusters. For large clusters Cys(H2O)6+ and Cys(H2O)6H+ the less computationally demanding statistical Microcanonical Metropolis Monte-Carlo method (M3C) is used to model the experimental fragmentation patterns. We are able to detail the energy redistribution in clusters upon collision activation. In the present case, about two thirds of the collision energy redistribute via an ergodic process, while the remaining one third is transferred into a non-ergodic channel leading to ejection of a single water molecule from the cluster. In contrast to molecular fragmentation, which can be well described by statistical models, modelling of collision-induced activation of weakly bound clusters requires inclusion of non-ergodic processes.

Ergodic Process of online Cultural Consumer Behaviors in China: A complement of AIINDAS model

Introduced the concept of “ergodic theory” and “cultural omnivore”, this article takes Masters In Forbidden City—a representative documentary film—as an example, analyses online cultural consumption in China and We-Media consumer behavior from cultural studies and frame analysis perspectives. By doing case study and qualitative research, this article first suggests that although derived from popular culture, online cultural consumers can not only be seen as extensions of popular culture consumers, nor unitary sub-culture groups. Youth sub-cultural groups were supposed to be differentiated and exclusive, but nowadays they are becoming increasingly “mainstreamed” due to the development of internet enterprises and cultural omnivores. Secondly, online cultural consumers have their own decision-making and information-processing system based on virtual community culture criteria. This system distinguishes internet cultural consumer behaviors from traditional offline ones. Therefore, the model of AIINDAS may be adjusted accordingly: “investigating” can be simplified into “filtrating”, “experience” can be complicated as “ergodicity”, and accurate information can be replaced by dynamic information.

Robust kernel regression function with uncertain scale parameter for high dimensional ergodic data using $ k $-nearest neighbor estimation

&lt;abstract&gt;&lt;p&gt;In this paper, we consider a new method dealing with the problem of estimating the scoring function $ \gamma_a $, with a constant $ a $, in functional space and an unknown scale parameter under a nonparametric robust regression model. Based on the $ k $ Nearest Neighbors ($ k $NN) method, the primary objective is to prove the asymptotic normality aspect in the case of a stationary ergodic process of this estimator. We begin by establishing the almost certain convergence of a conditional distribution estimator. Then, we derive the almost certain convergence (with rate) of the conditional median (scale parameter estimator) and the asymptotic normality of the robust regression function, even when the scale parameter is unknown. Finally, the simulation and real-world data results reveal the consistency and superiority of our theoretical analysis in which the performance of the $ k $NN estimator is comparable to that of the well-known kernel estimator, and it outperforms a nonparametric series (spline) estimator when there are irrelevant regressors.&lt;/p&gt;&lt;/abstract&gt;

Generalization of the Glivenko–Cantelli theorem to finite-dimensional distributions of ergodic homogeneous random fields

Let X(t),t∈Zm×Rk, be an ergodic stationary random processes or an ergodic homogeneous random field (k+m≥1). We prove that the distribution function of each random vector (X(t1),…,X(ts)) can be a.s. arbitrary fine uniformly approximated by the empirical distribution functions, if, e.g., in their construction increasing bounded convex sets Tn with infinitely increasing intrinsic diameters are used. We consider also the case when X is observed on a mixing homogeneous countable random set Sm(ω)⊂Rm (e.g., on a Poisson random set). These results open a way to consistent statistical inference on finite-dimensional distribution functions of ergodic processes and fields.