Ergodic Process Research Articles

Random telegraph processes with nonlocal memory.

We study two-state (dichotomous, telegraph) random ergodic continuous-time processes with dynamics depending on their past. We take into account the history of the process in an explicit form by introducing integral nonlocal memory term into conditional probability function. We start from an expression for the conditional transition probability function describing additive multistep binary random chain and show that the telegraph processes can be considered as continuous-time interpolations of discrete-time dichotomous random sequences. An equationinvolving the memory function and the two-point correlation function of the telegraph process is analytically obtained. This integral equationdefines the correlation properties of the processes with given memory functions. It also serves as a tool for solving the inverse problem, namely for generation of a telegraph process with a prescribed pair correlation function. We obtain analytically the correlation functions of the telegraph processes with two exactly solvable examples of memory functions and support these results by numerical simulations of the corresponding telegraph processes.

A local and hierarchical Koopman spectral analysis of fluid dynamics

AbstractA local and hierarchical Koopman spectral analysis is proposed to extend Koopman spectral analysis typically used in a linear system or an ergodic process to its application in general nonlinear dynamics. The continuous and analytic Koopman eigenfunctions and eigenvalues, derived from operator perturbation theory, are capable of dealing with a nonlinear transition process with mathematical rigorousness. A proliferation rule is identified to derive high‐order eigenvalues and eigenfunctions from lower‐order ones, thus various spectral patterns may be generated through recursive proliferations. The locally linear map around each state constructs base local Koopman eigenvalues and eigenfunctions from an algebraic eigenvalue problem, and high‐order ones are generated via the proliferation rule to express the systematic nonlinearity. The aforementioned hierarchy simplifies the Koopman spectral analysis and is verified by studying the development of Kármán vortex streets. The triangular chain and the lattice distribution of Koopman eigenvalues confirm the critical role of the proliferation rule and the hierarchy structure of Koopman eigenvalues. The local spectral analysis on the transition process shows that the periodic flow forms as the growth rates of the critical Koopman modes reduce to zero, and meanwhile, the Koopman modes at the same frequency superpose on each other to form the well‐known Fourier or Floquet modes, where the latter are the enhanced nonlinear motions due to the alignment of Koopman eigenvalues with the critical ones.

Ultimate boundedness and stability of highly nonlinear neutral stochastic delay differential equations with semi-Markovian switching signals

The ultimate boundedness and stability of highly nonlinear neutral stochastic delay differential equations (NSDDEs) are investigated in this paper. Different from many previous works, the highly nonlinear NSDDEs with semi-Markov switching signals are considered for the first time in this paper. Meanwhile, the time delay function in this paper is only required to meet much more relaxed restrictions, which can invalidate plentiful methods with requirements on its derivatives for the stability of NSDDEs. To overcome this difficulty, several novel techniques to tackle NSDDEs with such a delay are developed. Furthermore, a crucial property on the ergodic semi-Markov process is established, which plays a key role in the proof on the existence and boundedness of global solution. The generalized Khasminskii-type theorems are established for the existence and uniqueness of global solution by applying new methods and this property, and the criteria for boundedness and stability are also provided. In particular, feasible measures are provided to deal with the neutral term in our stability criteria. Finally, the effectiveness is verified by a numerical example.

Impact of Range Bar and Ergodic Process on Early Price Trend Detection Using Evidence From USD/CNY Currency

In this paper, range bars and the ergodic theory are combined to investigate early price movements for the USD/CNY currency pair in China from 2015 till 2019. The main findings suggested that early price trend detection may be accomplished within two standard deviations of the mean. During the duration of trading that was restricted to a narrow range, the sample revealed that at least 68 percent of the frequency mean result indicated that range bars solved price trend creation. This paper is subjected to additional analysis in order to acquire a correlation coefficient sample of at least 0.8. This sample revealed that ergodic theory prevented overpriced hedge trends. The goal of this paper is to improve the detection of price trends at an earlier stage than the methodology of trend tracking. Because of this, the authorities need to seriously consider putting early price trend detection models into place in order to boost the liquidity of the local currency.

Simulation of multivariate ergodic stochastic processes using adaptive spectral sampling and non-uniform fast Fourier transform

The simulation of multivariate ergodic stochastic processes is critical for structural dynamic analysis and reliability evaluation. Although the traditional spectral representation method (SRM) has a wide application in many areas, it is highly inefficient in simulating stochastic processes with many simulation points or long durations due to the significant computational cost associated with matrix factorizations concerning frequency. To address the encountered challenge, this paper presents an efficient approach for simulating ergodic stochastic processes with limited frequencies. Central to this approach is a fusion of the adaptive spectral sampling and the non-uniform fast Fourier transform (NUFFT) techniques. The adaptive spectral sampling of the envelope spectrum enables the determination of limited non-equispaced frequencies, which are randomly sampled according to a uniform distribution. Thus, the Cholesky decomposition is only required at limited specific frequencies, which dramatically reduces the computational cost of matrix factorizations. Since the randomly sampled frequencies are not equispaced, utilizing FFT to accelerate the summation of trigonometric functions becomes impractical. Then, the NUFFT that adapts the non-equispaced sampling points is employed instead to expedite this process with the non-uniform increment approximated through reduced interpolation. By taking the wind field simulation of a long-span suspension bridge as an example, a parametric analysis is conducted to investigate the effect of random frequencies on the simulation error of the developed approach and the convergence of spectra. Finally, the developed approach is further validated by focusing on the spectra and probabilistic density functions of the simulated wind samples, and the simulation performance is compared with that of the traditional approach. The analytical results demonstrate the efficiency and accuracy of the developed approach in simulating ergodic stochastic processes.

Monotonicity of the logarithmic energy for random matrices

It is well known that the semi-circle law, which is the limiting distribution in the Wigner theorem, is the minimizer of the logarithmic energy penalized by the second moment. A very similar fact holds for the Girko and Marchenko–Pastur theorems. In this work, we shed the light on an intriguing phenomenon suggesting that this functional is monotonic along the mean empirical spectral distribution in terms of the matrix dimension. This is reminiscent of the monotonicity of the Boltzmann entropy along the Boltzmann equation, the monotonicity of the free energy along ergodic Markov processes, and the Shannon monotonicity of entropy or free entropy along the classical or free central limit theorem. While we only verify this monotonicity phenomenon for the Gaussian unitary ensemble, the complex Ginibre ensemble, and the square Laguerre unitary ensemble, numerical simulations suggest that it is actually more universal. We obtain along the way explicit formulas of the logarithmic energy of the models which can be of independent interest.

Parameter inference for degenerate diffusion processes

We study parametric inference for ergodic diffusion processes with a degenerate diffusion matrix. Existing research focuses on a particular class of hypo-elliptic Stochastic Differential Equations (SDEs), with components split into ‘rough’/‘smooth’ and noise from rough components propagating directly onto smooth ones, but some critical model classes arising in applications have yet to be explored. We aim to cover this gap, thus analyse the highly degenerate class of SDEs, where components split into further sub-groups. Such models include e.g. the notable case of generalised Langevin equations. We propose a tailored time-discretisation scheme and provide asymptotic results supporting our scheme in the context of high-frequency, full observations. The proposed discretisation scheme is applicable in much more general data regimes and is shown to overcome biases via simulation studies also in the practical case when only a smooth component is observed. Joint consideration of our study for highly degenerate SDEs and existing research provides a general ‘recipe’ for the development of time-discretisation schemes to be used within statistical methods for general classes of hypo-elliptic SDEs.

Randomized limit theorems for stationary ergodic random processes and fields

Using the randomization approach, introduced by A. Tempelman in Randomized multivariate central limit theorems for ergodic homogeneous random fields, Stochastic Processes and their Applications. 143 (2022), 89–105, we prove: (a) a randomized version of the invariance principle (the functional CLT); (b) a version the Glivenko–Cantelli theorem; (c) a randomized theorem about convergence of empirical processes to the Brownian bridge. We also weaken the moment condition in the randomized CLTs, proved in the mentioned article. The main point of our work is that most of our theorems are valid for all ergodic homogeneous random fields on Zm and Rm,m≥1.

Research of stochastic stability of constructions parametric vibrations by the Monte-Carlo method

The stochastic parametric vibrations of the elastic systems behave to the statistical dynamics section of the nonlinear systems. Under stochastic parametric influence the vibrations of the elastic systems, which is a random process more frequent are not stabilized, but their amplitudes are fading or unlimited are growing. Therefore important is stability of stochastic vibrations, which can be examined as stability by probability, on average or by the moment functions of different orders. The mathematical models which describe the stochastic parametric vibrations of the elastic systems are building by analytical or numeral approaches. Researches of stochastic stability of parametric vibrations are executing by probabilistic methods. Stability of parametric vibrations of modern constructions under the action of operating random loads are not enough investigated. In the article the numeral method of research of dynamic stability of the elastic systems under stochastic parametric influence was presented. The mathematical model of stochastic parametric vibrations of construction in the form of a reduсed finite element model was formed. Matrixes of the reduced model were obtained using calculated procedures of finite element analysis software NASTRAN. The dynamic loading as stationary ergodic random process with the spectral density was presented in the form of a finite amount of harmonic functions. Stochastic stability of the constructions was investigated using the Monte Carlo method and the Runge Kutta method of direct numeral integration of the equations of reduсed model. Stochastic stability as stability by probability of appearance of tendency to fading or unlimited growth of amplitude of parametric vibrations was examined during the time interval in space of the random parameters of loading. Dynamic stability of a I-beam with corrugated wall under narrow-band parametric influence was investigated using the numeral method.

On the failure of Ornstein theory in the finitary category

We show the invalidity of finitary counterparts for three classification theorems: The preservation of being a Bernoulli shift through factors, Sinai’s factor theorem, and the weak Pinsker property. We construct a finitary factor of an i.i.d. process which is not finitarily isomorphic to an i.i.d. process, showing that being finitarily Bernoulli is not preserved through finitary factors. This refutes a conjecture of M. Smorodinsky [Finitary isomorphism of m m -dependent processes, Amer. Math. Soc., Birkhauser, Providence, RI, 1992, pp. 373–376], which was first suggested by D. Rudolph [A characterization of those processes finitarily isomorphic to a Bernoulli shift, Birkhäuser, Boston, Mass., 1981, pp. 1–64]. We further show that any ergodic system is isomorphic to a process none of whose finitary factors are i.i.d. processes, and in particular, there is no general finitary Sinai’s factor theorem for ergodic processes. Another consequence of this result is the invalidity of a finitary weak Pinsker property, answering a question of G. Pete and T. Austin [Math. Inst. Hautes Études Sci. 128 (2018), 1–119].

Regression estimation for continuous-time functional data processes with missing at random response

In this paper, we are interested in nonparametric kernel estimation of a generalised regression function based on an incomplete sample ( X t , Y t , ζ t ) t ∈ [ 0 , T ] copies of a continuous-time stationary and ergodic process ( X , Y , ζ ) . The predictor X is valued in some infinite-dimensional space, whereas the real-valued process Y is observed when the Bernoulli process ζ = 1 and missing whenever ζ = 0 . Uniform almost sure consistency rate as well as the evaluation of the conditional bias and asymptotic mean square error are established. The asymptotic distribution of the estimator is provided with a discussion on its use in building asymptotic confidence intervals. To illustrate the performance of the proposed estimator, a first simulation is performed to compare the efficiency of discrete-time and continuous-time estimators. A second simulation is conducted to discuss the selection of the optimal sampling mesh in the continuous-time case. Then, a third simulation is considered to build asymptotic confidence intervals. An application to financial time series is used to study the performance of the proposed estimator in terms of point and interval prediction of the IBM asset price log-returns. Finally, a second application is introduced to discuss the usage of the initial estimator to impute missing household-level peak electricity demand.

Efficient drift parameter estimation for ergodic solutions of backward SDEs

AbstractWe derive consistency and asymptotic normality results for quasi‐maximum likelihood methods for drift parameters of ergodic stochastic processes observed in discrete time in an underlying continuous‐time setting. The special feature of our analysis is that the stochastic integral part is unobserved and nonparametric. Additionally, the drift may depend on the (unknown and unobserved) stochastic integrand. Our results hold for ergodic semi‐parametric diffusions and backward SDEs. Simulation studies confirm that the methods proposed yield good convergence results.

Approximation with ergodic processes and testability

AbstractWe show that stationary time series can be uniformly approximated over all finite time intervals by mixing, non-ergodic, non-mean-ergodic, and periodic processes, and by codings of aperiodic processes. A corollary is that the ergodic hypothesis—that time averages will converge to their statistical counterparts—and several adjacent hypotheses are not testable in the non-parametric case. Further Baire category implications are also explored.

Стационарное распределение в системах массового обслуживания с переменной структурой

t. The construction of discrete ergodic Markov processes with continuous time is using to construct a mathematical model of a queuing system with a variable structure. This model is a combination of several models of queuing systems. The set of states of the combined system is a combination of sets of states of the combined systems, unlike classical queuing systems, in which the set of states of the system is a direct product of the sets of states of the combined systems. The transition intensities between the states of different systems are choosing so that the limit distribution of the combined system is a mixture of the limit distributions of the combined systems with different weights determined by the introduced transition intensities. As a result, the process of functioning of the combined system is obtaining by switching intensities corresponding to different combined systems at certain points in time.

Modeling of ergodic processes in dynamic graphs of computer networks

Summary: For a mathematical model of mobile wireless computer networks, the random process of alternating network connectivity and disconnection was studied using several mobility models: isotropic random walk with reflection from the boundaries of the network location area, partially directed random walk, random walk with elements of vortex motion. The probability distribution functions of the durations of periods of network connectivity and disconnection are found, the ergodicity of the process and the uncorrelated nature of the period durations are shown. A large positive asymmetry of the distributions was noted, indicating the presence of a relatively small number of periods of long duration in the process. The analysis was carried out using simulation methods. The results of the analysis are presented as dependency plots.

Rényi Entropy Rate of Stationary Ergodic Processes

Rényi Entropy Rate of Stationary Ergodic Processes

Controlling Uncertainty of Empirical First-Passage Times in the Small-Sample Regime.

We derive general bounds on the probability that the empirical first-passage time τ[over ¯]_{n}≡∑_{i=1}^{n}τ_{i}/n of a reversible ergodic Markov process inferred from a sample of n independent realizations deviates from the true mean first-passage time by more than any given amount in either direction. We construct nonasymptotic confidence intervals that hold in the elusive small-sample regime and thus fill the gap between asymptotic methods and the Bayesian approach that is known to be sensitive to prior belief and tends to underestimate uncertainty in the small-sample setting. We prove sharp bounds on extreme first-passage times that control uncertainty even in cases where the mean alone does not sufficiently characterize the statistics. Our concentration-of-measure-based results allow for model-free error control and reliable error estimation in kinetic inference, and are thus important for the analysis of experimental and simulation data in the presence of limited sampling.

Variance optimality in constrained and unconstrained stochastic differential games

The purpose of this paper is to extend the variance optimality criterion to the settings of constrained and unconstrained two-person stochastic differential games. We give conditions that ensure the existence of Nash equilibria with minimal variance. This criterion is a complement to that of the average long-run expected reward of an ergodic Markov process. Our main contribution is that we give sufficient conditions to define and ensure the existence of relaxed strategies that optimize the limiting variance of a constrained performance index of each player in the continuous-time framework. To the best of our knowledge, this theoretical task has been approached only for unconstrained Markov decision problems in a discrete-time context. The applications of our research range from predator–prey systems to actuarial paradigms and cell growth modeling in patients diagnosed with cancer.

Asymptotic behavior of a class of multiple time scales stochastic kinetic equations

We consider a class of stochastic kinetic equations, depending on two time scale separation parameters ɛ and δ: the evolution equation contains singular terms with respect to ɛ, and is driven by a fast ergodic process which evolves at the time scale t/δ2. We prove that when (ɛ,δ)→(0,0) the spatial density converges to the solution of a linear diffusion PDE. This result is a mixture of diffusion approximation in the PDE sense (with respect to the parameter ɛ) and of averaging in the probabilistic sense (with respect to the parameter δ). The proof employs stopping times arguments and a suitable new perturbed test functions approach which is adapted to consider the general regime ɛ≠δ.

Optical design algorithm utilizing continuous-discrete variables grounded on stochastic processes

This work proposes an optimization algorithm in optical design based on the concepts of ergodic and stochastic processes in statistical mechanics. In mixed-variable optimization problems, pseudo-random number and discrete-to-continuous variable conversion dramatically increase the speed at which the system solves for the optimal solution. Pseudo-random numbers are mainly applied in two important steps in the optimization algorithm: determining the combination of glasses involved and the order in which the successive glass parameters are replaced by real glasses. After two series of stochastic processes, the merit function value decreases rapidly along the steepest descent path, and thus the optical system approaches the optimal solution within a very short duration of time. By using the method proposed in this paper, a plan apochromatic objective with a long working distance was optimized, and finally, a high-quality optical system was obtained.