A functional central limit theorem for empirical processes under a strong mixing condition

This paper introduces a functional central limit theorem for empirical processes endowed with real values from a strictly stationary random field that satisfies an interlaced mixing condition. We proceed by using a common technique from Billingsley (Convergence of probability measures, Wiley, New York, 1999), by first obtaining the limit theorem for the case where the random variables of the strictly stationary ???-mixing random field are uniformly distributed on the interval [0, 1]. We then generalize the result to the case where the absolutely continuous marginal distribution function is not longer uniform. In this case we show that the empirical process endowed with values from the ???-mixing stationary random field, due to the strong mixing condition, doesn??t converge in distribution to a Brownian bridge, but to a continuous Gaussian process with mean zero and the covariance given by the limit of the covariance of the empirical process. The argument for the general case holds similarly by the application of a standard variant of a result of Billingsley (1999) for the space D(???, ??).     

Random processes with common cotransition probabilities

The paper is devoted to the investigation of equivalent Markov diffusion processes. Analogues of the fundamental facts of the discrete theory are obtained.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 184, pp. 169–181, 1990.     

On averaging principle for diffusion processes with null-recurrent fast component

An averaging principle is proved for diffusion processes of type (X_ε(t),Y_ε(t)) with null-recurrent fast component X_ε(t). In contrast with positive recurrent setting, the slow component Y_ε(t) alone cannot be approximated by diffusion processes. However, one can approximate the pair (X_ε(t),Y_ε(t)) by a Markov diffusion with coefficients averaged in some sense.     

Random walks on graphs with a strong isoperimetric property

A random walk on a graph is a Markov chain whose state space consists of the vertices of the graph and where transitions are only allowed along the edges. We study (strongly) reversible random walks and characterize the class of graphs where then-step transition probabilities tend to zero exponentially fast (geometric ergodicity). These characterizations deal with an isoperimetric property, norm inequalities for certain associated operators, and eigenvalues of the Laplace operator. There is some (strong) similarity with the theory of (non)amenable groups.     

On the uniform strong Lusin condition

Assume that f is a function defined on some interval I ? ? ^m . Literature offers several equivalences of the type: f has a property P (like absolute continuity, bounded variation, etc.) on I if and only if f has P on each (closed) null subset of I. Such results feature a pretty important role in the integration theory. We make a brief review of these results and then provide an example showing that they can break down if considered in the uniform version, that is, for sequences of functions instead of a particular function.     

Nonparametric regression with warped wavelets and strong mixing processes

We consider the situation of a univariate nonparametric regression where either the Gaussian error or the predictor follows a stationary strong mixing stochastic process and the other term follows an independent and identically distributed sequence. Also, we estimate the regression function by expanding it in a wavelet basis and applying a hard threshold to the coefficients. Since the observations of the predictor are unequally distant from each other, we work with wavelets warped by the density of the predictor variable. This choice enables us to retain some theoretical and computational properties of wavelets. We propose a unique estimator and show that some of its properties are the same for both model specifications. Specifically, in both cases the coefficients are unbiased and their variances decay at the same rate. Also the risk of the estimator, measured by the mean integrated square error is almost minimax and its maxiset remains unaltered. Simulations and an application illustrate the similarities and differences of the proposed estimator in both situations.

     

Two strong limit theorems for processes with independent increments

Two related almost sure limit theorems are obtained in connection with a stochastic process {ξ(t), ?∞ < t < ∞} with independent increments. The first result deals with the existence of a simultaneous stabilizing function H(t) such that $\text{(ξ(t) ? ξ(0))}\text{H(t)}\text{→ 0}$ for almost all sample functions of the process. The second result deals with a wide-sense stationary process whose random spectral distributions is ξ. It addresses the question: Under what conditions does $\text{(2T)}{}^{\mathrm{?1}}\text{∫}{}_{\mathrm{?T}}{}^{\mathrm{T}}\text{X(t)}\text{X(t + τ)}\text{dt}$ converge as T → ∞ for all τ for almost all sample functions?     

Random processes in Sobolev-Orlicz spaces

We establish conditions under which the trajectories of random processes from Orlicz spaces of random variables belong with probability one to Sobolev-Orlicz functional spaces, in particular to the classical Sobolev spaces defined on the entire real axis. This enables us to estimate the rate of convergence of wavelet expansions of random processes from the spaces L _p (Ω) and L ₂ (Ω) in the norm of the space L _q (ℝ). __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 10, pp. 1340–1356, October, 2006.     

Random self-decomposability and autoregressive processes

We introduce the notion of random self-decomposability and discuss its relation to the concepts of self-decomposability and geometric infinite divisibility. We present its connection with time series autoregressive schemes with a regression coefficient that randomly turns on and off. In particular, we provide a characterization of random self-decomposability as well as that of marginal distributions of stationary time series that follow this scheme. Our results settle an open question related to the existence of such processes.     

Random graph approach to Gibbs processes with pair interaction

This study was motivated by the observation that, in a broad class of cases, the distribution of classical Gibbs point processes in R ^d governed by pair potential, can be obtained as the equilibrium distribution of a Markov chain of point processes in R ^d. Our analysis of this Markov chain is based on its imbedding in an infinite random graph. A condition of ergodicity of the chain is given in terms of the absence of percolation in the graph, and this can be checked in simpler cases. The embedding also suggests a stochastic construction for the equilibrium distribution in question.These constructions (which can also be of independent interest) are related to Gibbs processes by means of the results obtained in a recent paper of R. V. Ambartzumian and H. S. Sukiasian [1] where the existence of a new class of stationary point processes in R ^d was established which have density (correlation) functions of the form % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaqdbaWexLMBb50ujb% qegi0BVTgib5gDPfxDHbacfiGae8NKbmOae8hkaGIae8hEaG3aaSba% a4qaaiaabgdaaeqaa0Gaaiilaiaac6cacaGGUaGaaiOlaiaacYcaca% WG4bWaaSbaa4qaaiaad6gaaeqaa0GaaiykaiaabccacqGH9aqpcaqG% GaGaamOyamaaCaaaoeqabaGaamOBaaaanmaarababaGaamiAaiaacI% cacaWG4bWaaSbaa4qaaiaadMgaaeqaaaqaaiaad6gaaeqaniabg+Gi% vdGaeyOeI0IaamiEamaaBaaaoeaacaWGQbaabeaaniaacMcaaaa!56B6!\[f(x_{\text{1}} ,...,x_n ){\text{ }} = {\text{ }}b^n \prod\nolimits_n {h(x_i } - x_j )\] (here and below, % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaqdbaWaaebaaeaada% WgaaGdbaGaamOBaaqabaaabeqab0Gaey4dIunaaaa!38CA!\[\prod {_n } \] denotes a product taken over all two-subsets {i, j} {1,..., n}).     

Regime-switching diffusion processes: strong solutions and strong Feller property

Abstract

We investigate the existence and uniqueness of strong solutions for state-dependent regime-switching diffusion processes in an infinite state space with singular coefficients. Non-explosion conditions are given by using the Zvonkin’s transformation. The strong Feller property is proved by further assuming that the diffusion in each fixed environment generates a strong Feller semigroup, and our results can also be applied to irregular or degenerate situations.     

The strong approximation of extremal processes

Summary If X ₁, X ₂, ..., are i.i.d. random variables and Y _n =Max(X ₁, ..., X _n ); if for some sequences A _n , B_n, n=1, 2, ..., E _n (t)=A_nY_[nt]+B_n is such that E _n (1) weakly converges to a non degenerate limit distribution, then we prove that it is possible to construct a sequence of replicates of extremal processes E ⁽ⁿ⁾(t) on the same probability space, such that d(E _n (.), E ⁽ⁿ⁾(.))0 a.s., with the Levy metric. We give the rates of consistency of the approximations.