Search

We construct a stationary ergodic process $X_1, X_2, \ldots $ such that each $X_t$ has the uniform distribution on the unit square and the length $L_n$ of the shortest path through the points $X_1, X_2, \ldots,X_n$ is not asymptotic to a constant times the square root of $n$. In other words, we show that the Beardwood, Halton and Hammersley theorem does not extend from the case of independent uniformly distributed random variables to the case of stationary ergodic sequences with uniform marginal distributions., 24 pages, 1 figure

In this work a method for statistical analysis of time series is proposed, which is used to obtain solutions to some classical problems of mathematical statistics under the only assumption that the process generating the data is stationary ergodic. Namely, three problems are considered: goodness-of-fit (or identity) testing, process classification, and the change point problem. For each of the problems a test is constructed that is asymptotically accurate for the case when the data is generated by stationary ergodic processes. The tests are based on empirical estimates of distributional distance., Conference version in: D. Ryabko, B. Ryabko, On hypotheses testing for ergodic processes, in Proceedgings of Information Theory Workshop, 2008, Porto, Portugal, pp. 281-283

The class of generalized reflectionless Schrodinger potentials was introduced by Marchenko-Lundina and was analyzed by Kotani. We state and prove various results concerning those stationary ergodic processes of Schrodinger potentials which are contained in this class.

Two series of binary observations x 1,x 1,… and y 1,y 2,… are presented: x n and y n are given at each time n∈ℕ. It is assumed that the sequences are generated independently of each other by two B-processes. The question of interest is whether the sequences represent a typical realization of two different processes or of the same one. It is demonstrated that this is impossible to decide, in the sense that every discrimination procedure is bound to err with non-negligible frequency when presented with sequences from some B-processes. This contrasts with earlier positive results on B-processes, in particular, those showing that there are consistent $\bar{d}$ -distance estimates for this class of processes, and on ergodic processes, in particular, those establishing consistent change point estimates.

Two series of binary observations x 1 ; x 1 ,… and y 1 , y 2 ,… are presented: at each time n ∈ ℕ we are given x n and y n . It is assumed that the sequences are generated independently of each other by two stochastic processes. We are interested in the question of whether the sequences represent a typical realization of two different processes or of the same one. We demonstrate that this is impossible to decide in the case when the processes are B-processes. It follows that discrimination is impossible for the set of all (finite-valued) stationary ergodic processes in general. This result means that every discrimination procedure is bound to err with non-negligible frequency when presented with sequences from some of such processes. It contrasts earlier positive results on B-processes, in particular those showing that there are consistent d-distance estimates for this class of processes.

We introduce a framework for the study of nonlinear homogenization problems in the setting of stationary continuous processes in compact spaces. The latter are functions f ○ T : R n × Q → Q with f ○ T ( x , ω ) = f ( T ( x ) ω ) where Q is a compact (Hausdorff topological) space, f ∈ C ( Q ) and T ( x ) : Q → Q , x ∈ R n , is an n-dimensional continuous dynamical system endowed with an invariant Radon probability measure μ. It can be easily shown that for almost all ω ∈ Q the realization f ( T ( x ) ω ) belongs to an algebra with mean value, that is, an algebra of functions in BUC ( R n ) containing all translates of its elements and such that each of its elements possesses a mean value. This notion was introduced by Zhikov and Krivenko [V.V. Zhikov, E.V. Krivenko, Homogenization of singularly perturbed elliptic operators, Mat. Zametki 33 (1983) 571–582, English transl. in Math. Notes 33 (1983) 294–300]. We then establish the existence of multiscale Young measures in the setting of algebras with mean value, where the compactifications of R n provided by such algebras plays an important role. These parametrized measures are useful in connection with the existence of correctors in homogenization problems. We apply this framework to the homogenization of a porous medium type equation in R n with a stationary continuous process as a stiff oscillatory external source. This application seems to be new even in the classical context of periodic homogenization.

Some further applications of a General Rate Conservation Law (GRCL) for stationary semimartingales are presented. GRCL is used to characterize the distribution of the reflection of Levy processes with arbitrary jumps. The stationary distribution of a general jump-diffusion with a reflecting boundary is derived and an interesting stochastic decomposition result is obtained. A multidimensional version of GRCL is used to examine the stationary behaviour of a multidimensional reflected Levy process. The relationship between GRCL and pathwise rate conservation is reviewed.

The main result in this paper is an invariance principle for the law of the iterated logarithm for square integrable martingales subject to fairly mild regularity conditions on the increments. When specialized to the case of identically distributed increments the result contains that of Stout [16] as well as the invariance principle for independent random variables of Strassen [17]. The martingale result is also used to obtain an invariance principle for the iterated logarithm law for a wide class of stationary ergodic sequences and a corollary is given which extends recent results of Oodaira and Yoshihara [10] on $\phi$-mixing processes.