It is not known if every finitary factor of a Bernoulli scheme is finitarily isomorphic to a Bernoulli scheme (is finitarily Bernoulli). In this paper, for any Bernoulli scheme X, we define a metric on the finitary factor maps from X. We show that for any finitary map f: X → Y, there exists a sequence of finitary maps fₙ: X → Y(n) that converges to f, where each Y(n) is finitarily Bernoulli. Thus, the maps to finitarily Bernoulli factors are dense. Let (X(n)) be a sequence of Bernoulli schemes such...

We introduce a fractional Langevin equation with α-stable noise and show that its solution $Y_{\kappa }\left(t\right),t\ge 0$ is the stationary α-stable Ornstein-Uhlenbeck-type process recently studied by Taqqu and Wolpert. We examine the asymptotic dependence structure of $Y_{\kappa }\left(t\right)$ via the measure of its codependence r(θ₁,θ₂,t). We prove that $Y_{\kappa }\left(t\right)$ is not a long-memory process in the sense of r(θ₁,θ₂,t). However, we find two natural continuous-time analogues of fractional ARIMA time series with long memory in the framework of the Langevin...

Fine regularity of stochastic processes is usually measured in a local way by local Hölder exponents and in a global way by fractal dimensions. In the case of multiparameter Gaussian random fields, Adler proved that these two concepts are connected under the assumption of increment stationarity property. The aim of this paper is to consider the case of Gaussian fields without any stationarity condition. More precisely, we prove that almost surely the Hausdorff dimensions of the range and the graph...

A problem of heredity of mixing properties (α-mixing, β-mixing and ρ-mixing) from a stationary point process on ℝ × ℝ₊ to a sequence of some of its points called 'seeds' is considered. Next, using the mixing properties, several versions of functional central limit theorems for the distances between seeds and the process of the number of seeds are obtained.

Let $\left\{W_t\right\}=\left\{{(X_{t^{\text{'}}}^{\text{'}},Y_t^{\text{'}})}^{\text{'}}\right\}$ be vector ARMA $(m,n)$ processes. Denote by ${\widehat{X}}_t\left(a\right)$ the predictor of $X_t$ based on $X_{t-a},X_{t-a-1},...$ and by ${\widehat{X}}_t(a,b)$ the predictor of $X_t$ based on $X_{t-a},X_{t-a-1},...,Y_{t-b},Y_{t-b-1},...$. The accuracy of the predictors is measured by ${\Delta }_X\left(a\right)=\text{E}[X_t-{\widehat{X}}_t\left(a\right)]{[X_t-{\widehat{X}}_t\left(a\right)]}^{\text{'}}$ and ${\Delta }_X(a,b)=\text{E}[X_t-{\widehat{X}}_t(a,b)]{[X_t-{\widehat{X}}_t(a,b)]}^{\text{'}}$. A general sufficient condition for the equality ${\Delta }_X\left(a\right)={\Delta }_X(a,a)]$ is given in the paper and it is shown that the equality ${\Delta }_X\left(1\right)={\Delta }_X(1,1)]$ implies ${\Delta }_X\left(a\right)={\Delta }_X(a,a)]$ for all natural numbers $a$.

A one-to-one correspondence between locally square integrable periodically correlated (PC) processes and a certain class of infinite-dimensional stationary processes is obtained. The correspondence complements and clarifies Gladyshev's known result [3] describing the correlation function of a continuous periodically correlated process. In contrast to Gladyshev's paper, the procedure for explicit reconstruction of one process from the other is provided. A representation of a PC process as a unitary...

For a binary stationary time series define ${\sigma }_n$ to be the number of consecutive ones up to the first zero encountered after time $n$, and consider the problem of estimating the conditional distribution and conditional expectation of ${\sigma }_n$ after one has observed the first $n$ outputs. We present a sequence of stopping times and universal estimators for these quantities which are pointwise consistent for all ergodic binary stationary processes. In case the process is a renewal process with zero the renewal state...

We give some estimation schemes for the conditional distribution and conditional expectation of the the next output following the observation of the first $n$ outputs of a stationary process where the random variables may take finitely many possible values. Our schemes are universal in the class of finitarily Markovian processes that have an exponential rate for the tail of the look back time distribution. In addition explicit rates are given. A necessary restriction is that the scheme proposes an...

We introduce and study a notion of Orlicz hypercontractive semigroups. We analyze their relations with general F-Sobolev inequalities, thus extending Gross hypercontractivity theory. We provide criteria for these Sobolev type inequalities and for related properties. In particular, we implement in the context of probability measures the ideas of Maz'ja's capacity theory, and present equivalent forms relating the capacity of sets to their measure. Orlicz hypercontractivity efficiently describes the...