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Abstract  

Let {X t : 0 ≦ t ≦ 1} be a centered stationary Gaussian process, with correlation function satisfying the condition ρ(t) = 1 − t β L(t), 0 < β < 2, and let L be a slowly varying function at zero. Observing the process at points i/N, i = 0,1,..., N and considering ¦X i/NX (i-1)/N¦p with p > 0, we study the properties of the Donsker line associated with p-th order variations

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\sum\limits_{i = 1}^{[N{\text{ }}t]} {|X_{i/N} } - {\text{ }}X_{(i - 1)/N} |^p$$ \end{document}
. We also study the relationship between the number of crossings of a regularization of the initial process and the local time of the initial process. The results depend on the values of β.

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Рассматривается мно гомерный стохастиче ский процесс диффузионного типа, з аданный уравнением (1). Доказана теорема 3, согласно кот орой в нестационарном случ ае, соответствующем условиюξ (φ)=φ, преобра зование Лапласа (8) распределения доста точных статистик имеет простую форму (16). Это дает обобщение од ного результата Новикова [1], относящегося к одномерному случаю. В стационарном случа е (28) доказана формула (31) (те орема 5), которая обобщает один резуль тат Арато-Бенцура (см. [3]).

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Abstract  

We consider an empirical process based upon ratios of selected pairs of spacings, generated by independent samples of arbitrary sizes. As a main result, we show that when both samples are uniformly distributed on (possibly shifted) intervals of equal lengths, this empirical process converges to a mean-centered Brownian bridge of the form B C(u) = B(u)−6Cu(1−u) Σ0 1 B(s)ds, where B(·) denotes a Brownian bridge, and C, a constant. The investigation of the class of Gaussian processes {B C(·): C ∈ ℝ} leads to some unexpected distributional identities such as B 2(·)

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\underline{\underline d}$$ \end{document}
B(·). We discuss this and similar results in an extended framework.

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Abstract  

This paper investigates weighted approximations for Studentized U-statistics type processes, both with symmetric and antisymmetric kernels, only under the assumption that the distribution of the projection variate is in the domain of attraction of the normal law. The classical second moment condition E|h(X 1, X 2)|2 < ∞ is also relaxed in both cases. The results can be used for testing the null assumption of having a random sample versus the alternative that there is a change in distribution in the sequence.

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. Barber. 1998: Bayesian classification with Gaussian processes. IEEE Transactions on Pattern Analysis and Machine Intelligence 20: 1342–1351. Barber D. Bayesian classification with

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Abstract  

In this paper the third order long-range dependence (LRD) is defined in terms of the bispectrum and third order cumulants (bicovariances). Two particular non-Gaussian processes with second order LRD are considered together with their bispectra and bicovariances.

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Abstract  

Strongly inspired by the result due to Carr-Ewald-Xiao that the arithmetic average of geometric Brownian motion is an increasing process in the convex order, we extend this result to integrals of Lévy processes and Gaussian processes. Our method consists in finding an appropriate sheet associated to the original Lévy or Gaussian process, from which the one-dimensional marginals of the integrals will appear to be those of a martingale, thus proving the increase in the convex order property.

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Abstract  

Large deviation results for Gaussian processes are presented. As an application, we obtain a functional limit result for small increments of a fractional Brownian motion. Lvy's modulus of continuity for a fractional Brownian motion is obtained as a special case.

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Abstract  

Let M n denote the n-th moment space of the set of all probability measures on the interval [0, 1], P n the uniform distribution on the set M n and r n + 1 the maximal range of the (n + 1)-th moments corresponding to a random moment point C n with distribution P n on M n. We study several asymptotic properties of the stochastic process (r nt⌋+1)t∈[0,T] if n → ∞. In particular weak convergence to a Gaussian process and a large deviation principle are established.

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