Approximation of weak sense stationary stochastic processes from local averages

We show that a weak sense stationary stochastic process can be approximated by local averages. Explicit error bounds are given. Our result improves an early one from Splettstosser.     

On the angle between past and future for multivariate stationary stochastic processes

A characterization for the positivityof the angle between past and future of multivariate stationary stochastic processes is established. In order to prove the results a lemma is proved which is of independent interest, and which is very useful in other areas of prediction theory as well.     

Discrete wave-analysis of continuous stochastic processes

The behaviour of a continuous-time stochastic process in the neighbourhood of zero-crossings and local maxima is compared with the behaviour of a discrete sampled version of the same process.For regular processes, with finite crossing-rate or finite rate of local extremes, the behaviour of the sampled version approaches that of the continuous one as the sampling interval tends to zero. Especially the zero-crossing distance and the wave-length (i.e., the time from a local maximum to the next minimum) have asymptotically the same distributions in the discrete and the continuous case. Three numerical illustrations show that there is a good agreement even for rather big sampling intervals.For non-regular processes, with infinite crossing-rate, the sampling procedure can yield useful results. An example is given in which a small irregular disturbance is superposed over a regular process. The structure of the regular process is easily observable with a moderate sampling interval, but is completely hidden with a small interval.     

On restricted linear estimation for regression in stochastic processes

In this paper the problem of restricted linear estimation for regression in stochastic processes is analyzed from different viewpoints, using RKHS methods. Of special interest is a relationship with an extended regression problem. Applications of the results to finite dimensional situations are also given.     

Boson fock representations of stochastic processes

A classification theory of quantum stationary processes similar to the corresponding theory for classical stationary processes is presented. Our main result is the classification of those pairs of classical stationary processes that admit a joint boson Fock canonical representation. Translated fromMatematicheskie Zametki, Vol. 67, No. 1, pp. 3–14, January, 2000.     

QUANTUM GAUSSIAN PROCESSES

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Spectral multiplicity of some stochastic processes

In this paper we consider the connection between the canonical and the weak-canonical representations for the given second-order stochastic process in a separable Hilbert space and we extend a well-known theorem of H. Cramer concerning sufficient conditions for a process to be of multiplicity one.

     

On functional central limit theorems for certain continuous time parameter stochastic processes

Weak invariance principles for certain continuous time parameter stochastic processes (including martingales and reverse martingales) are considered. Weak convergence in the sup-norm metric is also studied.     

Generalized stochastic processes in algebras of generalized functions

Stochastic processes with paths in a generalized function algebra are defined and it is shown that there exists an embedding of generalized functional stochastic processes into such ones. Gaussian stochastic processes with paths in an algebra of generalized functions are characterized by their first and second moments and an application to stochastic differential equations is given.     

On stochastic processes generated by quadric operators

This paper is a continuation of our earlier paper (J. Theoret. Prob.3, 51–70). The existence and uniqueness of solutions of equations for quadric stochastic processes will be studied in this paper.     

On the Airy Reproducing Kernel, Sampling Series, and Quadrature Formula

We determine the class of entire functions for which the Airy kernel (of random matrix theory) is a reproducing kernel. We deduce an Airy sampling series and quadrature formula. Our results are analogues of well known ones for the Bessel kernel. The need for these arises in investigating universality limits for random matrices at the soft edge of the spectrum. Research supported by NSF grant DMS0400446 and US-Israel BSF grant 2004353.     

Asymptotic expansions of posterior expectations,distributions and densities for stochastic processes

Asymptotic expansions are derived for Bayesian posterior expectations, distribution functions and density functions. The observations constitute a general stochastic process in discrete or continuous time.     

Lévy processes in semisimple Lie groups and stability of stochastic flows

We study the asymptotic stability of stochastic flows on compact spaces induced by Levy processes in semisimple Lie groups. It is shown that the Lyapunov exponents can be determined naturally in terms of root structure of the Lie group and there is an open subset whose complement has a positive codimension such that, after a random rotation, each of its connected components is shrunk to a single moving point exponentially under the flow.

     

Chaos expansion methods of stochastic processes for Malliavin-type equations

In this short survey we present the application of the Askey-scheme of orthogonal polynomials to define several discrete and continuous distribution types in frame-work of T. Hidaʼs white noise analysis. The results are applied to define fractional versions of the distributions and to solve stochastic differential equations involving the Malliavin derivative, Skorokhod integral and Ornstein-Uhlenbeck operator.     

加权再生核空间中信号的平均采样与重构

主要讨论L_v~p的加权再生核子空间中信号的平均采样与重构.首先,针对两种平均采样泛函建立了采样稳定性;其次,基于概率测度给出一个一般的迭代算法,将迭代逼近投影算法和迭代标架算法统一起来;最后,针对被白噪声污染的平均样本给出了信号重构的渐进点态误差估计.     

Gel'fand三元组上Lévy过程的随机积分

本文研究了算子值过程关于Gel’fand三元组EHE*上Lévy过程的随机积分.利用再生核Hilbert空间上柱Lévy过程的随机积分,定义一类算子值过程关于E*-值Lévy过程的随机积分。