Local times for functions and Gaussian processes

Some necessary and sufficient conditions that a Gaussian process with continuous time have a local time are discussed.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 85, pp. 104–112, 1979.The author is grateful to the participants of the seminar on local times and to Yu. A. Davydov in particular.     

Gaussian approximation of local empirical processes indexed by functions

Summary. An extended notion of a local empirical process indexed by functions is introduced, which includes kernel density and regression function estimators and the conditional empirical process as special cases. Under suitable regularity conditions a central limit theorem and a strong approximation by a sequence of Gaussian processes are established for such processes. A compact law of the iterated logarithm (LIL) is then inferred from the corresponding LIL for the approximating sequence of Gaussian processes. A number of statistical applications of our results are indicated. Received: 11 January 1995/In revised form: 12 July 1996     

Upper functions for positive random functionals. I. General setting and Gaussian random functions

In this paper we are interested in finding upper functions for a collection of real-valued random variables {Ψ(χ _θ ), θ ∈ Θ}. Here {χ _θ , θ ∈ Θ} is a family of continuous random mappings, Ψ is a given sub-additive positive functional and Θ is a totally bounded subset of a metric space. We seek a nonrandom function U: Θ → ?₊ such that sup _θ∈Θ{Ψ(χ _θ ) ? U(θ)}₊ is “small” with prescribed probability. We apply the results obtained in the general setting to the variety of problems related to Gaussian random functions and empirical processes.     

Traces of oscillating functions

We show that the trace of an indefinitely oscillating function on a subspace of ^d is not always indefinitely oscillating. In the periodic case, the number of oscillations of the trace depends on the regularity of the function. In the general case, we exhibit a definitive counter-example.     

Gaussian OS-positive processes

Summary Gaussian processes satisfying Osterwalder-Schrader positivity are studied. A representation of the (generalized) covariance function of an OS-positive process as the Laplace transform of an operator-valued probability measure is given. It is shown that every Gaussian OS-positive process has a unique Gaussian canonical Markov extension. An explicit application is made to the generalized free Euclidean fields.Partially supported by the National Science Foundation under grant MCS-76 06332     

Gaussian stochastic processes

The survey is devoted to works appearing in the last 3–5 years and pertaining mainly to local properties of the trajectories of Gaussian processes, the behavior of trajectories in the uniform metric, and properties of level sets. Some new results are also presented.     

Extremal ranks and transformation of variables for extremes of functions of multivariate Gaussian processes

The exit rate from a ‘safe region’ plays an important role in dynamic reliability theory with multivariate random loads. For Gaussian processes the exit rate is simply calculated only for spherical or linear boundaries. However, many smooth boundaries, not of any of these types, are asymptotically spherical in variables of lower dimension, having a greater curvature in the remaining variables. As is shown in this paper, the asymptotic exit rate is then simply expressed as the exit rate from a sphere for a process of the lower dimensions, corrected by an explicit factor.The procedure circumvents the need to calculate complicated exit rate integrals for general boundaries, reducing the problem to a Gaussian probability integral for independent variables.A result of independent interest relates the tail distribution for a sum of a noncentral χ²-variable and a weighted sum of squares of noncentral normal variables, to the tail distribution of the χ²-variable only.     

Oscillation of Gaussian processes

The properties of the oscillations of Banach-valued Gaussian processes are investigated. The oscillations of several Gaussian sequences are computed. The obtained results are used for the investigation of the properties of the trajectories of one-dimensional Gaussian processes.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 177, pp. 92–97, 1989.     

On asymptotic normality of estimates for correlation functions of stationary Gaussian processes in the space of continuous functions

We establish conditions of the weak convergence of the empirical correlogram of a stationary Gaussian process to some Gaussian process in the space of continuous functions. We prove that such a convergence holds for a broad class of stationary Gaussian processes with square integrable spectral density.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 11, pp. 1485–1497, November, 1995.This work was financially supported by the Ukrainian State Committee on Science and Technology.     

Linear prediction for generalized oscillating stochastic processes

Translated from Matematicheskie Modeli i Optimizatsiya Vychislitel'nykh Algoritmov, pp. 123–130, 1994.     

Gaussian and sparse processes are limits of generalized Poisson processes

The theory of sparse stochastic processes offers a broad class of statistical models to study signals, far beyond the more classical class of Gaussian processes. In this framework, signals are represented as realizations of random processes that are solution of linear stochastic differential equations driven by Lévy white noises. Among these processes, generalized Poisson processes based on compound-Poisson noises admit an interpretation as random L-splines with random knots and weights. We demonstrate that every generalized Lévy process—from Gaussian to sparse—can be understood as the limit in law of a sequence of generalized Poisson processes. This enables a new conceptual understanding of sparse processes and suggests simple algorithms for the numerical generation of such objects.     

Upper bounds for Gaussian stochastic programs

We present a construction which gives deterministic upper bounds for stochastic programs in which the randomness appears on the right–hand–side and has a multivariate Gaussian distribution. Computation of these bounds requires the solution of only as many linear programs as the problem has variables. Received December 2, 1997 / Revised version received January 5, 1999? Published online May 12, 1999     

Extremes of vector-valued Gaussian processes

The seminal papers of Pickands (Pickands, 1967; Pickands, 1969) paved the way for a systematic study of high exceedance probabilities of both stationary and non-stationary Gaussian processes. Yet, in the vector-valued setting, due to the lack of key tools including Slepian’s Lemma, there has not been any methodological development in the literature for the study of extremes of vector-valued Gaussian processes. In this contribution we develop the uniform double-sum method for the vector-valued setting, obtaining the exact asymptotics of the high exceedance probabilities for both stationary and n on-stationary Gaussian processes. We apply our findings to the operator fractional Brownian motion and Ornstein–Uhlenbeck process.     

Some comparisons for Gaussian processes

Extensions and variants are given for the well-known comparison principle for Gaussian processes based on ordering by pairwise distance.