Publisher Summary This chapter focuses on the inequalities, small ball probabilities, and application of Gaussian processes. It is well-known that the large deviation result plays a fundamental role in studying the upper limits of Gaussian processes, such as the Strassen type law of the iterated logarithm. However, the complexity of the small ball estimate is well-known, and there are only a few Gaussian measures for which the small ball probability can be determined completely. The small ball probability is a key step in studying the lower limits of the Gaussian process. It has been found that the small ball estimate has close connections with various approximation quantities of compact sets and operators, and has a variety of applications in studies of Hausdorff dimensions, rate of convergence in Strassen's law of the iterated logarithm, and empirical processes.

Gaussian processes: Inequalities, small ball probabilities and applications

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Isoperimetry and Gaussian analysis

A precise link proved by Kuelbs and Li relates the small ball behavior of a Gaussian measure $\mu$ on a Banach space $E$ with the metric entropy behavior of $K_\mu$, the unit ball of the reproducing kernel Hilbert space of $\mu$ in $E$. We remove the main regularity assumption imposed on the unknown function in the link. This enables the application of tools and results from functional analysis to small ball problems and leads to small ball estimates of general algebraic type as well as to new estimates for concrete Gaussian processes. Moreover, we show that the small ball behavior of a Gaussian process is also tightly connected with the speed of approximation by “finite rank” processes.

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Approximation, metric entropy and small ball estimates for Gaussian measures

The properties of several multifractal formalisms based on wavelet coefficients are compared from both mathematical and numerical points of view. When it is based directly on wavelet coefficients, the multifractal formalism is shown to yield, at best, the increasing part of the weak scaling exponent spectrum. The formalism has to be based on new multiresolution quantities, the wavelet leaders, in order to yield the entire and correct spectrum of Holder singularities. The properties of this new multifractal formalism and of the alternative weak scaling exponent multifractal formalism are investigated. Examples based on known synthetic multifractal processes are illustrating its numerical implementation and abilities.

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Wavelet Leaders in Multifractal Analysis

Anisotropic Gaussian random flelds arise in probability theory and in various applications. Typical examples are fractional Brownian sheets, operator-scaling Gaussian flelds with stationary increments, and the solution to the stochastic heat equation. This paper is concerned with sample path properties of anisotropic Gaussian random flelds in general. Let X = fX(t); t 2 R N g be a Gaussian random fleld with values in R d and with parameters H1;:::;HN. Our goal is to characterize the anisotropic nature of X in terms of its parameters explicitly. Under some general conditions, we establish results on the modulus of continuity, small ball probabilities, fractal dimensions, hitting probabilities and local times of anisotropic Gaussian random flelds. An important tool for our study is the various forms of strong local nondeterminism.

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Sample Path Properties of Anisotropic Gaussian Random Fields

We estimate small ball probabilities for locally nondeterministic Gaussian processes with stationary increments, a class of processes that includes the fractional Brownian motions. These estimates are used to prove Chung type laws of the iterated logarithm.

Small values of Gaussian processes and functional laws of the iterated logarithm

denote the first hitting time of {x}. We will assume that P~ for t>0, and that 0 is regular for {0}, i.e. that P~ Under these assumptions P~ {Tx 0 for all x ([3]). In Section 5 of [1] Blumenthal and Getoor investigate the absolute continuity of the L6vy measure v of the pathwise inverse to the local time of a general Markov process X at zero. Their main result is that v is absolutely continuous under the assumption that for each t > 0 and each x~R the transition function Pt(x, dy)=P~{Xt~dy} is abso/utely continuous with respect to Lebesgue measure. In order to prove this they have to show that the hitting time distributions PY{ T O ~dt} are absolutely continuous for y4=0. However, the proof they give for this fact is incorrect.

Ditlev Monrad

Papers

Small values of Gaussian processes and functional laws of the iterated logarithm

Lévy processes: Absolute continuity of hitting times for points