Small Ball Probabilities of Fractional Brownian Sheets via Fractional Integration Operators

We investigate the small ball problem for d-dimensional fractional Brownian sheets by functional analytic methods. For this reason we show that integration operators of Riemann–Liouville and Weyl type are very close in the sense of their approximation properties, i.e., the Kolmogorov and entropy numbers of their difference tend to zero exponentially. This allows us to carry over properties of the Weyl operator to the Riemann–Liouville one, leading to sharp small ball estimates for some fractional Brownian sheets. In particular, we extend Talagrand's estimate for the 2-dimensional Brownian sheet to the fractional case. When passing from dimension 1 to dimension d2, we use a quite general estimate for the Kolmogorov numbers of the tensor products of linear operators.     

Small Ball Constants and Tight Eigenvalue Asymptotics for Fractional Brownian Motions

In this paper we prove rigorous large n asymptotics for the Karhunen–Loeve eigenvalues of a fractional Brownian motion. From the asymptotics of the eigenvalues the exact constants for small L ₂ ball estimates for fractional Brownian motions follows in a straightforward way.     

Functional Quantization and Small Ball Probabilities for Gaussian Processes

Quantization consists in studying the L ^r -error induced by the approximation of a random vector X by a vector (quantized version) taking a finite number n of values. We investigate this problem for Gaussian random vectors in an infinite dimensional Banach space and in particular, for Gaussian processes. A precise link proved by Fehringer⁽⁴⁾ and Dereich et al. ⁽³⁾ relates lower and upper bounds for small ball probabilities with upper and lower bounds for the quantization error, respectively. We establish a complete relationship by showing that the same holds for the direction from the quantization error to small ball probabilities. This allows us to compute the exact rate of convergence to zero of the minimal L ^r -quantization error from logarithmic small ball asymptotics and vice versa.     

A Note on the Fractional Integrated Fractional Brownian Motion

We characterize the lower classes of the fractional integrated fractional Brownian motion by an integral test.     

Small Ball Probabilities Around Random Centers of Gaussian Measures and Applications to Quantization

Let be a centered Gaussian measure on a separable Hilbert space (E, ). We are concerned with the logarithmic small ball probabilities around a -distributed center X. It turns out that the asymptotic behavior of –log (B(X,)) is a.s. equivalent to that of a deterministic function _R (). These new insights will be used to derive the precise asymptotics of a random quantization problem which was introduced in a former article by Dereich, Fehringer, Matoussi, and Scheutzow.⁽⁸⁾     

On the Link Between Small Ball Probabilities and the Quantization Problem for Gaussian Measures on Banach Spaces

Let be a centered Gaussian measure on a separable Banach space E and N a positive integer. We study the asymptotics as N of the quantization error, i.e., the infimum over all subsets of E of cardinality N of the average distance w.r.t. to the closest point in the set . We compare the quantization error with the average distance which is obtained when the set is chosen by taking N i.i.d. copies of random elements with law . Our approach is based on the study of the asymptotics of the measure of a small ball around 0. Under slight conditions on the regular variation of the small ball function, we get upper and lower bounds of the deterministic and random quantization error and are able to show that both are of the same order. Our conditions are typically satisfied in case the Banach space is infinite dimensional.     

A Functional LIL and Some Weighted Occupation Measure Results for Fractional Brownian Motion

Weighted occupation measure results are obtained for fractional Brownian motion. Proofs depend on small ball probability estimates of the sup-norm for these processes, which are then used to obtain a functional law of the iterated logarithm. The occupation measure results are consequences of the law of the iterated logarithm.     

Small ball probabilities for Gaussian processes with stationary increments under Hölder norms

Small ball probabilities are estimated for Gaussian processes with stationary increments when the small balls are given by various Hölder norms. As an application we establish results related to Chung's functional law of the iterated logarithm for fractional Brownian motion under Hölder norms. In particular, we identify the points approached slowest in the functional law of the iterated logarithm.Supported in part by NSF Grant DMS-9024961.     

A Turning-Band Method for the Simulation of Anisotropic Fractional Brownian Fields

In this article, we propose a method for simulating realizations of two-dimensional anisotropic fractional Brownian fields (AFBF) introduced by Bonami and Estrade. The method is adapted from a generic simulation method called the turning-band method (TBM) due to Matheron. The TBM reduces the problem of simulating a field in two dimensions by combining independent processes simulated on oriented bands. In the AFBF context, the simulation fields are constructed by discretizing an integral equation arising from the application of the TBM to nonstationary anisotropic fields. This guarantees the convergence of simulations as the step of discretization is decreased. The construction is followed by a theoretical study of the convergence rate (the detailed proofs are available in the online supplementary materials). Another key feature of this work is the simulation of band processes. Using self-similarity properties, processes are simulated exactly on bands with a circulant embedding method, so that simulation errors are exclusively due to the field approximation. Moreover, we design a dynamic programming algorithm that selects band orientations achieving the optimal trade-off between computational cost and precision. Finally, we conduct a numerical study showing that the approximation error does not significantly depend on the regularity of the fields to be simulated, nor on their degree of anisotropy. Experiments also suggest that simulations preserve the statistical properties of the original field.     

广义布朗单的容度估计

研究了既没有平稳增量性,也没有scaling性质的N指标d维广义布朗单的容度问题．证明了广义布朗单“好象”一个局部平稳增量过程,应用Cairoli极大不等式和多参数鞅的方法得到了广义布朗单的碰撞概率与容度之间的关系,给出了其碰撞概率的确切容度估计．所得结果包含了布朗单和可加布朗运动的相应结果．     

Fast and Exact Simulation of Fractional Brownian Surfaces

Fractional Brownian surfaces are commonly used as models for landscapes and other physical processes in space. This work shows how to simulate fractional Brownian surfaces on a grid efficiently and exactly by embedding them in a periodic Gaussian random field and using the fast Fourier transform. Periodic embeddings are given that are proven to yield positive definite covariance functions and hence yield exact simulations for all possible densities of the simulation grid. Numerical results show these embeddings can sometimes be made more efficient in practice. Further numerical results show how the ideas developed for simulating fractional Brownian surfaces can be used for simulating other Gaussian random fields. The simulation methodology is used to study the behavior of a simple estimator of the parameters of a fractional Brownian surface.     

Small Ball Estimates for Gaussian Processes under Sobolev Type Norms

A sharp small ball estimate under Sobolev type norms is obtained for certain Gaussian processes in general and for fractional Brownian motions in particular. New method using the techniques in large deviation theory is developed for small ball estimates. As an application the Chung's LIL for fractional Brownian motions is given in this setting.     

Estimates of Certain Exit Probabilities for p-Adic Brownian Bridges

Journal of Theoretical Probability - For each prime p, a diffusion constant together with a positive exponent specify a Vladimirov operator and an associated p-adic diffusion equation. The...     

Small Ball Estimates for Quasi-Norms

This note contains two types of small ball estimates for random vectors in finite-dimensional spaces equipped with a quasi-norm. In the first part, we obtain bounds for the small ball probability of random vectors under some smoothness assumptions on their density function. In the second part, we obtain Littlewood–Offord type estimates for quasi-norms. This generalizes results which were previously obtained in Friedland and Sodin (C R Math Acad Sci Paris 345(9):513–518, 2007), and Rudelson and Vershynin (Commun Pure Appl Math 62(12):1707–1739, 2009).     

Estimates for transition probabilities on a compact manifold

The purpose of this note is to describe a procedure for transferring familiar estimates for transition probabilities on RN to transition probabilities on compact manifolds.     

Small Ball Probabilities for a Shifted Poisson Process

We investigate the probabilities of hitting shifted small balls by sample paths of a centered Poisson process and find the exact range of parameters for which the Wiener approximation of these probabilities is valid. Towards this aim, we introduce the Skorokhod density technique. For the Poisson process, this technique plays a role similar to that of the Cameron-Martin formula in the construction of associated laws for a Gaussian measure. Bibliography: 20 titles.     

Small Deviations for Some Multi-Parameter Gaussian Processes

We prove some general lower bounds for the probability that a multi-parameter Gaussian process has very small values. These results, when applied to a certain class of fractional Brownian sheets, yield the exact rate for their so-called small ball probability. We show by example how to use such results to compute the Hausdorff dimension of some exceptional sets determined by maximal increments.     

Logarithmic Level Comparison for Small Deviation Probabilities

Log-level comparisons of the small deviation probabilities are studied in three different but related settings: Gaussian processes under the L ² norm, multiple sums motivated by tensor product of Gaussian processes, and various integrated fractional Brownian motions under the sup-norm. Wenbo V. Li: Supported in part by NSF Grant DMS-0505805.     

Logarithmic Level Comparison for Small Deviation Probabilities

Log-level comparisons of the small deviation probabilities are studied in three different but related settings: Gaussian processes under the L² norm, multiple sums motivated by tensor product of Gaussian processes, and various integrated fractional Brownian motions under the sup-norm. An erratum to this article can be found at