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1.
Functional Quantization and Small Ball Probabilities for Gaussian Processes   总被引:1,自引:0,他引:1  
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.  相似文献   

2.
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)  相似文献   

3.
We obtain some new estimates for the small ball behavior of the d-dimensional fractional Brownian sheet under Hölder and Orlicz norms. For d=2, these bounds are sharp for the Orlicz and the sup-norm. In addition, we give bounds for the Kolmogorov and entropy numbers of some operators satisfying an L2-Hölder-type condition.  相似文献   

4.
Let μ be a Gaussian measure on a separable Banach space. We prove a tight link between the logarithmic small ball probabilities of μ and certain moment generating functions. Based upon this link we provide a new lower bound for the distortion-rate function (DRF) against the small ball function. This allows us to use results of the theory of small ball probabilities to deduce lower bounds for the DRF. In particular, we obtain the correct weak asymptotics of the distortion rate function in many important cases (e.g. Brownian motion).  相似文献   

5.
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.  相似文献   

6.
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  相似文献   

7.
Small Deviations for Some Multi-Parameter Gaussian Processes   总被引:1,自引:0,他引:1  
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.  相似文献   

8.
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.  相似文献   

9.
Let {X(t), 0t1} be a Gaussian process with mean zero and stationary increments. Let 2(h) =EX 2(h) be nondecreasing and concave on (0,1). A sharp bound on the small ball probability ofX(·) is given in this paper.Research supported by Charles Phelps Taft Post-doctoral Fellowship of the University of Cincinnati and by the Fok Yingtung Education Foundation of China.  相似文献   

10.
By estimating small ball probabilities for l^P-valued Gaussian processes, a Chung-type law of the iterated logarithm of l^P-valued Gaussian processes is given.  相似文献   

11.
The correspondence between Gaussian stochastic processes with values in a Banach space E and cylindrical processes which are related to them is studied. It is shown that the linear prediction of an E-valued Gaussian process is an E-valued random variable as well as the spectral measure of an E-valued Gaussian stationary process is a Gaussian random measure.  相似文献   

12.
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.  相似文献   

13.
We consider a Banach space valued diffusion process corresponding to a stochastic evolution equation with strongly nonlinear drift. Sufficient conditions are given for the existence of a unique martingale solution and existence of an invariant measure. The resulting diffusion process is shown to be strongly Feller and irreducible. These properties yield uniqueness of invariant measure and ergodicity of the process. We also show that the invariant measure is equivalent to the invariant measure of the diffusion without drift. The main tool to show these results is the Girsanov Transformation.  相似文献   

14.
Let hR denote an L normalized Haar function adapted to a dyadic rectangle Rd[0,1]. We show that for choices of coefficients α(R), we have the following lower bound on the L norms of the sums of such functions, where the sum is over rectangles of a fixed volume:
  相似文献   

15.
Let {X(t); 0t1} be a real-valued continuous Gaussian Markov process with mean zero and covariance (s, t) = EX(s) X(t) 0 for 0<s, t<1. It is known that we can write (s, t) = G(min(s, t)) H(max(s, t)) with G>0, H>0 and G/H nondecreasing on the interval (0, 1). We show that
In the critical case, i.e. this integral is infinite, we provide the correct rate (up to a constant) for log P(sup0<t1 |X(t)|<) as 0 under regularity conditions.  相似文献   

16.
We sharpen a classical result on the spectral asymptotics of boundary-value problems for self-adjoint ordinary differential operator. Using this result, we obtain the exact L 2-small ball asymptotics for a new class of zero-mean Gaussian processes. This class includes, in particular, the integrated generalized Slepian process, integrated centered Wiener process, and integrated centered Brownian bridge. Partially supported by RFFR grant No.07-01-00159 and by grant NSh-227.2008.1.  相似文献   

17.
We define the index of solvability, a topological characteristic, whose difference from zero provides the existence of a solution for variational inequalities of Stampacchia’s type with S +-type and pseudo-monotone multimaps on reflexive separable Banach spaces. Some applications to a minimization problem and to a problem of economical dynamics are presented. The work is supported by the Russian FBR Grants 05-01-00100 and 07-01-00137 and by the NATO Grant ICS.NR.CLG 981757.  相似文献   

18.
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.  相似文献   

19.
This note discusses the topological necessary and sufficient conditions for a locally compact connected group to admit a Gaussian measure that is absolutely continuous with respect to Haar measure.  相似文献   

20.
在Banach空间上,给出集值测度的扩张定理并借助集测度的扩张给出了模糊数测度的扩张定理。  相似文献   

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