Exact L small balls of Gaussian processes
نویسنده
چکیده
We prove a comparison theorem, extending Li [6], and develop a complexanalytic approach to treat L2 small ball probabilities of Gaussian processes. We demonstrate the techniques for the m-times integrated Brownian motions, and in examples where one can not apply Li’s comparison theorem. ∗Department of Mathematics, University of Idaho, Moscow, ID 83844-1103, [email protected] †Department of Statistics, Colorado State University, Ft. Collins, CO 80523-1877, [email protected] ‡Mathematics Department, University of Maryland, College Park, MD 20742-4015, (301) 405-5086, [email protected] §Department of Mathematical Sciences, Johns Hopkins University, Baltimore, MD 21218-2682, [email protected]
منابع مشابه
Gaussian Processes : Karhunen - Loève Expansion , Small Ball
In this dissertation, we study the Karhunen-Loève (KL) expansion and the exact L small ball probability for Gaussian processes. The exact L small ball probability is connected to the Laplace transform of the Gaussian process via Sytaja Tauberian theorem. Using this technique, we solved the problem of finding the exact L small ball estimates for the Slepian process S(t) defined as S(t) = W (t+a)...
متن کاملCubature formulae for spheres, simplices and balls
We obtain in explicit form the unique Gaussian cubature for balls (spheres) in Rn based on integrals over balls (spheres), centered at the origin, that integrates exactly all m-harmonic functions. In particular, this formula is exact for all polynomials in n variables of degree 2m − 1. A Gaussian cubature for simplices is also constructed. Upper bounds for the errors for certain smoothness clas...
متن کاملProbabilities of Randomly Centered Small Balls and Quantization in Banach
the random small ball function (RSBF), when ε is small. We will see that some typical features of the SBF are true as well for the RSBF but the exact asymptotics of the two functions do not coincide. Beyond structural properties of Gaussian measures, the research is motivated by a close link to so-called random strategies in quantization problems, that we briefly recall now. Let E be a space of...
متن کاملExact L2-small ball asymptotics of Gaussian processes and the spectrum of boundary value problems with ”non-separated” boundary conditions
We sharpen a classical result on the spectral asymptotics of the boundary value problems for self-adjoint ordinary differential operator. Using this result we obtain the exact L2-small ball asymptotics for a new class of zero mean Gaussian processes. This class includes, in particular, integrated generalized Slepian process, integrated centered Wiener process and integrated centered Brownian br...
متن کاملSmall ball estimates for Brownian motion and the Brownian sheet
Small ball estimates for Brownian motion and the Brownian sheet are obtained when the balls are given by certain H61der norms. As an application, we establish some lira inf results for Gaussian samples of these processes, as well as Chung's LIL in this setting, but now we introduce some notation. Let I = [0, 1 ], with Co(I) the continuous real-valued function f ( . ) on I such that f ( 0 ) = 0....
متن کامل