Small Ball Problem via Wavelets for Gaussian Processes

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)...

Exact 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...

Solving infinite horizon optimal control problems of nonlinear interconnected large-scale dynamic systems via a Haar wavelet collocation scheme

We consider an approximation scheme using Haar wavelets for solving a class of infinite horizon optimal control problems (OCP's) of nonlinear interconnected large-scale dynamic systems. A computational method based on Haar wavelets in the time-domain is proposed for solving the optimal control problem. Haar wavelets integral operational matrix and direct collocation method are utilized to find ...

Gaussian Processes: Inequalities, Small Ball Probabilities and Applications

High resolution coding of stochastic processes and small ball probabilities

In this thesis, we study the high resolution coding problem for stochastic processes. The general coding problem concerns the finding of a good representation of a random signal, the original, within a class of allowed representations. The class of allowed representations is defined through restrictions on the information content of these representations. Mostly we consider three different inte...