Theoretical aspects of the decorrelation property of the discrete wavelet transform when applied to stochastic processes have been studied exclusively from the increasing-domain perspective, in which the distance between neighboring observations stays roughly constant as the number of observations increases. To understand the underlying data-generating process and to obtain good interpolations, fixed-domain asymptotics, in which the number of observations increases in a fixed region, is often more appropriate than increasing-domain asymptotics. In the fixed-domain setting, we prove that, for a general class of inhomogeneous covariance functions, with suitable choice of wavelet filters, the wavelet transform of a nonstationary process has mostly asymptotically uncorrelated components.

Gaussian intrinsic random functions with power law generalized covariance functions, which in one dimension essentially correspond to fractional and integrated fractional Brownian motions, form a class of self-affine models for random fields with a wide range of smoothness properties. These random fields are nonstationary, but appropriately filtered versions of them are stationary. This work proves that most such random functions are locally stationary in a certain well-defined sense. This result yields an efficient and exact method of simulating all fractional and integrated fractional Brownian motions.

An extension of a general theorem by J.A. Bucklew (ibid., vol. IT-31, 677-679, 1985) on the asymptotic optimality of a linear predictor based on an incorrect covariance function is given. The result is applied to the problem of predicting a small time lag into the future to obtain an easily verifiable condition under which the Taylor series predictor given by Bucklew is nearly optimal. The critical condition of the theorem is as follows: Gaussian measures corresponding to the covariance function used to obtain the predictors and the actual covariance function must be equivalent probability measures (i.e., mutually absolutely continuous measures).>

In circuit design with component values that are subject to variation from circuit to circuit, we often wish to find a statistical distribution for these component values that yields an acceptable level for the mean of some cost function. Statistical methods for solving this problem have become popular in recent years, especially in circuits with large numbers of varying components, as deterministic methods are hard to use in these cases. In using these methods, we are often faced with the problem of estimating the cost under one distribution given that we have sampled under another distribution. However, if the sampling distribution and the distribution under which the expected cost is to be estimated differ too much, then a poor estimator of the expected cost may be obtained. A new method of doing further sampling only in regions where the original sampling distribution under-sampled is introduced. By using this procedure, large reductions in variance are possible with only a small amount of additional sampling.