A self-similar process Z(t) has stationary increments and is invariant in law under the transformation Z(i)--c-HZ(ct), c[greater-or-equal, slanted]0. The choice ensures that the increments of Z(t) exhibit a long range positive correlation. Mandelbrot and Van Ness investigated the case where Z(t)...

Recently, Rosinski and Woyczynski have given necessary and sufficient conditions for the existence of the double integral with respect to a symmetric stable process of index [alpha] in [1, 2). In their approach the double integral is defined as an iterated Itô-type integral. We show here that...

In this paper, we study almost sure central limit theorems for sequences of functionals of general Gaussian fields. We apply our result to non-linear functions of stationary Gaussian sequences. We obtain almost sure central limit theorems for these non-linear functions when they converge in law...

The multifractional Brownian motion (MBM) processes are locally self-similar Gaussian processes. They extend the classical fractional Brownian motion processes by allowing their self-similarity parameter H[set membership, variant](0,1) to depend on time. Two types of MBM processes were...

Let (X1, X2) be a symmetric [alpha]-stable random vector with 0 [alpha] 2. Its distribution is characterized by a finite measure [GH] on the unit circle called the spectral measure. It is known that if [GH] satisfies some integrability condition then the conditional moment E[X2pX1 = x] can...

Suppose that f is a deterministic function, is a sequence of random variables with long-range dependence and BH is a fractional Brownian motion (fBm) with index . In this work, we provide sufficient conditions for the convergencein distribution, as m--[infinity]. We also consider two examples....

Billingsley developed a widely used method for proving weak convergence with respect to the sup-norm and J1-Skorohod topologies, once convergence of the finite-dimensional distributions has been established. Here we show that Billingsley's method works not only for J oscillations, but also for M...

Suppose that Xt = [summation operator][infinity]j=0cjZt-j is a stationary linear sequence with regularly varying cj's and with innovations {Zj} that have infinite variance. Such a sequence can exhibit both high variability and strong dependence. The quadratic form 89 plays an important role in...

This paper expands on the multigraph method for expressing moments of non-linear functions of Gaussian random variables. In particular, it includes a list of regular multigraphs that is needed for the computation of some of these moments. The multigraph method is then used to evaluate...

Many econometric quantities such as long-term risk can be modeled by Pareto-like distributions and may also display long-range dependence. If Pareto is replaced by Gaussian, then one can consider fractional Brownian motion whose increments, called fractional Gaussian noise, exhibit long-range...