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...

We obtain limit theorems for a class of nonlinear discrete-time processes X(n) called the kth order Volterra processes of order k. These are moving average kth order polynomial forms: X(n)=∑0i1,…,ik∞a(i1,…,ik)ϵn−i1…ϵn−ik, where {ϵi} is i.i.d. with Eϵi=0, Eϵi2=1, where a(⋅)...

Consider the sum Z=∑n=1∞λn(ηn−Eηn), where ηn are independent gamma random variables with shape parameters rn0, and the λn’s are predetermined weights. We study the asymptotic behavior of the tail ∑n=M∞λn(ηn−Eηn), which is asymptotically normal under certain conditions. We...

We introduce a broad class of self-similar processes {Z(t),t≥0} called generalized Hermite processes. They have stationary increments, are defined on a Wiener chaos with Hurst index H∈(1/2,1), and include Hermite processes as a special case. They are defined through a homogeneous kernel g,...

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...

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...

We extend results of Maejima (1984) concerning the time that a two-dimensional stationary Gaussian process spends in an elliptical domain. Here: (a) the process may be cross-correlated while the domain is elliptical; (b) the cross-correlations do not vanish asymptotically; (c) a functional limit...

Jointly [alpha]-stable random variables with index 0 [alpha] 2 have only finite moments of order less than [alpha], but their conditional moments can be higher than [alpha]. We provide conditions for this to happen and use the existence of the conditional moments to study the regression...

We develop a pathwise construction of stochastic integrals relative to continuous martingales. The key to the construction is an almost sure approximation technique which associates a sequence of finitely generated filtrations ("skeleton filtrations") and a sequence of simple stochastic...