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 review the notion of linearity of time series, and show that ARCH or stochastic volatility (SV) processes are not only non-linear: they are not even weakly linear, i.e., they do not even possess a martingale representation. Consequently, the use of Bartlett’s formula is unwarranted in...

Using the CRSP (Center for Research in Security Prices) daily stock return data, we revisit the question of whether or not actual stock market prices exhibit long-range dependence. Our study is based on an empirical investigation reported in Teverovsky, Taqqu and Willinger [33] of the modified...

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(⋅)...

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...

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,...

The Rosenblatt distribution appears as limit in non-central limit theorems. The generalized Rosenblatt distribution is obtained by allowing different power exponents in the kernel that defines the usual Rosenblatt distribution. We derive an explicit formula for its third moment, correcting the...