Limit theorems, scaling of moments and intermittency for integrated finite variance supOU processes

Abstract

Superpositions of Ornstein-Uhlenbeck type (supOU) processes provide a rich class of stationary stochastic processes for which the marginal distribution and the dependence structure may be modeled independently. We show that they can also display intermittency, a phenomenon affecting the rate of growth of moments. To do so, we investigate the limiting behavior of integrated supOU processes with finite variance. After suitable normalization four different limiting processes may arise depending on the decay of the correlation function and on the characteristic triplet of the marginal distribution.To show that supOU processes may exhibit intermittency, we establish the rate of growth of moments for each of the four limiting scenarios. The rate change indicates that there is intermittency, which is expressed here as a change-point in the asymptotic behavior of the absolute moments.holds with convergence in the sense of convergence of all finite dimensional distributions as T → ∞, then Z is H-self-similar for some H > 0, that is, for any constant c > 0, the finite dimensional distributions of Z(ct) are the same as those of c H Z(t). Brownian motion for example is self-similar with H = 1/2. Moreover, the normalizing sequence is of the form A T = ℓ(T )T H for some ℓ slowly varying at infinity. For self-similar process, the moments evolve as a power function of time since E|Z(t)| q = E|Z(1)| q t Hq and therefore the scaling function of Z is τ Z (q) = Hq. Hence (4) does not hold for self-similar processes. But it may not hold either for the process X * in (5) because one would expect that E|X * (T t)| q A q T → E|Z(t)| q , ∀t ≥ 0,

Weak dependence and GMM estimation of supOU and mixed moving average processes

“…This leads to select the parameter m = r in both proofs. Moreover, (19) is equal to (31) and then η X (r) = θ X (r). The same observations hold when comparing the results in Corollary 3.3 or Corollary 3.4 with Corollary 3.10 or Corollary 3.11.…”

“…Moreover for the MMA stochastic volatility models, we show the θ-weak dependence of the return process and the distributional limit of its sample moments. In [29,30,31], the limiting behavior of integrated and partial sums of supOU processes is analyzed in relation to the growth rate of their moments, called intermittency when the grow rate is fast. This leads to some conclusions regarding their asymptotic finite dimensional distributions and to identify different limiting theorems depending on the short or long memory shown by the supOU process.…”

Intermittency and infinite variance: the case of integrated supOU processes

“…A suitably normalized integrated process exhibits complex limiting behavior. Indeed, if the underlying supOU process has finite variance, then four classes of processes may arise in a classical limiting scheme ( [19]). Namely, the limit process may be Brownian motion, fractional Brownian motion, a stable Lévy process or a stable process with dependent increments.…”

“…Namely, the limit process may be Brownian motion, fractional Brownian motion, a stable Lévy process or a stable process with dependent increments. The type of limit depends on whether the Gaussian component is present in (1.1) or not, on the behavior of π in (1.1) near the origin and on the growth of the Lévy measure µ in (1.1) near the origin (see [19] for details). In the infinite variance case, the limiting behavior is even more complex as the limit process may additionally depend on the regular variation index of the marginal distribution (see [20] for details).…”

The almost-sure asymptotic behavior of the solution to the stochastic heat equation with Lévy noise

“…Let us mention that a weak (i.e., moment-based) version of additive intermittency has been introduced in a series of papers [13,14,15] on superpositions of Ornstein-Uhlenbeck processes. The term "additive intermittency" itself was coined by Murad S. Taqqu in private communication with the first author discussing the references above.…”