Central limit theorems for stationary random fields under weak dependence with application to ambit and mixed moving average fields

Abstract

We obtain central limit theorems for stationary random fields which are based on the use of a novel measure of dependence called θ-lex weak dependence. We discuss hereditary properties for θ-lex and η-weak dependence and illustrate the possible applications of the weak dependence notions to the study of the asymptotic properties of stationary random fields. Our general results are applied to mixed moving average fields (MMAF in short) and ambit fields. We show general conditions such that MMAF and ambit fields, with the volatility field being an MMAF or a p-dependent random field, are weakly dependent. For all the aforementioned models, we give a complete characterization of their weak dependence coefficients and sufficient conditions to obtain asymptotic normality of their sample moments. Finally, we give explicit computations of the weak dependence coefficients in the case of MSTOU and CARMA fields.

Statistical inference for continuous-time locally stationary processes using stationary approximations

“…Using Remark 3.4 we also obtain consistency if (O2) holds. Indeed (c * ) holds, since the contrast Φ LS is of finite memory (for n = 1), [11,Theorem 3.36] one can show that Ỹu is θ-weakly dependent with exponentially decaying θ-coefficients θ(h). Then, the stated result follows from Remark 3.4.…”

“…Remark 4.5. The process Ỹu from (11) is the unique stationary solution to the stochastic differential equation…”

“…Lemma 7.1. Let Ỹu be a Lévy-driven Ornstein-Uhlenbeck process as given in (11) such that γ + |x|>1 xν(dx) = 0 and |x|>1 x 4 ν(dx) < ∞. Then, for any real numbers…”