Moving averages

Abstract

In this chapter we will apply the results of Chapter 4 to define one-sided and two-sided multivariate moving average processes with random coefficients. In Sections 15.1 and 15.2, we will recall the conditions for existence and the regular variation properties. We will compute the tail process and the candidate extremal index. The conditions of existence allow for a m-dependent tail equivalent approximations.

15.5 Bibliographical notes

The results of Sections 15.1 and 15.2, in such generality, seem to be new. The formulas (15.2.4a), (15.3.11b) for the extremal index were obtained in [CHM91, Proposition 2.1] in the case of a linear process with deterministic coefficients. [MS10] obtained the tail process of a moving average with values in a Banach space with deterministic operator weights.

Point process convergence for infinite order moving averages with stable innovations was studied for the first time in [Roo78]. It seems that the Single big jump heuristic was introduced to this area for the first time in that reference. Point process convergence for linear processes with deterministic weights has been established by m-dependent approximation, usually under conditions that rule out long memory, following closely the ideas of the previous reference. See [DR85a]. Using the nearly optimal results of [HS08] on the convergence of series allows the use of m-dependent approximations even in the long memory case.

Point process convergence immediately gives converges of partial sums when $\alpha \in (0,1)$. In the case $1<\alpha < 2$, convergence to a stable law is usually concluded directly by m-dependent approximation, instead of using point process convergence. See [DH95]. The case $\alpha =1$ seem to have been overlooked in the literature. [AT00, Theorem 2.1 and 2.2] prove convergence of partial sums and partial sums of squares in the $M_1$ topology in the long and short-memory cases, respectively. [Kri19] proves functional convergence for moving averages with random coefficients.

Sample covariances for $\alpha \in (0,2)$ are considered in [DR85a] by applying directly point process convergence. See also [DR85b]. The case $\alpha \in [2,4)$ is treated in [DR86].

Consistency of the Hill estimator for moving averages with deterministic weights was proved by [RS97, RS98] under conditions which exclude long memory. The consistency of the Hill estimator for long memory moving averages seems to have been unnoticed.

The asymptotic variance for the Hill estimator in case of infinite order moving averages with deterministic weights appears in [Hsi91b, Theorem 4.5] (m-dependent case) and [Dre03, Dre08]. In both cases, the asymptotic theory is obtained under $\beta$-mixing conditions.

The asymptotic variance for the blocks estimator of the extremal index in case of finite order moving averages is given in [Hsi91a, Example 4.7].

The $\beta$-mixing result of Lemma 15.3.1 is from [PT85]. Mixing properties of ARMA models were obtained by [Mok88, Mok90]. We are not aware of references which provide conditions for $\beta$-mixing of general moving averages with random coefficients in the framework of Theorem 15.2.2.