Long memory processes

Abstract

In this chapter, we will introduce models for which the standard extreme value theory developed in the previous chapters fail in some aspects. These models belong to the loosely defined class of long memory or long-range dependent processes. The first long-range dependent models were studied for Gaussian processes, so a natural definition is through the covariance function.

16.4 Bibliographical notes

Long-range dependent Gaussian processes have been studied for at least 50 years. For a complete picture and comprehensive bibliography, see [BFGK13] and [PT17]. Long-range dependence as a phase transition was introduced by [Sam06]. See also [Sam16].

Section 16.1

More about the Lévy fractional stable motion $L_{\alpha , H}$ can be found in [ST94, Chapters 7 and 12] and [Sam16, Chapter 3]. A proof of Theorem 16.1.2 can be found in Chapter 9 of the latter reference. Fractional ARIMA models with regularly varying innovations are discussed in [KT95, KT96].

Theorem 16.1.2 and similar results were obtained earlier by [KM86, KM88, AT92], the latter reference also showing that $J_1$ convergence is impossible for moving averages and that $M_1$ convergence holds if the weights are non-negative.

Results on convergence of partial sums and sample covariances in case of long memory linear processes with regularly varying innovations can be found in [HK08]. A comprehensive overview can be found in [BFGK13, Sections 4.3 and 4.4.3].

A related field which is unfortunately untouched here is the extremal theory of non-Gaussian stable processes and fields. See [Roy17] for a review and further references.

Section 16.2

The results of Section 16.2.1 on Gaussian processes are taken from [LLR83]. The condition $\rho _n\log (n) \rightarrow 0$, introduced in [Ber64], is usually referred to as Berman’s condition. For models which do not satisfy this condition, see, for instance, [Shi15] (branching random walks) and [CCH16] (Gaussian free fields).

The Hermite process appeared first in the context of limits of functions of Gaussian long memory processes simultaneously in [DM79] and [Taq79]. See the monograph [PT17] for more on multiple stochastic integrals and Hermite processes.

The best reference for Theorem 16.2.3 is [Arc94]. It is perhaps also the best and most elegant paper on long memory Gaussian processes. It is impossible not to pay tribute to the late author by quoting (the also late) Sándor Csörgő’s review in MathSciNet: “This is a brilliant paper. It extends virtually the whole existing asymptotic distribution theory of partial sums of sequences of random variables that are functions of a real stationary Gaussian sequence to the case when the governing Gaussian sequence consists of vectors; in fact it does more. ... The proofs operate through reduction to Hermite polynomials, a covariance lemma of independent interest, virtuoso use of the diagram formula throughout and, in the long-range dependent case, the Dobrushin-Major method of multiple Wiener-Itô integrals.“

Theorem 16.2.7 is due to [Dav83, Lemma 6] for the case $\alpha \in (0,1)$ and [SH08] for the case $\alpha \in (1,2)$. It must be noted that in both references the convergence of the point process of exceedences is claimed to be proved, but in [Dav83] the function h is implicitly assumed to be non-decreasing and the proof of [SH08, Proposition 2.1] is difficult to follow. It is not obvious that the same proof will work for increasing functions h as in Example 16.2.4 or for functions h with a pole at 0 as in Example 16.2.5. Nevertheless credit must be given where it is due: the idea to use hypercontractivity in [SH08] is brilliant and justifies the paper alone. The case $\alpha =1$ of Theorem 16.2.7 is stated without proof in [SH08, Theorem 1.2].

Section 16.3

Extremes of stochastic volatility models were first considered in [BD98]. This section essentially follows [KS11, KS12]. The results in Section 16.3 can be extended in several directions. First, the point process convergence of Theorem 16.3.2 does not require the specific Assumption 16.3.1 on the volatility sequence. It suffices that the volatility sequence ${\{\sigma _{j}, j\in \mathbb {Z}\}}$ is ergodic and the moment condition (16.3.2) holds. The point process convergence can be obtained via m-dependent approximation, see [DM01]. [KS12] allows for dependence between ${\{\sigma _{j}, j\in \mathbb {Z}\}}$ and ${\{Z_{j}, j\in \mathbb {Z}\}}$ in order to account for leverage effects. Partial sums convergence (16.3.3) can be extended to convergence of powers and models with leverage ([KS12, Theorem 4.1]) as well as to limits for sample covariances ([DM01, Theorem 4.1] and [KS12, Theorems 5.2 and 5.3]). In the latter reference, it is shown that the leverage may have a peculiar influence on the limiting behavior of sample covariances.

The limiting result for the tail empirical process and the Hill estimator is taken from [KS11]. Extensions in a multivariate setting are given in [KS13], while [BBIK19] allows for leverage and study the tail empirical process, the Hill estimator, and some extensions.

A second-order condition is given in [KS11] which ensures the bias condition (16.3.16) and the proof of Corollary 16.3.8 is given there.

Stochastic volatility models that allow for heavy tails in volatility are considered in [MR13] and [JD16] (in the short-memory case) and in [KS15]. Some of these models allow for extremal dependence.