Point process convergence

Abstract

In this chapter we will introduce one important tool in the study of the extremes of regularly varying time series, namely the point process of exceedences. The point process of exceedences records the location and value of the observations over a high threshold. Its convergence to a point process is the main object of this chapter. This convergence in turn will be the starting point of the limit theorems for the partial maxima and partial sum processes to be obtained in Chapter 8.

7.7 Bibliographical notes

The classical monograph [Res87] deals with i.i.d. sequences and uses the more usual vague convergence on the set $[-\infty ,\infty ]^d\setminus \{{\varvec{0}}\}$. Section 7.1 is essentially based on chapters 1–4 of the monograph [Kal17], complemented by [BP19] and [Zha16]. In particular, the last necessary and sufficient condition for weak convergence of random measures in Theorem 7.1.17 is [BP19, Proposition 4.6].

The study of the convergence of the point process of exceedences has a long story that we will not recall exhaustively here. In the earliest references, the point process of exceedences refers to the point process $N'_n(\cdot \times (1,\infty ))$ on $[0,\infty )$ which counts the number of exceedences but does not record their values. The convergence of the point process $N'$ is called complete Poisson convergence in [LLR83]. The convergence to a simple Poisson point process was established in [LLR83, Theorem 5.2.1] under two conditions. The first one is a mixing condition tailored to extremes, called Condition D, which is implied by strong mixing and holds for Gaussian stationary processes whose covariance decays sufficiently fast to zero. See [LLR83, Section 4.4]. The second one is the anticlustering Condition $D'$ which is related to extremal independence and prevents the clustering of points in the limiting point process. It is also satisfied by Gaussian processes under the same condition. In [HHL88], Condition $D'$ is dropped which results in clustering in the limiting point process.

The ideas developed in this chapter for the study of the convergence of the point process of exceedences come from the extremely deep paper [DH95]. The sequence $\varvec{Q}$ appears therein in relation to the representation of infinitely divisible point processes. Some results of that paper were translated into the language of the tail process by [BS09] which also introduced the candidate extremal index and proved that it is equal to the extremal index under the anticlustering condition ${\mathcal {A}\mathcal {C}(r_{n}, c_{n})}$ and 7.3.14 (without the time component). The convergence of the functional point process of exceedences $N'_n$ under 7.3.14 was established partially in [BKS12] and the proof was completed in [BT16]. [Kri16] proved that Condition (7.3.14) holds for an alpha-mixing time series. These ideas were further developed by [BPS18] to prove the convergence of the point process of clusters under conditions ${\mathcal {A}\mathcal {C}(r_{n}, c_{n})}$ and (7.3.3).

The ideas from [DH95] are extended in [DM98b] to point processes of lagged vectors in order to study the limiting behavior of sample covariances. This is the approach which is used in Problem 7.11. [BDM99] applies this general approach to bilinear processes and [BDM02b] to GARCH processes.

There are many references which establish the convergence of the functional point process of exceedences $N'_n$ for specific models by ad-hoc techniques. See, e.g. [DR85a, DR85b] (infinite order moving averages), [DR96] (bilinear processes), [Kul06] (moving averages with random coefficients), [DM01] and [KS12] (stochastic volatility models, possibly with long memory).

The concept of extremal index dates back to 1950s. A summary is given in [LLR83, Section 3.7]. Results under various mixing conditions are given to [O’B87] and [Roo88]. The idea of obtaining the extremal index as the limit of extremal indices of tail equivalent m-dependent approximation seems to be due to [CHM91] which used condition (7.4.1) and other involved conditions which are replaced here efficiently by the use of the tail process. This technique has been used by [Til18] for shot-noise type processes.