1. Introduction
A point process on a space
S is a stochastic process composed of a time series of indications of a specific event, which provides an elegant formulation of extremal behaviors of a stochastic process. The
block maxima model and the
peak-over-threshold model as two important approaches in extreme value theory can be formulated as applications of point processes. The important role of point processes has been widely discussed in various monographs such as [
1,
2,
3,
4], to name a few.
Let
be an iid sequence of random variables. Define the partial maxima
. According to the Fisher–Tippett theorem (see for example Theorem 3.2.3 in [
2]), if there exist constants
,
and a non-degenerate distribution
H such that
then
H is of the same type as one of the three distributions: Fréchet, Weibull, or Gumbel. By introducing the parameter
, the
generalized extreme value distribution (GEV) has the distribution function
where
u and
are two parameters, and
and
are the lower and upper endpoints of
G, respectively. When
,
corresponds to the Fréchet distribution; when
,
corresponds to the Gumbel distribution; and when
,
corresponds to the Weibull distribution. Define the marked point process
where
is the indicator function such that
According to Theorem 7.1 in [
3], we have
The process
N is a Poisson process with intensity measure
, where
is the Lebesgue measure on
and
. Moreover, Theorem 5.2.4 in [
2] ensures that (
4) holds if and only if
where
stands for the vague convergence; please refer to [
1] for more details on the vague convergence.
The event
is equivalent to the event
, which is the connection to the block maxima model. The peak-over-threshold model considers the probability of the form
which is equivalent to
For more details on the connections between the two models and point processes, please also refer to Section 7.4, [
3].
Extreme value theory takes heavy-tailed phenomena as the main objects to study, and it focuses on events that are bounded away from the origin. The
-convergence introduced in [
5] considers the convergence of measures on the space without the origin point, which uses a similar idea to the
-convergence studied in the monographs [
6,
7] on point processes. A gap between the
-convergence and the weak convergence of point processes exists, and one goal of this paper is to fill this gap. In the meantime, point processes provide a tool to analyze functional data, which might not be from a Hilbert space. As the review paper [
8] explains, the Karhunen–Loeve expansion is the main tool to convert an infinite-dimensional curve into a finite-dimensional vector, which plays an important role in functional data analysis and requires the curves defined in a Hilbert space. The objects exhibit heavy-tailed features such as extreme temperature curves, and high-frequency stock prices, to name a few. They are more reasonable to be viewed as a random element from a Banach space than from a Hilbert space. Thus, the framework of functional data analysis described in [
8] is not feasible to analyze these objects. Alternatively, point processes can be used for analyzing heavy-tailed functional data due to their connection to extreme value theory.
The paper will be organized as follows. In
Section 2, we introduce the space of measures
and the corresponding convergence, the
-convergence. The properties of point processes are studied in
Section 3, in which the equivalence between the weak convergence of point processes and the
-convergence is proved. Some interesting examples of metric spaces allow for the construction of the space
.
2. The Space of Measures and the
-Convergence
The
-convergence is a special case of the
-convergence when the set
is a singleton. The space
and the
-convergence have a natural connection with the concept of
hidden regular variation, and they are introduced and studied in [
9]. Here, we will briefly introduce some key results of the
-convergence.
2.1. The Metric Space
Let
be a complete and separable metric space, and the Borel
-field on
is denoted
, which is generated by open balls
for
. Let
be a closed subset of
and let
. We further assume that
is equipped with a scalar multiplication; see
Section 3.1 [
9].
Definition 1. A scalar multiplication on is a map satisfying the following properties:
- 1.
for all and ;
- 2.
for ;
- 3.
The map is continuous with respect to the product topology;
- 4.
if and if , then .
The set is assumed to be a cone, i.e., for . We shall prove that for all , the point . Trivially, for . Choose an arbitrary . We have for and, moreover, , which leads to a contradiction to (iv) in Definition 1 when and . Therefore, for all . The underlying space we consider in the paper is the complete and separable metric space , which is equipped with scalar multiplication and as a closed cone. Later in the paper, we will write or instead of for convenience.
Remark 1. It is possible that when and is not a singleton. Here is an example. Consider a space and a real line . Let . Take two elements from , and , where and . Define and for . It is easy to verify that is complete and separately equipped with a scalar multiplication and as a closed cone. If , even when .
2.2. The Space of Measures on and
Their Convergences
Let be the -algebra, and let be a collection of real-valued, non-negative, bounded continuous functions f on vanishing on for some . We say a set is bounded away from if for some .
Let
be the space of Borel measures on
that are bounded on the complements of
,
. As discussed in [
9], the space
is complete and separable with a proper choice of the metric
. For a measure
, we must have
for all
, and thus there is, at most, countable
such that
. Here,
stands for the boundary of a set
A. Moreover, for any
, we can find
such that
. The set
is said to be
μ-smooth if
.
The
-convergence is characterized by functions in
. Suppose that there is a sequence of measures
with
and a measure
. We say that
in
or
if
as
for all
. A Portmanteau theorem for the
-convergence is given as Theorem 2.1 in [
9], and we will present useful parts of this theorem for the paper.
Theorem 1. (Portmanteau theorem). Let . The following statements are equivalent:
- 1.
in as .
- 2.
for all .
- 3.
for all with .
2.3. The Counting Measures
Let be the space of all measures such that is a non-negative integer for all and . A measure N is a counting measure if .
Proposition 1. The space is a closed subsect of .
Proof. It is enough to show that the limit of a converging sequence in
is still in
. Let
be a sequence of counting measures and
in
. Let
y be an arbitrary point in
. Since
, for all but a countable set of values of
,
. We can find a decreasing sequence
such that
and
,
. By Portmanteau theorem, we have that for
,
Since
are non-negative integers,
are also non-negative integers, and thus
N is a counting measure by Lemma A1 in
Appendix A. □
3. Properties of Point Processes
We are interested in showing the equivalence between the weak convergence of point processes and the
-convergence as the equivalence between (
4) and (
5).
3.1. Random Measures and Point Processes
Definition 2. 1. A random measure ξ with state space is a measurable mapping from a probability space into .
2. A point process on is a measurable mapping from into .
A realization of a random measure has the value on the Borel set . For each fixed A, is a function mapping into . The following theorem provides a convenient way to examine whether a mapping is a random measure or a process is a point process.
Theorem 2. Let ξ be a mapping from a probability space into . Then, ξ is a random measure if and only if is a random variable taking values from for each . Similarly, N is a point process if and only if is a random variable taking values from non-negative integers for each .
Proof. Let
be the
-algebra of
. Let
be the
-algebra of subsets of
whose inverse images under
are events, and let
denote the mapping taking a measure
into
. Because
,
If is a random variable, and we have by definition. This implies that and thus is a random measure. Conversely, if is a random measure, for and hence . This shows that is a random variable.
Similarly, we can prove the property for N; thus, the details are omitted. □
3.2. Laplace Functionals
Let
and let
be a random measure. The Laplace functional of
is given by
We are interested in two important properties of Laplace functionals, which are listed in the two propositions.
Proposition 2. The Laplace functions uniquely determine the distribution of a random measure ξ.
To prove Proposition 2, we need the following lemma, which is derived directly from Theorem 3.3 [
10].
Lemma 1. The distribution of a random measure is completely determined by the fidi distributions for all finite families of disjoint sets generating .
A point process N is a random measure, and if Proposition 2 holds, uniquely determines the distribution of N.
Proof of Proposition 2. For
and Borel sets
bounded away from
and
,
, the function
is given by
Then for each realization
,
and
which is the joint Laplace transform of the random vector
. The uniqueness of Laplace transform for random vectors yields that
uniquely determines the law of
and Lemma 1 completes the proof. □
The following proposition shows that the convergence of Laplace functionals is equivalent to the convergence of random measures, which will be useful in the proofs.
Proposition 3. Let and ξ be random measures defined in Definition 2. The Laplace functional as for all if and only if as .
Proof. Think of the simple functions of the form , where k is a finite positive integer, and are a family of Borel sets with . The convergence of distributions of the integrals is equivalent to the finite-dimensional convergence for every finite k. Following a classical argument, we can find satisfying that and holds uniformly for every as . Proposition A1 implies finite-dimensional convergence and hence weak convergence. □
Since point processes and N are also random measures, the convergence of as is equivalent to the convergence as for all .
3.3. Poisson Processes and Marked Point Processes
As an important example of point processes, we shall give a definition of Poisson processes.
Definition 3. Given a random measure , a point process N is called a Poisson process or Poisson random measure (PRM) with mean measure μ if it satisfies the following conditions:
- 1.
For and a non-negative integer k, - 2.
For , if are mutually disjoint Borel sets in , then , are independent random variables.
We will write a Poisson process with mean measure as PRM .
Proposition 4. For a measure , PRM exists and its law is determined by two conditions in Definition 3. Moreover, the Laplace functional of PRM is given byand conversely, a point process N with Laplace functional of the form (6) must be PRM . Proof. Following the lines in the proof of Proposition 3.6, [
1] and choosing
for
and
, the Laplace functional
has the form (
6). Let
where
,
and
are disjoint sets bounded away from
. Similarly, it can be shown that the Laplace function
has the form (
6). Then, for any
, there are simple functions
of the form (
7) such that
with
as
. By Proposition 3.6 in [
1], we have that the Laplace function
has the form (
6). Moreover, it is easy to prove by following the lines in the proof of Proposition 3.6 [
1].
The proof of the existence of PRM(
) is through construction. We will use the same trick as in the proof of Lemma A1 to divide
into countable disjoint subspaces
,
. Then, let
for
. Using the arguments in the proof of Proposition 3.6, [
1], it is easy to construct PRM(
) for
, named
. Let
. For
,
This shows that PRM() exists. □
As discussed before, the block maxima and peak-over-threshold models have close relations to marked point processes.
Theorem 3. Let be iid copies of a random variable X taking values from and a measure . Suppose is a sequence of point processes with state space and N is PRM . Then, in if and only if holds.
4. Examples
According to Theorem 3, the applications of marked point processes to the block maxima and peak-over-threshold models require that the underlying metric space is complete, separable, and equipped with scalar multiplication. In this section, we will provide some interesting examples.
4.1. The Space as a Complete and Separable Space
The space
consists of continuous functions on the unit interval
, and the distance between
is given by
We choose the zero function
as
. As shown in Chapter 2, [
11], the space
is complete and separable. Moreover, with natural scalar multiplication, the space
satisfies the conditions in
Section 2. Therefore, the results in
Section 3 are applicable to the space
.
In Chapter 9 of [
12], extreme value theory in
is studied. Some of the results therein can be easily shown by an application of Theorem 3. We take Theorem 9.3.1 of [
12], for example. Given that
are iid stochastic processes in
, where
. Theorem 9.3.1 of [
12] states that if
then
with
. Define the marked point process as
. The event
is equivalent to the event
with
as the complement of the set
A. According to Theorem 3, the weak convergence
leads to
. In this example, the
-convergence is equivalent to the vague convergence, i.e,
. Furthermore, two convergences (
11) and (
12) are equivalent due to Theorem 3, which extend the results in [
12].
4.2. The Space of Sequences in
The
-convergence is a special case of the
-convergence when
is a singleton with exactly the zero point, which has been used to study the regular variation of sequences. Suppose that
is the underlying space satisfying the conditions in
Section 2. Let
be the space of all sequences
with elements in
, and the corresponding metric
is given by
The space is complete and separable. By choosing , scalar multiplication is defined componentwise.
Let
B be a Banach space with a countable base
. For
, we have
with
. We further assume that
for
. Assume that the norm in
B,
is a Hilbert-Schimdt norm
Then, the space
is the underlying space satisfying the conditions in
Section 2. It allows for the study of functional data from a Banach space instead of a Hilbert space by using point processes, which might be of interest in the context of extreme value theory.