Subordinated Compound Poisson processes of order $k$

In this article, the compound Poisson processes of order $k$ (CPPoK) is introduced and its properties are discussed. Further, using mixture of tempered stable subordinator (MTSS) and its right continuous inverse, the two subordinated CPPoK with various distributional properties are studied. It is also shown that space and tempered space fractional versions of CPPoK and PPoK can be obtained, which generalize the results in the literature.


Introduction
The Poisson distribution has been the conventional model for count data analysis, and due to its popularity and applicability various researchers have generalized it in several directions; e.g. compound Poisson processes, fractional (time-changed) versions of Poisson processes (see [4,9,1], and references therein). A handful of researchers have also studied the distributions and processes of order k (see [8,5]). In particular, the discrete distribution of order k, introduced by Philippou et al. (see [15]), include binomial, geometric and negative binomial distributions of order k. These distributions play an important role in several areas, such as, reliability, statistics, finance, actuarial risk analysis (see [12,16,13]).
In risk theory, the total claim amount is usually modelled by using a compound Poisson process, say Z t = N (t) i=1 Y i , where the compounding random variables Y i are iid and the number of claims N(t), independent of {Y i } i≥1 , follow Poisson distribution. But, due to the restriction of single arrival in each inter-arrival time, the model is not suitable to use. Kostadinova and Minkova [7] introduced a Poisson process of order k (PPoK), which allows us to model arrival in a group of size k. Recently, a time-changed version of Poisson processes of order k is studied by [16] which allows group arrivals and also the case of extreme events, which is not covered by [7]. In spite of its applicability, this model is still not suitable for underdispersed dataset. Therefore, a generalization of this model is essential and is proposed in this article.
To the best of our knowledge, such a generalization is not yet studied. Therefore, we introduce the compound Poisson process of order k (CPPoK) with the help of the Poisson process of order k (PPoK) and study its distributional properties. Then, we time-change CPPoK with a special type of Lévy subordinator known as mixture of tempered stable subordinator, and its right continuous inverse, and analyze some properties of these timechanged processes.
The article is organized as follows. In section 2, we introduce CPPoK and derive some of its general properties along with martingale characterization property. In section 3, we introduce two types of CPPoK with the help of MTSS and its right continuous inverse, and derive some important distributional properties.

Compound Poisson Process of order k and its properties
In this section, we introduce CPPoK and derive its distributional properties. First, we define the Poisson distribution of order k.
Definition 1 (Philippou [5]). Let N (k) ∼ P oK(λ), the Poisson distribution of order k (PoK) with rate parameter λ > 0, then the probability mass function (pmf ) of N (k) is given by where the summation is taken over all non-negative integers x 1 , x 2 , . . . , x k such that Philippou [5] showed the existence of PoK as a limiting distribution of negative binomial distribution of order k. Kostadinova and Minkova [7] later generalized PoK to evolve over time, in terms of, a process which can be defined as follows.
However, the clumping behavior associated with random phenomenon cannot be handled by PPoK [7]. Hence, there is a need to generalize this notion as well. We propose the following generalization of PPoK. Definition 3. Let {N (k) (t)} t≥0 be the P P oK(λ) and {Y i } i≥1 be a sequence of IID random variables, independent of N (k) (t), with cumulative distribution function (CDF) H. Then the process {Z(t)} t≥0 defined by Y i is called the compound Poisson process of order k (CPPoK) and is denoted by CP P oK(λ, H).
From the definition, it is clear that: (i) for k = 1, {Z(t)} t≥0 is CP P (λ, H) the usual compound Poisson process.
Next, we present a characterization of CP P oK(λ, H), in terms of, the finite dimensional distribution (FDD).
Proof. Let 0 = t 0 ≤ t 1 ≤ . . . ≤ t n = t be the partition of [0, t]. Since, the increments of {N (k) (t)} are independent and stationary, we can write The mean and variance of the process {Z(t)} t≥0 can be expressed as

Index of dispersion
In this subsection, we discuss the index of dispersion of CP P oK(λ, H).
Definition 4 (Maheshwari and Vellaisamy [11]). The index of dispersion for a counting process {Z(t)} t≥0 is defined by Then the stochastic process {Z(t)} t≥0 is said to be overdispersed if I(t) > 1, underdispersed if I(t) < 1, and equidispersed if I(t) = 1.
Alternatively, Definition 4 can be interpreted as follows. A stochastic process From the above definition, the following cases arise: (i) If Y ′ i s are over and equidispersed, then CP P oK(λ, H) exhibits overdispersion.

Long range dependence
In this subsection, we prove the long-range dependence (LRD) property for the CP P oK(λ, H). There are several definitions available in literature. We used the definition given in [10].
Proposition 2.1. The CP P oK(λ, H) has the LRD property.
, which decays like the power law t −1/2 . Hence CP P oK(λ, H) has LRD property.

Martingale characterization for CPPoK
It is well known that the martingale characterization for homogeneous Poisson process is called Watanabe theorem (see [3]). Now, we extend this theorem for CP P oK(λ, H), where H is discrete distribution with support on Z + and for this we need following two lemmas.
Proof. Let Z(t) be the F t adapted stochastic process. If Z(t) is a compound Poisson process of order k, then Hence, the process Y i . So the other part easily follows using [18, Theorem 5.2]. (5), it is proved that CPPoK is equal in distribution to {D(t)} t≥0 . Hence, CPPoK is also a Lévy process and hence infinitely divisible.

Remark 2.3. We know that CPP is a Lévy process and in
Remark 2.4. The characteristic function of CP P oK(λ, H) can be written as where, α j , j = 1, 2, . . . are as defined in Remark 2.2, and kλα j = ν j is called the Lévy measure of CP P oK(λ, H).

Main Results
In this section, we recall the definitions of Lévy subordinator and its first exit time. Further, we define the subordinated versions of CP P oK(λ, H) and discuss their properties.

Lévy Subordinator
A Lévy subordinator {D f (t)} t≥0 is a one-dimensional non-decreasing Lévy process whose Laplace transform (LT) can be expressed in the form (see [2]) where the function f : [0, ∞) → [0, ∞) is called the Laplace exponent and Here b is the drift coefficient and ν is a non-negative Lévy measure on positive half-line satisfying which ensures that the sample paths of D f (t) are almost surely (a.s.) strictly increasing. Also, the inverse subordinator {E f (t)} t≥0 is the first exit time of the Lévy subordinator {D f (t)} t≥0 , and it is defined as Next, we study CP P oK(λ, H) by taking subordinator as mixture of tempered stable subordinators (MTSS).
Then, by using Theorem 3.5, we have that In the similar manner, we can get the expression for variance.