VMSTA Modern Stochastics: Theory and Applications 2351-6054 2351-6046 2351-6046 VTeXMokslininkų g. 2A, 08412 Vilnius, Lithuania VMSTA165 10.15559/20-VMSTA165 Research Article Subordinated compound Poisson processes of order k SengarAyushi Singhayushisingh28nov@gmail.com UpadhyeNeelesh S.neelesh@iitm.ac.in∗ Department of Mathematics, Indian Institute of Technology Madras, Chennai-600036, Tamil Nadu, India Corresponding author. 2020 511202074395413 1052020 3182020 24102020 © 2020 The Author(s). Published by VTeX2020 Open access article under the CC BY license.

In this article, the compound Poisson process of order k (CPPoK) is introduced and its properties are discussed. Further, using mixture of tempered stable subordinators (MTSS) and its right continuous inverse, the two subordinated CPPoK with various distributional properties are studied. It is also shown that the space and tempered space fractional versions of CPPoK and PPoK can be obtained, which generalize the process defined in [Statist. Probab. Lett. 82 (2012), 852–858].

 Compound Poisson process of order k mixture of tempered stable subordinators martingale characterization 60G51 60G48 Introduction

The Poisson process 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, weighted Poisson distributions, fractional (time-changed) versions of Poisson processes (see [6, 21, 13, 2, 1], and references therein). A handful of researchers have also studied the distributions and processes of order k in [11, 23, 18]. In particular, the discrete distribution of order k, introduced by Philippou et al. (see [24]), includes binomial, geometric and negative binomial distributions of order k. These distributions play an important role in several areas, such as reliability, statistics, finance, and actuarial risk analysis (see [19, 29, 4]).

In risk theory, the total claim amount is usually modeled by using a compound Poisson process, say Zt=∑i=1N(t)Yi , where the compounding random variables Yi are iid and the number of claims N(t) , which are independent of {Yi}i≥1 , follows the Poisson distribution. Due to the restriction of single arrival in each inter-arrival time, the model is not suitable to use. Kostadinova and Minkova [10] introduced a Poisson process of order k (PPoK), which allows us to model the arrival in a group of size k. Recently, a time-changed version of Poisson processes of order k is studied by [29], which allows group arrivals and also accommodates the case of extreme events, which was not covered by [10]. In spite of its applicability, we observe that this model does not cover the phenomenon of underdispersion, where the variance is less than the mean (see [25, 32] and references therein). Therefore, a generalization of this model is essentially required 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. Recently, tempered stable processes and their inverses have been widely used for time-change (subordination) as their moments are finite and hence various real-life situations can be modeled easily (see [12, 26, 20]). Various versions of Poisson processes, using subordination like space and time-fractional Poisson processes have been studied in the literature (see [22, 17]). Hence, it is worth exploring the time-change of CPPoK with a special type of Lévy subordinator known as mixture of tempered stable subordinators, its right continuous inverse, and analyze some properties of these time-changed processes. These processes also generalize the process discussed in [22].

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. Further we derive some important distributional properties.

 Acronym
PoK    Poisson distribution of order k
PPoK    Poisson process of order k
PP    Poisson process
CPPoK    Compound Poisson process of order k
FDD    Finite-dimensional distribution
IID    Independent and identically distributed
MTSS    Mixture of tempered stable subordinators
TCPPoK   Time changed compound Poisson process of order k
pmf    Probability mass function
pgf    Probability generating function
LRD    Long-range dependence
Compound Poisson process of order <italic>k</italic> and its properties Compound Poisson process of order <italic>k</italic>

In this section, we introduce CPPoK and derive its distributional properties. First, we define the Poisson distribution of order k.

([<xref ref-type="bibr" rid="j_vmsta165_ref_023">23</xref>]).

Let N(k)∼PoK(λ) , the Poisson distribution of order k (PoK) with rate parameter λ>0 0$]]>, then the probability mass function (pmf) of N(k) is given by P[N(k)=n]=∑x1,x2,…,xk≥0∑j=1kjxj=ne−kλλx1+x2+..+xkx1!x2!...xk!,n=0,1,…, where the summation is taken over all non-negative integers x1,x2,…,xk such that x1+2x2+⋯+kxk=n .

Philippou [23] showed the existence of PoK as a limiting distribution of negative binomial distribution of order k. Kostadinova and Minkova [10] later generalized PoK to evolve over time, in terms of a process which can be defined as follows.

([<xref ref-type="bibr" rid="j_vmsta165_ref_010">10</xref>]).

Let {N(t)}t≥0 denote PP(kλ) , the Poisson process with rate parameter kλ , and {Xi}i≥1 be a sequence of independent and identically distributed (IID) discrete uniform random variables with support on {1,2,…,k} . Also, assume that {Xi}i≥1 and {N(t)}t≥0 are independent. Then {N(k)(t)}t≥0 , defined by N(k)(t)=∑i=1N(t)Xi is called the Poisson process of order k (PPoK) and is denoted by PPoK(λ) .

However, the clumping behavior associated with the random phenomenon in [8] cannot be accommodated by PPoK [10]. Hence, there is a need to generalize this notion as well. Now we propose the following generalization of PPoK.

Let {N(k)(t)}t≥0 be the PPoK(λ) and {Yi}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 Z(t)=∑i=1N(k)(t)Yi is called the compound Poisson process of order k (CPPoK) and is denoted by CPPoK(λ,H) .

From the definition, it is clear that:

for k=1 , {Z(t)}t≥0 is CPP(λ,H) the usual compound Poisson process.

for H=δ1 , the Dirac measure at 1, {Z(t)}t≥0 is PPoK(λ) .

for k=1 and H=δ1 , {Z(t)}t≥0 is PP(λ) .

 Next, we present a characterization of CPPoK(λ,H) , in terms of the finite-dimensional distribution (FDD).

Let {Z(t)}t≥0 be as defined in Definition 3. Then the FDD, denoted as FZ(t1),…,Z(tn)(y1,…,yn)=P[Z(t1)≤y1,…,Z(tn)≤yn] has the following form FZ(t1),…,Z(tn)(y1,…,yn)=∑j1,…jn ∏l=1npjl(Δtl)∫−∞v1…∫−∞vn ∏m=1nhY1∗jm(xm)dxm, where the summation is taken over all non-negative integers ji≥0,i=1,2,…,n , vk=yk−∑l=1k−1xl,k=1,…,n , Δtl=tl−tl−1 , h is the density/pmf of H, pjl is the pmf of PPoK(λ) , and (∗jn) represents the jn -fold convolution.

Let 0=t0≤t1≤⋯≤tn=t be the partition of [0,t] . Since, the increments of {N(k)(t)} are independent and stationary, we can write N(k)(ti)=∑l=1iN(k)(Δtl),i=1,…,n , and P[N(k)(t)=j]=pj(t),j=0,1,… . FZ(t1),…,Z(tn)(y1,…,yn)=P[ ∑i=1N(k)(t1)Yi≤y1,…, ∑i=1N(k)(tn)Yi≤yn]=∑j1,…jnP[ ∑i=1j1Yi≤y1,…, ∑i=1∑l=1njlYi≤yn] ∏l=1npjl(Δtl) Let us denote ∑i=1j1Yi=Y(j1),…,∑i=j1+⋯+jn−1+1j1+⋯+jnYi=Y(jn) , then it becomes FZ(t1),…,Z(tn)(y1,…,yn)=∑j1,…jn ∏l=1npjl(Δtl)P[Y(j1)≤y1,…, ∑l=1nY(jl)≤yn]=∑j1,…jn ∏l=1npjl(Δtl)∫−∞y1…∫−∞yn−∑l=1n−1xl ∏m=1nhY1∗jm(xm)dxm=∑j1,…jn ∏l=1npjl(Δtl)∫−∞v1…∫−∞vn ∏m=1nhY1∗jm(xm)dxm.  □

For n=1 , (1) reduces to P[Z(t)≤y]=∑j=0∞pj(t)∫−∞yhY1∗j(x)dx , which is the marginal distribution of CPPoK(λ, H).

For n=1 and H has discrete distribution, the probability mass function (pmf) of CPPoK(λ,H) , denoted as P[Z(t)=n]=Pn(t) is given as Pn(t)=P[Z(t)=n]= ∑m=0∞pm(t)hY1∗m(n), where pm(t)=P[N(k)(t)=m] , the pmf of PPoK(λ) and, hY1∗m is the m-fold convolution of pmf of Y1 . Difference-differential equation satisfied by the pmf of CPPoK(λ,H) is given as ddtPn(t)= ∑m=0∞hY1∗m(n)ddtpm(t)= ∑m=0∞hY1∗m(n)[−kλpm(t)+λ ∑j=1m∧kpm−j(t)] with the initial condition Pn(0)=hY1∗m(n),n=1,2,… and P0(0)=hY1∗m(0) and m∧k=min(m,k) .

Next, we present some examples of CPPoK(λ,H) by taking different distribution of H.

Let H has geometric distribution with parameter p∈(0,1] . Then the marginal distribution of CPPoK(λ,H) is given as FZ(t)(y)=P[Z(t)≤y]= ∑j=0∞pj(t)∫−∞yhY1∗j(x)dx= ∑j=0∞I1−p(j,y+1)pj(t), where Ix=B(x;a,b)B(a,b),0<x<1 is the regularized incomplete beta function, B(x;a,b)=∫0xta−1(1−t)b−1 is the incomplete beta function.

Let H has exponential distribution with parameter μ>0 0$]]>. Then the marginal distribution of CPPoK(λ,H) is given as FZ(t)(y)=P[Z(t)≤y]= ∑j=0∞pj(t)∫−∞yhY1∗j(x)dx= ∑j=0∞γ(j,μy)(j−1)!pj(t), where γ(s,x)=∫0xe−tts−1dt,x≥0 is the incomplete gamma function.

Here we plot the pmf of the considered process when Yi∼Geo(p) , p∈(0,1] . Fix t=1,λ=1,p=0.5,andk=1 .

Plot for P[Z(t)=n]

The mean, variance and covariance of the process {Z(t)}t≥0 can be expressed as

E[Z(t)]=E[Y]E[N(k)(t)] ,

Var[Z(t)]=Var(Y)E[N(k)(t)]+E[Y]2Var(N(k)(t)) .

Cov[Z(s),Z(t)]=Var(Y)E[N(k)(s∧t)]+E[Y]2Var[N(k)(s∧t)],wheres∧t=min(s,t) .

E[Z(t)]=E[∑i=1N(k)(t)Yi]=E[Y]E[N(k)(t)]=k(k+1)2λtE[Y] . Hence, part (i) is proved.

Next, we derive the expression for variance of Z(t) . Let pn(t) be the pmf of PPoK(λ) . Then Var[Z(t)]=E[Z(t)−E[Z(t)]]2= ∑n=0∞E[[Z(t)−E[Z(t)]]2|Nk(t)=n]pn(t)= ∑n=0∞ESn−E[Y]E[N(k)(t)]2pn(t),whereSn=∑i=1nYi= ∑n=0∞E[(Sn−nE[Y])+(nE[Y]−E[Y]E[N(k)(t)])]2pn(t)= ∑n=0∞Var(Sn)+E[Y]2E[n−E[N(k)(t)]]2pn(t)=Var(Y)E[N(k)(t)]+E[Y]2Var(N(k)(t)), which proves part (ii) .

Now, in order to derive the covariance term, first we evaluate E[Z(s)Z(t)] . So E[Z(s)Z(t)]=E[Z(s)]E[Z(t)−Z(s)]+E[Z(s)2]=E[Z(s)]E[Z(t)−Z(s)]+Var[Z(s)]+E[Z(s)]2 Therefore, Cov(Z(s),Z(t))=E[Z(s)Z(t)]−E[Z(s)]E[Z(t)]=Var(Y)E[N(k)(s∧t)]+E[Y]2Var[N(k)(s∧t)], hence, part (iii) is proved.  □

Now we present the mean and covariance formula for some specific cases of CPPoK(λ,H) that we discussed in Definition 3.
S. No. CPPoK(λ,H) Mean Covariance Variance
1. For H=δ1 E[N(k)(t)] Var[N(k)(s∧t)] Var[N(k)(t)]
2. For k=1 E[Y]E[N(t)] E[Y2]E[N(s∧t)] E[Y2]E[N(t)]
3. For H=δ1 and k=1 E[N(t)] Var[N(s∧t)] Var[N(t)]

From this table, we make the following observations.

For H=δ1 , it reduces to mean and covariance formula for PPoK(λ) (see [29]).

For k=1 , it reduces to mean and covariance formula for CPP(λ,H) .

For H=δ1 and k=1 , it reduces to mean and covariance formula for PP(λ) .

 Further we present the plots of mean and variance when Yi∼exp(μ) , μ>0 0$]]> for different setting of parameters. Fix t=10 .

Plots for mean and variance

 Index of dispersion

In this subsection, we are going to discuss the index of dispersion of CPPoK(λ,H) .

([<xref ref-type="bibr" rid="j_vmsta165_ref_016">16</xref>]).

The index of dispersion for a counting process {Z(t)}t≥0 is defined by I(t)=Var[Z(t)]E[Z(t)]. Then the stochastic process {Z(t)}t≥0 is said to be overdispersed if I(t)>1 1$]]>, underdispersed if I(t)<1 , and equidispersed if I(t)=1 .

Alternatively, Definition 4 can be interpreted as follows. A stochastic process {Z(t)}t≥0 is over(under)-dispersed if Var[Z(t)]−E[Z(t)]>0(<0) 0(<0)$]]>. Therefore, we first calculate Var[Z(t)]−E[Z(t)]=k(k+1)2λtVar[Y]−E[Y]+(2k+1)3E[Y]2=k(k+1)2λtE[Y2]−E[Y]+2k−23E[Y]2. From the above definition, the following cases arise:

If Yi′s are over and equidispersed, then CPPoK(λ,H) exhibits overdispersion.

If Yi′s are underdispersed with non-negative integer valued random variable i.e., [E(Y2)−E(Y)]≥0 , then CPPoK(λ,H) shows overdispersion.

If Yi′s are underdispersed, then CPPoK(λ,H) may show over, equi and underdispersion.

Further we present some examples to discuss the index of dispersion.

If Yi∼exp(μ) , where μ>0 0$]]>, then (2) becomes Var[Z(t)]−E[Z(t)]=k(k+1)2μλt1μ2k+43−1.

If 0<μ≤1 , then CPPoK(λ,H) exhibits overdispersion.

If 1<μ<2k+43 , then CPPoK(λ,H) shows overdispersion.

If μ=2k+43 , then CPPoK(λ,H) shows equidispersion.

If μ>2k+43 \frac{2k+4}{3}$]]>, then CPPoK(λ,H) shows underdispersion.

If Yi∼Geo(p) , where p∈(0,1] , then (2) becomes Var[Z(t)]−E[Z(t)]=k(k+1)λt21−pp21+2k+13. If 0<p<1 , then CPPoK(λ,H) exhibits overdispersion.

 Long range dependence

In this subsection, we prove the long-range dependence (LRD) property for the CPPoK(λ,H) . There are several definitions available in literature. We used the definition given in [15]. ([<xref ref-type="bibr" rid="j_vmsta165_ref_015">15</xref>]).

Let 0≤s<t and s be fixed. Assume a stochastic process {X(t)}t≥0 has the correlation function Corr[X(s),X(t)] that satisfies c1(s)t−d≤Corr[X(s),X(t)]≤c2(s)t−d, for large t,d>0,c1(s)>0andc2(s)>0 0,\hspace{2.5pt}{c_{1}}(s)>0\hspace{2.5pt}and\hspace{2.5pt}{c_{2}}(s)>0$]]>. i.e., limt→∞Corr[X(s),X(t)]t−d=c(s) for some c(s)>0 0$]]> and d>0 0$]]>. We say that, X(t) has the long-range dependence (LRD) property if d∈(0,1) and short-range dependence (SRD) property if d∈(1,2) .

The CPPoK(λ,H) has the LRD property.

Let 0≤s<t . Consider Corr[Z(s),Z(t)]=Cov[Z(s),Z(t)]Var[Z(s)]Var[Z(t)],=k(k+1)2λsVar(Y)+k(k+1)(2k+1)6λsE[Y]2k(k+1)2λtVar(Y)+k(k+1)(2k+1)6λtE[Y]2,from Theorem 2=c(s)t−1/2, where, 0<c(s)=k(k+1)2λsVar(Y)+k(k+1)(2k+1)6λsE[Y]2k(k+1)2λVar(Y)+k(k+1)(2k+1)6λE[Y]2 , which decays like the power law t−1/2 . Hence CPPoK(λ,H) has LRD property.  □

Let Yi≡1 , then CPPoK(λ,H) reduces to PPoK(λ) and Corr[Z(s),Z(t)] becomes s1/2t−1/2 . Hence LRD property also holds for PPoK (see [29, Lemma 3.1]).

 Martingale characterization for CPPoK

It is well known that the martingale characterization for homogeneous Poisson process is called Watanabe theorem (see [5]). Now, we extend this theorem for CPPoK(λ,H) , where H is discrete distribution with support on Z+ and for this, we need following two lemmas.

Let D(t)=∑j=1N(t)Xj,t≥0 be the compound Poisson process, where {N(t)}t≥0 is PP(kλ) and {Xj}j≥1 are non-negative IID random variable, independent from {N(t)}t≥0 , with pmf P(Xj=i)=αi,(i=0,1,2,…,j=1,2,…) . Then {D(t)}t≥0 can be represented as D(t)=d ∑i=1∞iZi(t),t≥0, where, Zi(t),i=1,2,… are independent PP(kλαi) , and the symbol =d denotes the equality in distribution.

We prove this lemma by showing that the probability generating function (pgf) of L.H.S. and R.H.S. coincides. Let GD(t)(u) is the pgf of {D(t)}t≥0 , then GD(t)(u)=E[uD(t)]= ∑n=0∞E[uX1]nP[N(t)=n]=exp[−λkt(1−E[uX])]=exp[λkt ∑j=1∞αj(uj−1)]. Now we compute the pgf of ∑i=1∞iZi(t) , i.e., E[u∑i=1∞iZi(t)]= ∏i=1∞E[uiZi(t)]= ∏i=1∞ ∑ni=0∞uiniP[Zi(t)=ni]= ∏i=1∞ ∑ni=0∞uinie−kλαit(kλαit)nini!=exp[λkt ∑i=1∞αi(ui−1)]. Hence, this lemma is proved.  □

The pgf of Z(t)=∑i=1N(k)(t)Yi,t≥0 has the following form GZ(t)(u)=expλkt ∑j1=1∞qj1(1)+qj1(2)+⋯+qj1(k)k(uj1−1), where Yi,i=1,2,… are non-negative integer valued IID random variables with P[Yi=n]=qn,n=0,1,… .

Let GN(k)(t)(u) is the pgf of {N(k)(t)}t≥0 , which is given by (see [29, Remark 2.2]) GN(k)(t)(u)=exp[−kλt+λt{u+u2+⋯+uk}]. The pgf of {Z(t)}t≥0 , denoted as GZ(t)(u)=E[uZ(t)] , is then given by GZ(t)(u)=exp[λt{GY(u)+⋯+GY∗k(u)}−kλt],whereGY(u)=E[uY]=expλt ∑j1=0∞qj1uj1+ ∑j1=0∞qj1∗2uj1+⋯+ ∑j1=0∞qj1∗kuj1−kλt=expλt ∑j1=0∞qj1uj1+⋯+ ∑j1=0∞… ∑jk=0jk−1qjk…qj1−j2uj1−kλtLet us denoteqj1(n)= ∏m=2n ∑jm=0jm−1qjnqjn−1−jn…qj1−j2,n=1,…,k.=expλt ∑j1=0∞qj1(1)(uj1−1)+⋯+λt ∑j1=0∞qj1(k)(uj1−1)=expλkt ∑j1=1∞qj1(1)+qj1(2)+⋯+qj1(k)k(uj1−1).  □

Set αj1=qj1(1)+⋯+qj1(k)k,j1=0,1,2,… . Substituting αj1 in Lemma 1, we get the following relation D(t)=d ∑i=1∞iZi(t)=d ∑i=1N(k)(t)Yi,t≥0, where Yi′s are non-negative integer valued random variables and N(k)(t) is PPoK(λ) .

Let {Z(t)}t≥0 be the Ft -adapted stochastic process, where {Ft} is non-decreasing family of sub-sigma algebras. Then {Z(t)}t≥0 is a CPPoK(λ,H) , where H is discrete distribution with support on Z+ iff process M(t)=Z(t)−k(k+1)2λtE[Y],t≥0 is an Ft martingale.

Let {Z(t)}t≥0 be the Ft adapted stochastic process. If {Z(t)}t≥0 is a compound Poisson process of order k, then E[M(t)|Fs]=E[Z(t)−k(k+1)2λtE[Y]|Fs],0≤s≤t.=E[Z(t)|Fs]−k(k+1)2λtE[Y]=E[Z(t)−Z(s)|Fs]+E[Z(s)|Fs]−k(k+1)2λtE[Y]=E[Z(t)−Z(s)]+Z(s)−k(k+1)2λtE[Y]=Z(s)−k(k+1)2λsE[Y]=M(s). Hence, the process {M(t)}t≥0 is Ft martingale.

Since in Remark 4 it is shown that ∑i=1∞iZi(t)=d∑i=1N(k)(t)Yi , then the other part easily follows using [31, Theorem 5.2].  □

We know that CPP is a Lévy process and in (4) 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.

The characteristic function of CPPoK(λ,H) can be written as E[eiwZ(t)]=exp[t ∑j=1∞(eiwj−1)kλαj], where, αj,j=1,2,… are as defined in Remark 4, and kλαj=νj is called the Lévy measure of CPPoK(λ,H) .

For K=1 , (5) reduces to E[eiwZ(t)]=exp[t∑j=1∞(eiwj−1)λαj] , which is the characteristic function of CPP(λ,H) .

For H=δ1 , (5) reduces to E[eiwZ(t)]=exp[λkt∑j=1k(eiwjk−1)] , which is the characteristic function of PPoK(λ) .

For H=δ1 and k=1 , (5) reduces to E[eiwZ(t)]=exp[λt(eiw−1)] , which is the characteristic function of PP(λ) .

Main results

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

Lévy subordinator

A Lévy subordinator {Df(t)}t≥0 is a one-dimensional non-decreasing Lévy process whose Laplace transform (LT) can be expressed in the form (see [3]) E[e−λDf(t)]=e−tf(λ),λ>0, 0,\]]]> where the function f:[0,∞)→[0,∞) is called the Laplace exponent and f(λ)=bλ+∫0∞(1−e−λx)ν(dx),b≥0. Here b is the drift coefficient and ν is a non-negative Lévy measure on positive half-line satisfying ∫0∞(x∧1)ν(dx)<∞andν([0,∞))=∞, which ensures that the sample paths of Df(t) are almost surely (a.s.) strictly increasing. Also, the inverse subordinator {Ef(t)}t≥0 is the first exit time of the Lévy subordinator {Df(t)}t≥0 , and it is defined as Ef(t)=inf{r≥0:Df(r)>t},t≥0. t\},\hspace{2.5pt}t\ge 0.\]]]> Next, we study CPPoK(λ,H) by taking subordinator as mixture of tempered stable subordinators (MTSS).

 CPPoK time changed by mixtures of tempered stable subordinators

The mixtures of tempered stable subordinators (MTSS) {Sα1,α2μ1,μ2(t)}t≥0 is a Lévy process with LT (see [7]) E[e−sSα1,α2μ1,μ2(t)]=exp{−t(c1((s+μ1)α1−μ1α1)+c2((s+μ2)α2−μ2α2))},s>0, 0,\]]]> where c1+c2=1,c1,c2≥0,μ1,μ2>0 0$]]> are tempering parameters and α1,α2∈(0,1) are stability indices. The function f(s)=c1((s+μ1)α1−μ1α1)+c2((s+μ2)α2−μ2α2) is the Laplace exponent of MTSS. It can also be represented as sum of two independent tempered stable subordinators Sα1μ1(t) and Sα2μ2(t) as Sα1,α2μ1,μ2(t)=Sα1μ1(c1t)+Sα2μ2(c2t),c1,c2≥0. The mean and variance of MTSS are given as E[Sα1,α2μ1,μ2(t)]=t(c1α1μ1α1−1+c2α2μ2α2−1), Var[Sα1,α2μ1,μ2(t)]=t(c1α1(1−α1)μ1α1−2+c2α2(1−α2)μ2α2−2).

Let {Sα1,α2μ1,μ2(t)}t≥0 be the Lévy subordinator satisfying E[Sα1,α2μ1,μ2(t)ρ]<∞ for all ρ>0 0$]]>. Then the time-changed CPPoK(λ,H) , denoted by TCPPoK(λ,H,Sα1,α2μ1,μ2) is defined as Z1(t)=Z(Sα1,α2μ1,μ2(t))= ∑i=1N(k)(Sα1,α2μ1,μ2(t))Yi,t≥0, where {Z(t)}t≥0 is CPPoK(λ,H) , independent from {Sα1,α2μ1,μ2(t)}t≥0 .

If α1=α2=α , and μ1=μ2=0 , then MTSS becomes α-stable subordinator, reducing Z1(t) to TCPPoK(λ,H,Sα) , which we call as space fractional CPPoK and written as Z(Sα(t))= ∑i=1N(k)(Sα(t))Yi,t≥0. Let Yi′s≡1 , then Z(Sα(t)) reduces to space fractional PPoK, denoted as PPoK(λ,Sα) . Its pgf is given as GZ1(t)(u)=e[−tλα(k−∑j=1kuj)α],t≥0. It can be seen as extension of space fractional Poisson process (see [22]).

If α1=α2=α , and μ1=μ,μ2=0 , then MTSS reduces to tempered α- stable subordinator and Z1(t) becomes tempered space fractional CPPoK, denoted as TCPPoK(λ,H,Sαμ) , can be written as Z(Sαμ(t))= ∑i=1N(k)(Sαμ(t))Yi,t≥0, where μ>0 0$]]> is tempering parameter. Substituting Yi′s≡1 , it becomes tempered space fractional PPoK, denoted as PPoK(λ,Sαμ) .

The finite dimensional distribution of TCPPoK(λ,H,Sα1,α2μ1,μ2) has the following form FZ1(t1),…,Z1(tn)(y1,…,yn)=∑j1,…jn ∏l=1nqjl(Δtl)∫−∞v1…∫−∞vn ∏m=1nhY1∗jm(xm)dxm, where the summation is taken over all non-negative integers ji≥0,i=1,2,…,n , Δtl=tl−tl−1 , vk=yk−∑l=1k−1xl,k=1,…,n , h is the density of H, and qj(t)=P[N(k)(Sα1,α2μ1,μ2(t))=j] , where {Sα1,α2μ1,μ2(t)}t≥0 is the MTSS and {N(k)(t)}t≥0 is PPoK(λ) .

The result easily follows from the proof of Theorem 1.  □

Now, we present some distributional properties of TCPPoK(λ,H,Sα1,α2μ1,μ2) .

Let 0<s≤t<∞ . Then the mean and covariance function of TCPPoK(λ,H,Sα1,α2μ1,μ2) are given as

E[Z1(t)]=E[Z(1)]E[Sα1,α2μ1,μ2(t)] ,

Cov[Z1(s),Z1(t)]=E[Z(1)]2Var[Sα1,α2μ1,μ2(s)]+E[Sα1,α2μ1,μ2(s)]Var[Z(1)] .

 On putting s=t in part (ii), we can get the expression for variance of TCPPoK(λ,H,Sα1,α2μ1,μ2) .

The proof follows from (see [14, Theorem 2.1]).  □

Here are some specific cases of mean and covariance for CPPoK(λ,H,Sα1,α2μ1,μ2) that we discussed in Definition 3.

For H=δ1 , (i) reduces to E[Z1(t)]=E[N(k)(1)]E[Sα1,α2μ1,μ2(t)] and (ii) reduces to Cov[Z1(s),Z1(t)]=E[N(k)(1)]2Var[Sα1,α2μ1,μ2(s)]+E[Sα1,α2μ1,μ2(s)]Var[N(k)(1)] which is mean and covariance of TCPPoK-I as discussed in (see [29, Theorem 3.2]).

For k=1 , (i) reduces to E[Z1(t)]=λE[Y]E[Sα1,α2μ1,μ2(t)] and (ii) reduces to Cov[Z1(s),Z1(t)]=(λE[Y])2Var[Sα1,α2μ1,μ2(s)]+E[Sα1,α2μ1,μ2(s)]λE[Y2] which is the mean and covariance of time-changed CPP(λ,H) .

For H=δ1 and k=1 , (i) reduces to E[Z1(t)]=E[N(1)]E[Sα1,α2μ1,μ2(t)] and (ii) reduces to Cov[Z1(s),Z1(t)]=E[N(1)]2Var[Sα1,α2μ1,μ2(s)]+E[Sα1,α2μ1,μ2(s)]Var[N(1)] which is the mean and covariance of time-changed PP(λ) .

Now, we discuss the index of dispersion for TCPPoK(λ,H,Sα1,α2μ1,μ2) . For this, we evaluate Var[Z1(t)]−E[Z1(t)]=E[Z(1)]2Var[Sα1,α2μ1,μ2(t)]+E[Sα1,α2μ1,μ2(t)]Var[Z(1)]−E[Z(1)]. Since {Sα1,α2μ1,μ2(t)}t≥0 is a Lévy subordinator, therefore E[Sα1,α2μ1,μ2(t)]>0 0$]]>. Thus the following cases arises:

If Z(1) is over/equidispersed, then TCPPoK(λ,H,Sα1,α2μ1,μ2) exhibits overdispersion.

If Z(1) is underdispersed, then TCPPoK(λ,H,Sα1,α2μ1,μ2) may show over, under and equidispersion.

 Moreover, we discuss the index of dispersion by taking some example of TCPPoK(λ,H,Sα1,α2μ1,μ2) .

When H has exponential distribution with parameter μ>0 0$]]>, (9) becomes Var[Z1(t)]−E[Z1(t)]=Var[Sα1,α2μ1,μ2(t)]k(k+1)λ2μ2+E[Sα1,α2μ1,μ2(t)]×k(k+1)λ2μ2k+43μ−1.

If 0<μ≤1 , then Z(1) is overdispersed. Since the sample paths of Lévy subordinator Sα1,α2μ1,μ2(t) are strictly increasing (see [28, Theorem 21.3]), so E[Sα1,α2μ1,μ2(t)] is positive, and both the terms of (10) is positive. Therefore Z1(t) shows overdispersion.

If μ=2k+43 , then Z(1) is equidispersed. Second term becomes zero but the first terms of (10) is positive. Therefore Z1(t) shows overdispersion.

If μ>2k+43 \frac{2k+4}{3}$]]>, then Z(1) is underdispersed. So the second term in (10) becomes negative and the following cases arises

if k(k+1)λ2μ2Var[Sα1,α2μ1,μ2(t)]>E[Sα1,α2μ1,μ2(t)]k(k+1)λ2μ1−2k+43μ \mathbb{E}[{S_{{\alpha _{1}},{\alpha _{2}}}^{{\mu _{1}},{\mu _{2}}}}(t)]\frac{k(k+1)\lambda }{2\mu }\left[1-\frac{2k+4}{3\mu }\right]$]]>, then Z1(t) shows overdispersion.

if k(k+1)λ2μ2Var[Sα1,α2μ1,μ2(t)]=E[Sα1,α2μ1,μ2(t)]k(k+1)λ2μ1−2k+43μ , then Z1(t) shows equidispersion.

if k(k+1)λ2μ2Var[Sα1,α2μ1,μ2(t)]<E[Sα1,α2μ1,μ2(t)]k(k+1)λ2μ1−2k+43μ , then Z1(t) shows underdispersion.

Long-range dependence

Now we analyze the LRD property for TCPPoK(λ,H,Sα1,α2μ1,μ2) .

Let {Z1(t)}t≥0 be the time-changed CPPoK(λ,H) . It has the LRD property.

We have E[Sα1,α2μ1,μ2(t)n]∼(c1α1λ1α1−1+c2α2λ2α2−1)ntn , as t→∞ , from [7]. Therefore Var[Z1(t)]∼E[Sα1,α2μ1,μ2(t)]Var[Z(1)]∼Kt, where K=(c1α1λ1α1−1+c2α2λ2α2−1)Var[Z(1)] .

Let 0<s<t<∞ , then Corr[Z1(s),Z1(t)]=Cov[Z1(s),Z1(t)]Var[Z1(s)]Var[Z1(t)],∼Cov[Z1(s),Z1(t)]t1/2K,from (11)=c(s)t−1/2, where c(s)=Cov[Z1(s),Z1(t)]K>0 0$]]>. Hence from the Definition 5, TCPPoK(λ,H,Sα1,α2μ1,μ2) captures LRD property.  □

 CPPoK time changed by the first exit time of mixtures of tempered stable subordinators

In this subsection, we consider CPPoK(λ,H) subordinated with first exit time of MTSS and discuss the asymptotic behavior of its moments.

The first exit time of Lévy subordinator Sα1,α2μ1,μ2(t) also known as inverse subordinator is defined as Eα1,α2μ1,μ2(t)=inf{r≥0:Sα1,α2μ1,μ2(r)>t},t≥0. t\},\hspace{2.5pt}t\ge 0.\]]]>

Let {Z(t)}t≥0 be the CPPoK(λ,H) as discussed in Definition 3, then the subordinated CPPoK with first exit time of MTSS, denoted as TCPPoK(λ,H,Eα1,α2μ1,μ2) is defined as Z2(t)=Z(Eα1,α2μ1,μ2(t))= ∑i=1N(k)(Eα1,α2μ1,μ2(t))Yi,t≥0 where the process {Z(t)}t≥0 is independent from {Eα1,α2μ1,μ2(t)}t≥0 .

If μ1=μ2=0 , then Eα1,α2λ1,λ2(t) becomes inverse mixed stable subordinator as discussed in [9], reducing Z2(t) to TCPPoK(λ,H,Eα1,α2) , which we call as mixed fractional CPPoK and written as Z2(t)= ∑i=1N(k)(Eα1,α2(t))Yi,t≥0.

The marginal distribution of TCPPoK(λ,H,Eα1,α2μ1,μ2) is given as FZ2(t)(y)=∫0∞FZ(x)(y)gE(x,t)dx, where FZ(x)(y)=∑j=0∞pj(x)∫−∞yhY1∗j(z)dz is the marginal distribution of CPPoK(λ,H) , and gE(x,t) is the density function of Eα1,α2μ1,μ2(t) .

We prove this proposition by using the conditioning argument on Eα1,α2μ1,μ2(t) . Then the rest follows from Theorem 1.  □

The mean and covariance function of TCPPoK(λ,H,Eα1,α2μ1,μ2) are given as

E[Z2(t)]=E[Z(1)]E[Eα1,α2μ1,μ2(t)] ,

Cov[Z2(s),Z2(t)]=Var[Z(1)]E[Eα1,α2μ1,μ2(s)]+E[Z(1)]2Cov[Eα1,α2μ1,μ2(s),Eα1,α2μ1,μ2(t)] .

The proof follows as given in [14, Theorem 2.1].  □

Now, we discuss the asymptotic behavior of moments of TCPPoK(λ,H,Eα1,α2μ1,μ2) . First we need the following Theorem (see [3, 30]).

(Tauberian Theorem).

Let l:(0,∞)→(0,∞) be a slowly varying function at 0 (respectively ∞) and let ρ≥0 . Then for a function U:(0,∞)→(0,∞) , the following are equivalent.

U(x)∼xρl(x)/Γ(1+ρ) , x→0 (respectively x→∞ ).

U˜(s)∼s−ρ−1l(1/s) , s→∞ (respectively s→0 ), where U˜(s) is the LT of U(x) .

We know that LT of pth order moment of inverse subordinator {Ef(t)}t≥0 is given by (see [16]) L[E(Ef(t))p]=Γ(1+p)s(f(s))p,p>0, 0,\]]]> where f(s) is the corresponding Bernstein function associated with Lévy subordinator.

The asymptotic form of mean and variance of {Z2(t)}t≥0 is given as E[Z2(t)]∼E[Z(1)]tc1α1λ1α1−1+c2α2λ2α2−1,ast→∞.Var[Z2(t)]∼Var[Z(1)]tc1α1λ1α1−1+c2α2λ2α2−1,ast→∞.

Let M˜(s) be the LT of E[Eα1,α2μ1,μ2(t)] . Then M˜(s)=L[E[Eα1,α2μ1,μ2(t)]]=1s(c1((s+λ1)α1−λ1α1)+c2((s+λ2)α2−λ2α2))∼1s2(c1α1λ1α1−1+c2α2λ2α2−1),ass→0, (see [7]). Then, by using Theorem 8, we have that E[Z2(t)]=E[Z(1)]E[Eα1,α2μ1,μ2(t)]∼E[Z(1)]tc1α1λ1α1−1+c2α2λ2α2−1,ast→∞. Now we compute the asymptotic behavior of variance of {Z2(t)}t≥0 . So L[E(Eα1,α2μ1,μ2(t))2]=2s(c1((s+λ1)α1−λ1α1)+c2((s+λ2)α2−λ2α2))2∼2s3(c1α1λ1α1−1+c2α2λ2α2−1)2,ass→0. Therefore VarZ2(t)=Var[Z(1)]E[Eα1,α2μ1,μ2(t)]+E[Z(1)]2Var[Eα1,α2μ1,μ2(t)]=Var[Z(1)]E[Eα1,α2μ1,μ2(t)]+E[Z(1)]2{E[Eα1,α2μ1,μ2(t)2]−E[Eα1,α2μ1,μ2(t)]2}∼Var[Z(1)]tc1α1λ1α1−1+c2α2λ2α2−1,ast→∞.  □

There are few studies available in the literature which consider underdispersion phenomenon such as [27] uses hyper-Poisson regression model, [25] uses weighted Poisson distribution, and [21] uses compound weighted Poisson distributions. In this work, we have proposed a model accomodating more number of parameters, which makes this model more flexible to use.

 Acknowledgments

The authors would like to thank Aditya Maheshwari (IIM, Indore) for being part of various discussions on this topic. Also, the authors are grateful to the referees for suggesting various improvements which certainly improved the presentation of the article.

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