Correlated fractional counting processes on a finite time interval
We present some correlated fractional counting processes
on a finite time interval. This will be done by considering a
slight generalization of the processes in
. The main case concerns a class
of space-time fractional Poisson processes and, when the
correlation parameter is equal to zero, the univariate
distributions coincide with the ones of the space-time fractional
Poisson process in . On the other
hand, when we consider the time fractional Poisson process, the
multivariate finite dimensional distributions are different from
the ones presented for the renewal process in
. Another case concerns a class of
fractional negative binomial processes.
Keywords: Poisson process, negative binomial
process, weighted process.
Mathematical Subject Classification: 60G22, 60G55, 60E05, 33E12.
Several fractional processes in the literature are defined by considering some known equations in terms of suitable fractional derivatives. In this paper we are interested in particular Lévy counting processes, as in the recent paper ; in particular we deal with Poisson and negative binomial processes. There is a wide literature on fractional Poisson processes: see e.g. , , , ,  and  (we also cite  and  where their representation in terms of randomly time-changed and subordinated processes was studied in detail). Some references with fractional negative binomial processes are  (see Example 3) and . Among the other fractional processes in the literature we recall the diffusive processes (see e.g. , ,  , ), the telegraph processes in  and the pure birth processes in .
Often the results for these fractional processes are given in terms of the Mittag-Leffler function
(see e.g. , page 17); we also recalled the generalized Mittag-Leffler function
is the rising factorial (also called Pochhammer symbol), and coincides with when .
In this paper we consider some processes on a finite time interval , for some . More precisely is defined by
where is a nonnegative integer valued random variable with probability generating function , i.e.
and is a sequence of random variables with (common) distribution function such that and , and independent of ; moreover the correlation coefficient between any pair of random variables and , with , is equal to a common value .
We have ; thus the distribution of does not depend on .
In this way we are considering a slight generalization of the processes presented in ; indeed we can recover several formulas in  by setting for some (which concerns a Poisson distributed random variable with mean ), and for , where . The case without correlation, i.e. the case , appears in ; see also  where that process is considered as a claim number process in insurance. Here, in view of what follows, we recall the following formulas (see e.g. (9) and (10) in ): we have the probability generating function
and the probability mass function
concerns the case (see (2.4) in ).
As pointed out in , this class of counting processes can be useful to tackle the problem of overdispersion and underdispersion in the analysis of count data where correlations between events are present. A possible application can be given for example in models of non-exponential extinction of radiation in correlated random media (see e.g. ). We also remark that, as far as the the marginal distribution of each random variable , in (2) we have a mixture between three probability mass functions, i.e. , and , and the weights are , and , respectively.
The aim of this paper is to present some correlated fractional counting processes by choosing in a suitable way the probability generating function and a distribution function above. In Section 2 we present a class of space-time fractional Poisson processes (in fact we have the same univariate distributions of the space-time fractional Poisson process in  when ). A class of fractional negative binomial processes is presented in Section 3.
Finally, since the presentation of the results in  refers to the concept of weighted Poisson processes (see also the previous reference  concerning the case ), in the final Section 4 we give some minor results on weighted processes. Even though this section seems to be disconnected from the other ones in this paper, in our opinion it is a small nice enrichment of the content of .
2 A class of correlated fractional Poisson processes
For the aims of this section, some preliminaries are needed. Firstly we consider the Caputo (left fractional) derivative of order (see e.g. in (2.4.14) and (2.4.15) in  with ; we use the notation ) defined by
note that, since here we consider , we have (see e.g. (2.4.17) in  with )
We also consider the (fractional) difference operator in ; more precisely is the identity operator, is the backward shift operator defined by and , and therefore
We now recall that Orsingher and Polito in  considered the space-time fractional Poisson process , for , whose probability mass functions solve the Cauchy problem
The explicit form of the probability generating function of this process has the form (see , equation (2.28))
In this section we consider a class of correlated space-time fractional Poisson processes on a finite time interval . For we consider such that the probability generating function of is
and the distribution function of the random variables is
In what follows we present the probability generating functions in Proposition 2.1 and the corresponding probability mass functions in Proposition 2.2. Moreover, in Proposition 2.3, we give an equation for the probability mass functions in Proposition 2.2 with respect to time .
The probability generating functions are
Proof. We have
by (1), and we conclude with some manipulations of the last term.
By Proposition 2.1, if we have the probability generating function
which coincides with the one presented in the last case of Table 1 in ; note that (6) is a generalization of (5) with instead of . Thus the univariate distributions of the random variables (for the case ) coincide with the ones of the random variables of the renewal process in  (restricted to the same finite time interval). On the other hand one can check that the multivariate finite dimensional marginal distributions are different from the ones in  (and, in fact, is not a renewal process). We explain this with a simple example where we take into account that
by (2.5) in . In fact, for , we have
by combining (11) and (14) in  (with and ) with (2) and (4) in the same reference, and
The probability mass functions are
Proof. Firstly we have
Then, by a well-known “Newton’s generalized binomial theorem”, we obtain
In view of the next Proposition 2.3 we remark that in a part of the proof we refer to Theorem 2 in  which can be derived by referring to a subordinated representation of the space-time fractional Poisson process in terms of both stable subordinator and its inverse (see also (3.20), together with (3.1), in the same reference).
Let be the probability mass functions in Proposition 2.2. Then we have the following equations: for ,
for all ,
In all cases we have the initial conditions and for all .
Proof. The initial conditions trivially hold. Throughout this proof we consider the notation
Moreover we have
by Theorem 2 in  and
(see e.g. (2.2.11) and (2.4.8) in , or a correction of (2.4.28) in the same reference); then we get
From now on we consider the cases and
Case . Firstly we have
by (4); therefore
then we get the desired equation by noting that
The proof is complete.
Finally we remark that, even if the equations in Proposition 2.3 have some analogies with other results for fractional Poisson processes in the literature, here some standard techniques do not work because we deal with a finite horizon time case (i.e. ).
3 A class of correlated fractional negative binomial processes
It is well-known that the negative binomial process can be seen as a suitable compound Poisson process with logarithmic distributed summands (see e.g. Proposition 1.1 in ). More precisely, for some and some integer , we have the probability generating function
is the probability generating function of a Poisson distributed random variable with mean ;
is the probability generating function of a logarithmic distributed random variable (obviously we have if ).
In this section we present a class of correlated fractional negative binomial processes on a finite time interval . More precisely we consider the same approach with the probability generating function of a space-time fractional Poisson distributed random variable; thus, for , we have
in place of (note that coincides with ), again with , and this meets in (5) with . Thus we have
As far as the distribution function is concerned, we argue as in Section 2 as follows: for all , we want to have the condition
so, if we divide both numerator and denominator by , we get
Moreover we have
and the function has to be a decreasing. We also give a particular example with a choice of , and we provide the corresponding distribution function .
In what follows we present the probability generating functions in Proposition 3.1 and, for only, the corresponding probability mass functions in Proposition 3.2 (for we have the -th convolution of the probability mass function of the case , but we cannot provide manageable formulas). Moreover, in Proposition 3.3, we give an equation for the probability generating functions in Proposition 3.1 for , and ; in this case we consider fractional derivatives with respect to their argument , and not with respect to time .
The probability generating functions are
Proof. This is an immediate consequence of
(1) and the formulas above.
is the Fox-Wright function (see e.g. (1.11.14) in ) under the convergence condition
(see e.g. (1.11.15) in ).
If , the probability mass functions are
where, for all ,
and (note that the convergence condition (12) holds because we have ).
Firstly, if as in the statement of the proposition, we have
and we immediately obtain (13) for . Moreover, if we prove that