Let {L(t),t≥0} be a Lévy process with representative random variable L(1) defined by the infinitely divisible logarithmic series distribution. The distribution of any Lévy process is completely determined by the distribution of its representative random variable, which is infinitely divisible [22]. The probability generating function (p.g.f.) of the random variable L(1) is expressed by the Gauss hypergeometric function 2F1(1,1;2;z) [10]. This makes the usage of enumerative combinatorics methods indispensable in this study [19]. Thus, using the partial Bell polynomials we obtain an explicit representation of the transition probability and Lévy measure of this process. But, first of all, we distinguish two logarithmic distributions.

The logarithmic series distribution supported by positive integers N={1,2,…} was firstly introduced by R.A. Fisher, A.S. Corbet and C.B. Williams (1943) [9]. It is not an infinitely divisible distribution. It is a Lévy measure for the well-known Negative-Binomial process. The paper [9] represents an impressive combination of empirical data and mathematical analysis, remaining a model for ecology today.

The logarithmic series distribution supported by nonnegative integers Z+={0,1,…} is a particular case of the Kemp generalized hypergeometric probability distribution (1956) [12]. Its infinite divisibility was proved by K. Katti (1967) [11]. Infinitely divisible random variables with values in Z+ were first studied by Feller (1968) [8], where it is shown that on Z+ the infinitely divisible distributions coincide with the compound Poisson distributions. A historical review on the origin of infinitely divisible distributions “from de Finetti’s problem to Lévy–Khintchine formula” is presented by F. Mainardi and S. Rogosin (2006) [15].

At the present time, there are many works related to the topics of infinite divisibility and discrete distributions. Some of them are monographs of F.W. Steutel and K. Van Harn [22], and N.L. Johnson, A.W. Kemp and S. Kotz [10]. Integer-valued Lévy processes and their application in financial econometrics are developed by O.E. Barndorff-Nielsen, D. Pollard and N. Shephard [1]. The compositions of Poisson and Gamma processes are investigated by K. Buchak and L. Sakhno in [4, 5]. The consecutive subordinations of Poisson and Gamma processes realized on two sequences containing only four new processes are studied in [17]. It is shown there how the additional randomness, caused by random time, is accumulated. The transformation of the Poisson random measure and the jump-structure of the subordinated process is described in [16]. Other interesting integer-valued Markov processes are derived from Markov branching processes. In the model of branching particle system with a random initial condition, we obtained distributions describing the number of particles at time t, corresponding to the compound Poisson processes over the radius of a flux of particles [18].

The subordination by Bochner is also developed in many books and articles related to applications in financial mathematics and functional analysis, see [20, 7]. There is a special Chapter 3 in [3] devoted to the subordinators. The Lévy measure and potential kernel are also considered in [2]. The properties of the Bernstein functions are studied in [21]. The theory of subordinators and inverse subordinators is applied to study risk processes in insurance models by N. Leonenko et al. in [13, 14].

Our work in the topic is described in the following sections. In Section 2 we introduce the infinitely divisible logarithmic series distribution and its m-fold convolution. Our main tools of investigation are the Gauss hypergeometric function and partial Bell polynomials, Stirling numbers and harmonic numbers. In Section 3 we present two methods defining the transition probability of the Lévy process L(t) – starting with the Lévy measure or starting with p.g.f. F(t,s)=E[sL(t)] and its Taylor expansion. Then, in the following two Sections 4 and 5 we consider the subordinated processes Y(t) and Z(t) . They are obtained respectively from the Negative-Binomial process X(t) directed by the Gamma one and the Logarithmic Lévy process L(t) directed by the Poisson one. In this study of subordinated processes, we proceed also by two methods – either integrating the transition probability of the ground process, as it is shown in [20], or constructing the compound Poisson process with a prior defined Lévy measure. The Bernstein functions of the processes L(t) and Y(t) contain the iterated logarithmic function. The Lévy measure of Z(t) is a shifted Lévy measure of X(t) . We compare the behaviour of all these processes in order to understand better the place of the Logarithmic Lévy process in the picture of compound Poisson processes. Several combinatorics identities arrive as auxiliary results. Finally, some applications derived from studied processes are explained and demonstrated in Section 6.

The Gauss hypergeometric function is defined in [10], page 20, for |z|<1 by 2F1(c,d;g;z)= ∑k=0∞[c]k↑[d]k↑[g]k↑zkk!, where the increasing factorial, known as Pochhammer’s symbol, is denoted as [c]k↑=c(c+1)⋯(c+k−1) , [c]0↑=1 . In particular, 2F1(1,1;2;z)= ∑k=0∞k!k!(k+1)!zkk!=−log(1−z)z. By definition, given in [22], Chapter 2, Example 11.7, the infinitely divisible random variable L(1) with logarithmic series distribution has the probability mass function (p.m.f.) supported by {0,1,2,…} and given as follows: (1) P(L(1)=k)=1Aαk+1(k+1),0<α<1,A=−log(1−α),k=0,1,…. The p.g.f. defined by F(1,s)=E[sL(1)] , |s|≤1 , is F(1,s)= ∑k=0∞αk+1k+1skA=αA ∑k=0∞(αs)kk+1=−log(1−αs)As and can be presented as follows: (2) F(1,s)=αA(1+G(s)),G(s)= ∑k=1∞(αs)kk+1,G(1)=A−αα. We remark that the following simple representation is a starting point of the Taylor expansion, 2F1(1,1;2;αs)=1+G(s). Let us denote the finite sum of random variables L1,L2,…,Lm , being independent copies of L(1) , as (3) L(m)= ∑j=1mLj. By convolution of p.m.f. we express the probability distribution of the random variable L(2) by means of the harmonic numbers as follows: P(L(2)=n)=αA22αnn+21+12+13+⋯+1n+1,n=0,1,2,…. Knowing the infinite divisibility of L(1) we write the p.g.f. of L(m) (3) by the power, F(m,s)={F(1,s)}m , namely: −log(1−αs)Asm=αA(1+G(s))m. We present here two methods on expanding F(m,s) as power series of s, expanding only (−log(1−αs))m , or with the binomial expanding of (αA(1+G(s)))m . For this purpose, the function G(s) is presented as an exponential generating function: (4) G(s)= ∑k=1∞gkskk!,gk=αkk!k+1. For convenience, we denote the sequences by bullet, as it is shown in [19], g∙=(g1,g2,…). Similarly, the sequences of the powers are expressed by a∙=(a,a2,…) , and in particular: 1∙=(1,1,…) . In both cases of expanding we use the Faa di Bruno formula and the partial Bell polynomials Bn,k [19], allowing to express the power [G(s)]k as follows: (5) (G(s))kk!=1k! ∑j=1∞gjsjj!k= ∑n=k∞Bn,k(g∙)snn!, where Bn,k(g∙)=∑(k1,k2,⋯,kn)n!g1k1…gnknk1!(1!)k1⋯kn!(n!)kn. The sum is over all partitions of n into k parts, that is over all nonnegative integer solutions (k1,k2,…,kn) of the equations: k1+2k2+⋯+nkn=n,k1+k2+⋯+kn=k. For example, Bn,1(x∙)=xn , Bn,n(x∙)=(x1)n and (6) Bn,k(a∙bx∙)=anbkBn,k(x∙). The falling factorials are defined as follows: [x]n↓=x(x−1)⋯(x−n+1)=Γ(x+1)Γ(x+1−n). Let us denote the Stirling numbers of the first kind by |s(n,k)| and s(n,k) , respectively, – unsigned |s(n,k)|:=Bn,k((∙−1)!)=Bn,k(0!,1!,2!,….), and signed s(n,k) depending on the parity of n−k given by s(n,k)=(−1)n−k|s(n,k)|. The Stirling numbers of the first kind transform the factorials into powers, (7) [x]n↑= ∑k=0n|s(n,k)|xk,[x]n↓= ∑k=0ns(n,k)xk. Thus, having these definitions and relations, the following useful lemma is formulated and proved.

The comparison of two expressions (8) and (9) leads to the following combinatorics identity: ∑k=1m∧n1(m−k)!Bn,k(c∙)=|s(m+n,m)|n!(m+n)!. Remark 1.

The harmonic numbers take part in the expansion of the hypergeometric functions. A complete review on summation formulas involving generalized harmonic numbers and Stirling numbers is given in [6]. The generalized harmonic numbers are defined as follows: Hn(k):=1+12k+13k+⋯+1nk,Hn(1)=Hn. For m=2,3,4,… , we use the following relations between Stirling numbers of the first kind and generalized harmonic numbers to calculate directly the convolution probability of L(m) and to confirm the previous combinatorics identity. For example, (10) |s(2+n,2)|=(n+1)!1+12+13+⋯+1n+1 and |s(n,3)|=(n−1)!2{(Hn−1)2−Hn−1(2)}, and |s(n,4)|=(n−1)!3!{(Hn−1)3−3Hn−1Hn−1(2)+2Hn−1(3)}. The general recurrence formula on this relation is given in [6].

The principal information on the behaviour of any Lévy process is given by the representative random variable and it is expressed by the canonical representation of the Bernstein function and the Lévy measure, [20, 21]. The Laplace transform of the process L(t) is given by E[e−λL(t)]=exp(−tψL(λ)),λ≥0, where the Laplace exponent is a Bernstein function defined by the random variable L(1) as follows: ψL(λ)=−logE[e−λL(1)],λ≥0. For integer-valued Lévy processes it is also convenient to work with probability generating functions, see [22], Chapter 2.

Let us denote the Lévy measure of the process L(t) by ΠL(n),n=1,2,… , its total mass by θL=∑n=1∞ΠL(n) , and its generating function by QL(s)= ∑n=1∞snΠL(n),|s|≤1. The normalised Lévy measure is denoted by Π˜L(n) and respectively, its p.g.f. as Q˜L(s)=QL(s)/θL . Then in these notations, the p.g.f. FL(t,s)=E[sL(t)] is given by FL(t,s)=exp{−tθL[1−Q˜L(s)]}=exp{−tθL+tQL(s)},|s|≤1, and the Bernstein function is in the form ψL(λ)= ∑k=1∞1−e−λkΠL(k),λ≥0. All these characteristics of the Logarithmic Lévy process L(t) are specified in the following lemma.

As a direct result of (11), the computations of several terms of the Lévy measure are simplified, such as ΠL(1)=α2,ΠL(2)=α22!512,ΠL(3)=α33!34,ΠL(4)=α44!251120,ΠL(5)=α55!9512.

It is well known from the [22] (see Theorem 4.4, Chapter 2), that the p.m.f. P(L(1)=n) , n=0,1,… , (1) is related to the sequence of the (canonical) Lévy measure ΠL(n),n=1,2,… , (11) by the following recurrence equation: (n+1)P(L(1)=n+1)= ∑k=0nP(L(1)=k)(n−k+1)ΠL(n−k+1). It is equivalent to the next combinatorical identity: n+1n+2= ∑k=0n1(k+1)(n−k)! ∑j=1n−k+1(−1)j−1(j−1)!Bn−k+1,j(c∙).

There are two ways to define the transition probability P(L(t)=n) , n=0,1,… , of the Lévy process L(t) . We could proceed either by starting with p.g.f. FL(t,s) and its Taylor expansion or by using the Lévy measure to define the compound Poisson process L(t) . We present these methods separately in two independent proofs.

In particular, it is easy to calculate several terms of the transition probability, directly from (13): P(L(t)=1)=αAtαt2,P(L(t)=2)=αAtα22!2t3+t(t−1)4,P(L(t)=3)=αAtα33!3!t4+t(t−1)3.2!2.3+t(t−1)(t−2)8,P(L(t)=4)=αAtα44!4!t5+133[t]2↓+[t]3↓+116[t]4↓.

We remark that in the matrix representation of partial Bell polynomials for composition function the numbers Bn,k(x∙),n≥k≥1 , are defined as product of matrices, [19], page 19. Let us denote by H(s) and G(s) respectively the exponential generating functions of both sequences (h∙) and (g∙) . Likewise, by (x∙) is denoted the sequence whose exponential generating function is the composition H(G(s)) , such as H(s)=log(1+s),H(G(s))=log(1+G(s)). Then the matrix associated with the sequence (x∙) is the product of the triangular matrices associated with (g∙) and (h∙) respectively: Bn,k(x∙)= ∑j=knBn,j(g∙)Bj,k(h∙),k≤j≤n. The sequence (h∙) defined by the function H(s)=log(1+s) is exactly the sequence hk=(−1)k−1(k−1)! and Bn,k(h∙)=s(n,k) , i.e. signed Stirling numbers of the first kind. Then, after applying formulas (6) and (7) and changing the order of summation, where k≤j , we confirm the equivalence of (13) and (14) as follows: ∑k=1ntkBn,k(x∙)=αn ∑j=1nBn,j(c∙) ∑k=1jtkBj,k(h∙)=αn ∑j=1nBn,j(c∙)[t]j↓.

The Lévy measure is the infinitesimal generator of the convolution semi-group given by the transition probability P(L(t)=n) , n=0,1,2,… , see [2], page 172. It is a limit in vague convergence, see [3], page 39, as follows: ΠL(n)=limt↓0PL(t,n)t,n=1,2,…. Then ΠL(n)=limt↓0(αA)tαnn! ∑k=1n[t]k↓tBn,k(c∙). Finally, we know that [t]k↓=[−t]k↑(−1)k . In this way, limt↓0[t]k↓t=(−1)k−1(k−1)!.

The concept of subordination was introduced by S. Bochner in 1955 for the Markov processes, Lévy processes, and corresponding semigroups, as randomization of the time parameter: Y(t)=X(T(t)) . There are two sources of randomness – the underlying process X(t) and a random time process T(t) , under the assumption of their independence. The time-change process T(t) is supposed to be a subordinator – the Lévy process with nonnegative increments, [3], Chapter 3. The independence of the ground process and the random time process ensures the preservation of Markov property and Lévy property for the subordinated process. The transformation of the main probabilistic characteristics, such as transition probability, Lévy measure and Laplace exponent, is stated and proved in [20], Chapter 6, Theorem 30.1. See also [7], Chapter 7, Theorem 6.2 and Theorem 6.18. They are our principal references.

In this paragraph, we study the effect of a random time-change for the Negative-Binomial process {X(t),t≥0} . The Lévy measure of a Negative-Binomial process is defined by a logarithmic series distribution supported by positive integers N={1,2,…} with the same parameter 0<α<1 as for the Logarithmic Lévy process L(t) . The Gamma subordinator {Tβ(t),t≥0} with selective parameter β>0 0$]]> can be considered as a random observation time, where the mathematical expectation E[Tβ(t)]=βt . The obtained results are formulated and proved in the following theorem.