Let

Let

The logarithmic series distribution supported by positive integers

The logarithmic series distribution supported by nonnegative integers

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 [

The subordination by Bochner is also developed in many books and articles related to applications in financial mathematics and functional analysis, see [

Our work in the topic is described in the following sections. In Section

The Gauss hypergeometric function is defined in [

Expanding only the logarithmic function

The comparison of two expressions (

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 [

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, [

Let us denote the Lévy measure of the process

Following representation (

The logarithmic function

As a direct result of (

It is well known from the [

There are two ways to define the transition probability

The transition probability

In particular, it is easy to calculate several terms of the transition probability, directly from (

Let the positive random variable

We remark that in the matrix representation of partial Bell polynomials for composition function the numbers

The Lévy measure is the infinitesimal generator of the convolution semi-group given by the transition probability

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:

In this paragraph, we study the effect of a random time-change for the Negative-Binomial process

The main assumption in the definition of subordination by Bochner is the independence of the ground process and the random time-change process. The methods of the Laplace transform and conditional probability for independent processes give the following convenient representations of the main characteristics, see [

Finally, we remark that for

The next studied process

Once again, the composition of two Bernstein functions is obvious:

In this situation, the range of the random time process

After replacing

But, if we take

We confirm the expression of the Lévy measure by the following limit:

An important problem in many applications is how to recognize the original process from the observation data when the registration is randomly perturbed. The problem is growing in cases when the process is composed of several different processes. We see that the probabilistic characteristics for the couples of processes

Selection A. Comparison between the Lévy measure

Selection A. Comparative plot of main Bernstein functions after rescaling with

Selection B: Comparative plot of main Bernstein functions after rescaling with

All these inequalities are related to the following:

The Negative-Binomial process in consideration can be constructed by the subordination of a Poisson process by a Gamma process. In this way, the process

In the

In the

In the general setting:

The authors are very thankful to the anonymous referees for the valuable comments on the paper.