We study the long-range dependence (LRD) of the increments of the fractional Poisson process (FPP), the fractional negative binomial process (FNBP) and the increments of the FNBP. We first point out an error in the proof of Theorem 1 of Biard and Saussereau [2] and prove that the increments of the FPP has indeed the short-range dependence (SRD) property, when the fractional index β satisfies 0 < β < 1 3 . We also establish that the FNBP has the LRD property, while the increments of the FNBP possesses the SRD property.

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In this article, we study the Poisson process of order k (PPoK) time-changed with an independent Lévy subordinator and its inverse, which we call respectively, as TCPPoK-I and TCPPoK-II, through various distributional properties, long-range dependence and limit theorems for the PPoK and the TCPPoK-I. Further, we study the governing difference-differential equations of the TCPPoK-I for the case inverse Gaussian subordinator. Similarly, we study the distributional properties, asymptotic moments and the governing difference-differential equation of TCPPoK-II. As an application to ruin theory, we give a governing differential equation of ruin probability in insurance ruin using these processes. Finally, we present some simulated sample paths of both the processes.2010 Mathematics Subject Classification. 60G55; 60G51.

The space-time fractional Poisson process (STFPP), defined by Orsingher and Poilto in [17], is a generalization of the time fractional Poisson process (TFPP) and the space fractional Poisson process (SFPP). We study the fractional generalization of the non-homogeneous Poisson process and call it the non-homogeneous space-time fractional Poisson process (NSTFPP). We compute their pmf and generating function and investigate the associated differential equation. The limit theorems and the law of iterated logarithm for the NSTFPP process are studied. We study the distributional properties, the asymptotic expansion of the correlation function of the non-homogeneous time fractional Poisson process (NTFPP) and subsequently investigate the long-range dependence (LRD) property of a special NTFPP. We investigate the limit theorem and the LRD property for the fractional non-homogeneous Poisson process (FNPP), studied by Leonenko et. al. (2016). Finally, we present some simulated sample paths of the NSTFPP process.

Stochastic modelling of fatigue (and other material's deterioration), as well as of cumulative damage in risk theory, are often based on compound sums of independent random variables, where the number of addends is represented by an independent counting process. We consider here a cumulative model where, instead of a renewal process (as in the Poisson case), a linear birth (or Yule) process is used. This corresponds to the assumption that the frequency of “damage” increments accelerates according to the increasing number of “damages”. We start from the partial differential equation satisfied by its transition density, in the case of exponentially distributed addends, and then we generalize it by introducing a space derivative of convolution type (i.e., defined in terms of the Laplace exponent of a subordinator). Then we are concerned with the solution of integro‐differential equations, under proper initial conditions, which, in a special case, reduce to a fractional one. Correspondingly, we analyze the related cumulative jump processes under a general infinitely divisible distribution of the (positive) jumps. Some special cases (such as the stable, tempered stable, gamma, and Poisson) are presented.