Most of the multivariate counting processes studied in the literature are regular processes, which implies, ignoring the types of the events, the non-occurrence of multiple events. However, in practice, several different types of events may occur simultaneously. In this paper, a new class of multivariate counting processes which allow simultaneous occurrences of multiple types of events is suggested and its stochastic properties are studied. For the modeling of such kind of process, we rely on the tool of superposition of seed counting processes. It will be shown that the stochastic properties of the proposed class of multivariate counting processes are explicitly expressed. Furthermore, the marginal processes are also explicitly obtained. We analyze the multivariate dependence structure of the proposed class of counting processes.

We study a collector's problem with K renewal arrival processes for different type items, where the objective is to collect complete sets. In particular, we derive the asymptotic distribution of the sequence of interarrival times between set completions.

The problem of a semi-infinite propagating crack in the piezoelectric material subjected to a dynamic anti-plane concentrated body force is investigated in the present study. It is assumed that between the growing crack surfaces there is a permeable vacuum free space, in which the electrostatic potential is nonzero. It is noted that this problem has characteristic lengths and a direct attempt towards solving this problem by transform and Wiener-Hopf techniques [1] is not applicable. This paper proposes a new fundamental solution for propagating crack in the piezoelectric material and the transient response of the propagating crack is determined by superposition of the fundamental solution in the Laplace transform domain. The fundamental solution represents the responses of applying exponentially distributed loadings in the Laplace transform domain on the propagating crack surface. Exact analytical transient solutions for the dynamic stress intensity factor and the dynamic electric displacement intensity factor are obtained by using the Cagniard-de Hoop method [2,3] of Laplace inversion and are expressed in explicit forms. Finally, numerical results based on analytical solutions are calculated and are discussed in detail.

The iteration of random tessellations in ℝd is considered, where each cell of an initial tessellation is further subdivided into smaller cells by so-called component tessellations. Sufficient conditions for stationarity and isotropy of iterated tessellations are given. Formulae are derived for the intensities of their facet processes, and for the expected intrinsic volumes of their typical facets. Particular emphasis is put on two special cases: superposition and nesting of tessellations. Bernoulli thinning of iterated tessellations is also considered.

In this paper a new multifractal stochastic process called Limit of theIntegrated Superposition of Diffusion processes with Linear differencialGenerator (LISDLG) is presented which realistically characterizes the networktraffic multifractality. Several properties of the LISDLG model are presentedincluding long range dependence, cumulants, logarithm of the characteristicfunction, dilative stability, spectrum and bispectrum. The model captureshigher-order statistics by the cumulants. The relevance and validation of theproposed model are demonstrated by real data of Internet traffic.


A generalization of Markov point processes is introduced in which interactions occur between connected components of the point pattern. A version of the Hammersley-Clifford characterization theorem is proved which states that a point process is a Markov interacting component process if and only if its density function is a product of interaction terms associated with cliques of connected components. Integrability and superpositional properties of the processes are shown and a pairwise interaction example is used for detailed exploration.

A general model for the actuarial risk-reserve process as a superposition of compound delayed-renewal processes is introduced and related to previous models which have been used in collective risk theory. It is observed that non-stationarity of the portfolio ‘age-structure' within this model can have a significant impact upon probabilities of ruin. When the portfolio size is constant and the policy age-distribution is stationary, the moderate- and large-deviation probabilities of ruin are bounded and calculated using the strong approximation results of Csörg et al. (1987a, b) and a large-deviation theorem of Groeneboom et al. (1979). One consequence is that for non-Poisson claim-arrivals, the large-deviation probabilities of ruin are noticeably affected by the decision to model many parallel policy lines in place of one line with correspondingly faster claim-arrivals.

Several different orderings for the comparison of point processes have been introduced and their relationships discussed in Whitt [9], Daley [2] and Deng [4]. It is of some interest to know whether these orderings, in general, are preserved under various operations on point processes. Some results concerning limit operations were given in Deng [4].

In the present paper, we first further introduce some convex and concave orderings for counting processes, and survey the relationships among all orderings mentioned in [9], [4] and this paper. Then we focus our attention on the study of the conditions for the preservation of orderings under the operations of superposition, thinning, shift, and random change of time.

Cross-covariances between the Bernoulli thinned processes of an arbitrary point process are determined. When the point process is renewal it is shown that zero correlation implies independence. An example is given to show that zero covariance between intervals does not imply zero covariance between counts. Mark-dependent thinning of Markov renewal processes is discussed and the results are applied to the overflow queue. Here we give an example of two uncorrelated but dependent renewal processes, neither of which is Poisson, which yield a Poisson process when superposed. Finally, we study Markov-chain thinning of renewal processes.

Theorem: A necessary and sufficient condition for the superposition of two ordinary renewal processes to again be a renewal process is that they be Poisson processes.

A complete proof of this theorem is given; also it is shown how the theorem follows from the corresponding one for the superposition of two stationary renewal processes.