Martin-Löf randomness was originally defined and studied in the context of the Cantor space 2ω . In [1] probability theoretic random closed sets (RACS) are used as the foundation for the study of Martin-Löf randomness in spaces of closed sets. Here we focus on the space of closed subsets of R and a particular family of measures on this space, the generalized Poisson processes. This gives a nove...

The paper proposes a formal estimation procedure for parameters of the fractional Poisson process (fPp). Such procedures are needed to make the fPp model usable in applied situations. The basic idea of fPp, motivated by experimental data with long memory is to make the standard Poisson model more flexible by permitting nonexponential, heavy-tailed distributions of interarrival times and differe...

We present a probabilistic model of events in continuous time in which each event triggers a Poisson process of successor events. The ensemble of observed events is thereby modeled as a superposition of Poisson processes. Efficient inference is feasible under this model with an EM algorithm. Moreover, the EM algorithm can be implemented as a distributed algorithm, permitting the model to be app...

Abstract We consider a random sphere covering model made of random balls with interacting random radii of the product form R(r, ω) = rG(ω), based on a Poisson random measure ω(dy, dr) on IR×IR+. We provide sufficient conditions under which the corresponding random ball counting processes are well-defined, and we study the fractional behavior of the associated random fields. The main results rel...

A kinetic model is derived to study the successive movements of particles, described by a Poisson process, as well as their generation. The irreversible thermodynamics of this system is also studied from the kinetic model. This makes it possible to evaluate the differences between thermodynamical quantities computed exactly and up to second-order. Such differences determine the range of validit...

Forecasting recruitment to clinical trials is often too simplistic and rarely statistical. This is unfortunate because modelling trial recruitment is simple using Poisson processes. The simplest case is to assume patients arrive randomly at a constant rate, using homogeneous Poisson processes to simulate sample recruitment curves. In contrast, time-varying complexity can be incorporated using n...

This paper analyzes statistical properties of the Poisson line Cox point process useful in the modeling of vehicular networks. The point process is created by a two-stage construction: a Poisson line process to model road infrastructure and independent Poisson point processes, conditionally on the Poisson lines, to model vehicles on the roads. We derive basic properties of the point process, in...

The polygons arising from a planar Poisson line process have an exponential distribution of their side lengths and are known to be more regular as their area, perimeter or number of sides increase. Local regions with higher line density have smaller polygon side lengths and conversely. Numerical analysis of computer generated Poisson line processes shows that when pairs of adjacent polygon side...

Consider the distance between two i.i.d. Poisson processes with arrival rate λ > 0 and respective arrival times X1, X2, . . . and Y1, Y2, . . . on a line. We give a closed analytical formula for the E [|Xk+r − Yk|] , for any integer k ≥ 1, r ≥ 0, when a is natural number. The expected distance to the power a between two i.i.d. Poisson processes we represent as the combination of the Pochhammer ...

Example: The stationary Poisson process has a constant rate λ > 0 of customer arrivals. This means the interarrival time A between two consecutive customers is an exponential random variable with the cumulative distribution function A(t) = 1 − e. In order to generate the times of arrivals t1, t2, . . . of customer 1, 2, . . . starting from time t0, we use the inverse transform A−1 recursively.