A stochastic process cumulating random increments at random moments is studied. We model it as a two-dimensional random point process and study advantages of such an approach. First, a rather general model allowing for the dependence of both components mutually as well as on covariates is formulated, then the case where the increments depend on time is analyzed with the aid of the multiplicative hazard regression model. Special attention is devoted to the problem of prediction of process behaviour....

In this paper, we prove that the laws of interacting brownian particles are characterized as Gibbs fields on pathspace associated to an explicit class of hamiltonian functionals. More generally, we show that a large class of Gibbs fields on pathspace corresponds to brownian diffusions. Some applications to time reversal in the stationary and non stationary case are presented.

In this paper, we prove that the laws of interacting Brownian particles are characterized as Gibbs fields on pathspace associated to an explicit class of Hamiltonian functionals. More generally, we show that a large class of Gibbs fields on pathspace corresponds to Brownian diffusions. Some applications to time reversal in the stationary and non stationary case are presented.

We prove that Poisson measures are invariant under (random) intensity preserving transformations whose finite difference gradient satisfies a cyclic vanishing condition. The proof relies on moment identities of independent interest for adapted and anticipating Poisson stochastic integrals, and is inspired by the method of Üstünel and Zakai (Probab. Theory Related Fields103 (1995) 409–429) on the Wiener space, although the corresponding algebra is more complex than in the Wiener case. The examples...

We consider a random walk on a homogeneous Poisson point process with energy marks. The jump rates decay exponentially in the $\alpha$-power of the jump length and depend on the energy marks via a Boltzmann-like factor. The case $\alpha =1$ corresponds to the phonon-induced Mott variable range hopping in disordered solids in the regime of strong Anderson localization. We prove that for almost every realization of the marked process, the diffusively rescaled random walk, with an arbitrary start point, converges to...

Models of random sets and of point processes are introduced to simulate some specific clustering of points, namely on random lines in ${ℝ}^2$ and ${ℝ}^3$ and on random planes in ${ℝ}^3$. The corresponding point processes are special cases of Cox processes. The generating distribution function of the probability distribution of the number of points in a convex set $K$ and the Choquet capacity $T\left(K\right)$ are given. A possible application is to model point defects in materials with some degree of alignment. Theoretical results...

En esta comunicación se establece una medida de la entropía contenida en un proceso puntual mediante el concepto de entropía de orden α y tipo β introducida por Sharma and Mittal (1975); quedando, de este modo, generalizada la entropía de McFadden. Una vez que se estudian las propiedades relativas a la tasa de cambio de la Entropía, se demuestra que el proceso de Poisson es el de Entropía máxima dentro de la clase de los procesos puntuales estacionarios.

Consider a stationary Boolean model $X$ with convex grains in ${ℝ}^d$ and let any exposed lower tangent point of $X$ be shifted towards the hyperplane $N_0=\{x\in {ℝ}^d:x_1=0\}$ by the length of the part of the segment between the point and its projection onto the $N_0$ covered by $X$. The resulting point process in the halfspace (the Laslett’s transform of $X$) is known to be stationary Poisson and of the same intensity as the original Boolean model. This result was first formulated for the planar Boolean model (see N. Cressie [Cressie])...

Observations are made on a point process $𝛯$ in ${ℝ}^d$ in a window $Q_{\lambda }$ of volume $\lambda$. The observation, or ‘score’ at a point $x$, here denoted $\xi (x,𝛯)$, is a function of the points within a random distance of $x$. When the input $𝛯$ is a Poisson or binomial point process, the large $\lambda$ limit theory for the total score ${\sum }_{x\in 𝛯\cap Q_{\lambda }}\xi (x,𝛯\cap Q_{\lambda })$, when properly scaled and centered, is well understood. In this paper we establish general laws of large numbers, variance asymptotics, and central limit theorems for the total score for Gibbsian input $𝛯$....

We study a Markov process on a system of interlacing particles. At large times the particles fill a domain that depends on a parameter ε > 0. The domain has two cusps, one pointing up and one pointing down. In the limit ε ↓ 0 the cusps touch, thus forming a tacnode. The main result of the paper is a derivation of the local correlation kernel around the tacnode in the transition regime ε ↓ 0. We also prove that the local process interpolates between the Pearcey process and the GUE minor process....

We show that any loop-free Markov chain on a discrete space can be viewed as a determinantal point process. As an application, we prove central limit theorems for the number of particles in a window for renewal processes and Markov renewal processes with Bernoulli noise.

The longitudinal regression model $Z_i^j=m({\theta }_0,{𝕏}_i\left(T_i^j\right))+{\epsilon }_i^j,$ where $Z_i^j$ is the $j$th measurement of the $i$th subject at random time $T_i^j$, $m$ is the regression function, ${𝕏}_i\left(T_i^j\right)$ is a predictable covariate process observed at time $T_i^j$ and ${\epsilon }_i^j$ is a noise, is studied in marked point process framework. In this paper we introduce the assumptions which guarantee the consistency and asymptotic normality of smooth $M$-estimator of unknown parameter ${\theta }_0$.

In the present paper we consider the problem of fitting parametric spatial Cox point process models. We concentrate on the moment estimation methods based on the second order characteristics of the point process in question. These methods represent a simulation-free faster-to-compute alternative to the computationally intense maximum likelihood estimation. We give an overview of the available methods, discuss their properties and applicability. Further we present results of a simulation study in...