Lectures on the Poisson Process

  title={Lectures on the Poisson Process},
  author={Mathew D. Penrose},
Structural properties of Gibbsian point processes in abstract spaces
In the language of random counting measures many structural properties of the Poisson process can be studied in arbitrary measurable spaces. We provide a similarly general treatise of Gibbs
Malliavin calculus for marked binomial processes: portfolio optimisation in the trinomial model and compound Poisson approximation
In this paper we develop a stochastic analysis for marked binomial processes, that can be viewed as the discrete analogues of marked Poisson processes. The starting point is the statement of a
Hyperuniform and rigid stable matchings
A stable partial matching of the (possibly randomized) $d$-dimensional lattice with a stationary determinantal point process $\Psi$ on $\mathbb{R}^d$ with intensity $\alpha>1$ with hyperuniformity and number rigid properties is studied.
On almost sure convergence of random variables with finite chaos decomposition
Under mild conditions on a family of independent random variables $(X_n)$ we prove that almost sure convergence of a sequence of tetrahedral polynomial chaoses of uniformly bounded degrees in the
Lace Expansion and Mean-Field Behavior for the Random Connection Model
The lace expansion is adapted to fit the framework of the underlying continuum-space Poisson point process to derive the triangle condition in sufficiently high dimension and furthermore to establish the infra-red bound.
Brownian Polymers in Poissonian Environment: a survey
We consider a space-time continuous directed polymer in random environment. The path is Brownian and the medium is Poissonian. We review many results obtained in the last decade, and also we present
Leaves on the line and in the plane
  • M. Penrose
  • Mathematics
    Electronic Journal of Probability
  • 2020
The Dead Leaves Model (DLM) provides a random tessellation of $d$-space, representing the visible portions of fallen leaves on the ground when $d=2$. For $d=1$, we establish formulae for the
Phase transitions for the Boolean model of continuum percolation for Cox point processes
We consider the Boolean model with random radii based on Cox point processes. Under a condition of stabilization for the random environment, we establish existence and non-existence of subcritical
Combined Degradation Process and Remaining Useful Life Improvement in an Actuator
The proposed degradation modeling and remaining useful life prediction are applied to a double-tank level control system and it is shown that the reliability evolution of the actuator can be estimated by probabilistic approach.
Zentrale Grenzwertsätze im Random Connection Model


Random Measures: Classical Lectures
  • Walter Warmuth Verlag
  • 2011
The Homogeneous Chaos
The Discrete Chaos
Stationäre zufällige Maße auf lokalkompakten Abelschen Gruppen
ZusammenfassungAls ein wichtiges Hilfsmittel zur Untersuchung stationärer zufälliger Ma\e auf einer lokalkompakten Ahelsehen Gruppe G wird in Abschnitt 2 der vorliegenden Arbeit das Palmsche Ma\
Sur les intégrales dont les éléments sont des variables aléatoires indépendantes
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Intensity variations in telephone traffic
Infinitely Divisible Point Processes
Stochastic processes
Random Measures, Theory and Applications