scholarly journals The Generating Function for the Airy Point Process and a System of Coupled Painlevé II Equations

2018 ◽  
Vol 140 (4) ◽  
pp. 403-437 ◽  
Author(s):  
Tom Claeys ◽  
Antoine Doeraene
Keyword(s):  
Point Process ◽  
Generating Function ◽  
Painlevé Ii ◽  
Airy Point Process

scholarly journals Airy point process at the liquid-gas boundary

2018 ◽  
Vol 46 (5) ◽  
pp. 2973-3013 ◽  
Author(s):  
Vincent Beffara ◽  
Sunil Chhita ◽  
Kurt Johansson
Keyword(s):  
Point Process ◽  
Airy Point Process

scholarly journals Gap probabilities in the bulk of the Airy process

2021 ◽  
pp. 2250022
Author(s):  
Elliot Blackstone ◽  
Christophe Charlier ◽  
Jonatan Lenells
Keyword(s):  
Point Process ◽  
Airy Process ◽  
Airy Point Process ◽  
Gap Probabilities

We consider the probability that no points lie on [Formula: see text] large intervals in the bulk of the Airy point process. We make a conjecture for all the terms in the asymptotics up to and including the oscillations of order [Formula: see text], and we prove this conjecture for [Formula: see text].

scholarly journals EXPLICIT COMPUTATIONS FOR A FILTERING PROBLEM WITH POINT PROCESS OBSERVATIONS WITH APPLICATIONS TO CREDIT RISK

2011 ◽  
Vol 25 (3) ◽  
pp. 393-418 ◽  
Author(s):  
Vincent Leijdekker ◽  
Peter Spreij
Keyword(s):  
Credit Risk ◽  
Point Process ◽  
Generating Function ◽  
Moment Generating Function ◽  
Cox Process ◽  
Risk Modeling ◽  
Conditional Moment ◽  
Credit Risk Modeling ◽  
Conditional Moment Generating Function ◽  
Filtering Problem

We consider the filtering problem for a doubly stochastic Poisson or Cox process, where the intensity follows the Cox–Ingersoll–Ross model. In this article we assume that the Brownian motion, which drives the intensity, is not observed. Using filtering theory for point process observations, we first derive the dynamics for the intensity and its moment-generating function, given the observations of the Cox process. A transformation of the dynamics of the conditional moment-generating function allows us to solve in closed form the filtering problem, between the jumps of the Cox process as well as at the jumps, which constitutes the main contribution of the article. Assuming that the initial distribution of the intensity is of the Gamma type, we obtain an explicit solution to the filtering problem for all t>0. We conclude the article with the observation that the resulting conditional moment-generating function at time t, after Nt jumps, corresponds to a mixture of Nt+1 Gamma distributions. Currently, the model that we analyze has become popular in credit risk modeling, where one uses the intensity-based approach for the modeling of default times of one or more companies. In this approach, the default times are defined as the jump times of a Cox process. In such a model, one only has access to observations of the Cox process, and thus filtering comes in as a natural technique in credit risk modeling.

A Versatile Packet Arrival Process and Its Second Order Properties

1993 ◽  
Vol 7 (4) ◽  
pp. 495-513 ◽  
Author(s):  
Teunis J. Ott
Keyword(s):  
Point Process ◽  
Generating Function ◽  
Discrete Time ◽  
Branching Process ◽  
Second Order ◽  
Arrival Process ◽  
Multitype Branching Process ◽  
Infected People ◽  
Index Of Dispersion ◽  
Packet Arrival

In this paper we describe a class of discrete time processes that can be used to model packet arrival streams in packetized communication. Mathematically, (K(t)) can be seen as a discrete time self-exciting point process, as a multitype branching process, or as an epidemic with immigration of infected people. The purpose of this paper is to show that this class of models simultaneously is quite useful and analytically more tractable than is obvious at first glance. It is shown that certain probabilities can reliably be computed using generating function methods, and expressions are given for the second order properties and for the asymptotic index of dispersion.

scholarly journals The generating function for the Bessel point process and a system of coupled Painlevé V equations

2019 ◽  
Vol 08 (03) ◽  
pp. 1950008 ◽  
Author(s):  
Christophe Charlier ◽  
Antoine Doeraene
Keyword(s):  
Point Process ◽  
Generating Function ◽  
Hilbert Problem ◽  
Fredholm Determinant ◽  
Probability Generating Function ◽  
Joint Probability ◽  
Hard Edge ◽  
Jump Discontinuities ◽  
Bessel Kernel ◽  
Gap Probability

We study the joint probability generating function for [Formula: see text] occupancy numbers on disjoint intervals in the Bessel point process. This generating function can be expressed as a Fredholm determinant. We obtain an expression for it in terms of a system of coupled Painlevé V equations, which are derived from a Lax pair of a Riemann–Hilbert problem. This generalizes a result of Tracy and Widom [C. A. Tracy and H. Widom, Level spacing distributions and the Bessel kernel, Commun. Math. Phys. 161(2) (1994) 289–309], which corresponds to the case [Formula: see text]. We also provide some examples and applications. In particular, several relevant quantities can be expressed in terms of the generating function, like the gap probability on a union of disjoint bounded intervals, the gap between the two smallest particles, and large [Formula: see text] asymptotics for [Formula: see text] Hankel determinants with a Laguerre weight possessing several jump discontinuities near the hard edge.

scholarly journals Generating function associated with the rational solutions of the Painlevé II equation

2002 ◽  
Vol 35 (16) ◽  
pp. L207-L211 ◽  
Author(s):  
Katsunori Iwasaki ◽  
Kenji Kajiwara ◽  
Toshiya Nakamura
Keyword(s):  
Generating Function ◽  
Rational Solutions ◽  
Painlevé Ii

Habitat use of toothed whales in a marine protected area based on point process models

2019 ◽  
Vol 609 ◽  
pp. 239-256 ◽  
Author(s):  
TL Silva ◽  
G Fay ◽  
TA Mooney ◽  
J Robbins ◽  
MT Weinrich ◽  
...  
Keyword(s):  
Habitat Use ◽  
Point Process ◽  
Protected Area ◽  
Marine Protected Area ◽  
Process Models ◽  
Point Process Models
Sign in / Sign up

Export Citation Format

Share Document