LARGE DEVIATIONS FOR THE LONGEST GAP IN POISSON PROCESSES

2019 ◽  
Vol 101 (1) ◽  
pp. 146-156 ◽  
Author(s):  
JOSEPH OKELLO OMWONYLEE
Keyword(s):  
Large Deviations ◽  
Laplace Transform ◽  
Large Deviation ◽  
Poisson Processes ◽  
Homogeneous Poisson Process ◽  
Arrival Times ◽  
Large Deviation Principles ◽  
The Laplace Transform ◽  
Rate Functions ◽  
Maximal Time

The longest gap $L(t)$ up to time $t$ in a homogeneous Poisson process is the maximal time subinterval between epochs of arrival times up to time $t$; it has applications in the theory of reliability. We study the Laplace transform asymptotics for $L(t)$ as $t\rightarrow \infty$ and derive two natural and different large-deviation principles for $L(t)$ with two distinct rate functions and speeds.

scholarly journals Laplace transform asymptotics and large deviation principles for longest success runs in Bernoulli trials

2016 ◽  
Vol 53 (3) ◽  
pp. 747-764 ◽  
Author(s):  
Takis Konstantopoulos ◽  
Zhenxia Liu ◽  
Xiangfeng Yang
Keyword(s):  
Large Deviations ◽  
Laplace Transform ◽  
Large Deviation ◽  
Moment Generating Function ◽  
Precise Asymptotics ◽  
Bernoulli Trials ◽  
Success Runs ◽  
Large Deviation Principles ◽  
Logarithmic Asymptotics ◽  
The Moment

AbstractThe longest stretch L(n) of consecutive heads in n independent and identically distributed coin tosses is seen from the prism of large deviations. We first establish precise asymptotics for the moment generating function of L(n) and then show that there are precisely two large deviation principles, one concerning the behavior of the distribution of L(n) near its nominal value log1∕pn and one away from it. We discuss applications to inference and to logarithmic asymptotics of functionals of L(n).

scholarly journals Ruin Probabilities for Two Classes of Risk Processes

2005 ◽  
Vol 35 (1) ◽  
pp. 61-77 ◽  
Author(s):  
Shuanming Li ◽  
José Garrido
Keyword(s):  
Laplace Transform ◽  
Ruin Probability ◽  
Risk Model ◽  
Compound Poisson Process ◽  
Ruin Probabilities ◽  
Counting Processes ◽  
Arrival Times ◽  
The Laplace Transform ◽  
Sparre Andersen Risk Model ◽  
Risk Processes

We consider a risk model with two independent classes of insurance risks. We assume that the two independent claim counting processes are, respectively, Poisson and Sparre Andersen processes with generalized Erlang(2) claim inter-arrival times. The Laplace transform of the non-ruin probability is derived from a system of integro-differential equations. Explicit results can be obtained when the initial reserve is zero and the claim severity distributions of both classes belong to the Kn family of distributions. A relation between the ruin probability and the distribution of the supremum before ruin is identified. Finally, the Laplace transform of the non-ruin probability of a perturbed Sparre Andersen risk model with generalized Erlang(2) claim inter-arrival times is derived when the compound Poisson process converges weakly to a Wiener process.

LARGE DEVIATIONS FOR SMALL PERTURBATIONS OF SDES WITH NON-MARKOVIAN COEFFICIENTS AND THEIR APPLICATIONS

2006 ◽  
Vol 06 (04) ◽  
pp. 487-520 ◽  
Author(s):  
FUQING GAO ◽  
JICHENG LIU
Keyword(s):  
Large Deviations ◽  
Large Deviation Principle ◽  
Large Deviation ◽  
Wiener Space ◽  
Small Perturbations ◽  
Deviation Principle ◽  
Large Deviation Principles ◽  
Sobolev Norms

We prove large deviation principles for solutions of small perturbations of SDEs in Hölder norms and Sobolev norms, where the SDEs have non-Markovian coefficients. As an application, we obtain a large deviation principle for solutions of anticipating SDEs in terms of (r, p) capacities on the Wiener space.

scholarly journals Nonstandard limit theorems and large deviations for the Jacobi beta ensemble

2014 ◽  
Vol 03 (03) ◽  
pp. 1450012 ◽  
Author(s):  
Jan Nagel
Keyword(s):  
Weak Convergence ◽  
Spectral Measure ◽  
Limit Theorems ◽  
Large Deviation ◽  
The Other ◽  
Limit Measure ◽  
Jacobi Ensemble ◽  
Large Deviation Principles ◽  
Beta Ensemble ◽  
Rate Functions

In this paper, we show weak convergence of the empirical eigenvalue distribution and of the weighted spectral measure of the Jacobi ensemble, when one or both parameters grow faster than the dimension n. In these cases, the limit measure is given by the Marchenko–Pastur law and the semicircle law, respectively. For the weighted spectral measure, we also prove large deviation principles under this scaling, where the rate functions are those of the other classical ensembles.

An invariance property of Poisson processes

1969 ◽  
Vol 6 (02) ◽  
pp. 453-458 ◽  
Author(s):  
Mark Brown
Keyword(s):  
Poisson Process ◽  
Point Processes ◽  
Random Variables ◽  
Poisson Processes ◽  
Invariance Property ◽  
Homogeneous Poisson Process ◽  
Arrival Times ◽  
Random Functions

In this paper we shall investigate point processes generated by random variables of the form 〈gi (Ti ]), i=± 1, ± 2, … 〉, where 〈Ti, i= ± 1, … 〉 is the set of arrival times from a (not necessarily homogeneous) Poisson process or mixture of Poisson processes, and 〈gi, i = ± 1, … 〉 is an independently and identically distributed (i.i.d.) or interchangeable sequence of random functions, independent of 〈Ti 〉.

scholarly journals Ruin Probabilities for Two Classes of Risk Processes

2005 ◽  
Vol 35 (01) ◽  
pp. 61-77 ◽  
Author(s):  
Shuanming Li ◽  
José Garrido
Keyword(s):  
Laplace Transform ◽  
Ruin Probability ◽  
Risk Model ◽  
Compound Poisson Process ◽  
Ruin Probabilities ◽  
Counting Processes ◽  
Arrival Times ◽  
The Laplace Transform ◽  
Sparre Andersen Risk Model ◽  
Risk Processes

We consider a risk model with two independent classes of insurance risks. We assume that the two independent claim counting processes are, respectively, Poisson and Sparre Andersen processes with generalized Erlang(2) claim inter-arrival times. The Laplace transform of the non-ruin probability is derived from a system of integro-differential equations. Explicit results can be obtained when the initial reserve is zero and the claim severity distributions of both classes belong to the Kn family of distributions. A relation between the ruin probability and the distribution of the supremum before ruin is identified. Finally, the Laplace transform of the non-ruin probability of a perturbed Sparre Andersen risk model with generalized Erlang(2) claim inter-arrival times is derived when the compound Poisson process converges weakly to a Wiener process.

Large deviations for departures from a shared buffer

1997 ◽  
Vol 34 (3) ◽  
pp. 753-766 ◽  
Author(s):  
Neil O'connell
Keyword(s):  
Large Deviations ◽  
Large Deviation ◽  
The State ◽  
Stationary Case ◽  
Service Capacity ◽  
Equilibrium System ◽  
Large Deviation Principles ◽  
Service Policy ◽  
Shared Buffer

In this paper we describe how the joint large deviation properties of traffic streams are altered when the traffic passes through a shared buffer according to a FCFS service policy with stochastic service capacity. We also consider the stationary case, proving large deviation principles for the state of the system in equilibrium and for departures from an equilibrium system.

scholarly journals Large deviations for a class of tempered subordinators and their inverse processes

2021 ◽  
pp. 1-21
Author(s):  
Nikolai Leonenko ◽  
Claudio Macci ◽  
Barbara Pacchiarotti
Keyword(s):  
Weak Convergence ◽  
Large Deviations ◽  
Large Deviation ◽  
Common Law ◽  
Random Variables ◽  
Tempered Distributions ◽  
One Dimensional ◽  
Large Deviation Principles ◽  
Time Changes ◽  
Non Gaussian

We consider a class of tempered subordinators, namely a class of subordinators with one-dimensional marginal tempered distributions which belong to a family studied in [3]. The main contribution in this paper is a non-central moderate deviations result. More precisely we mean a class of large deviation principles that fill the gap between the (trivial) weak convergence of some non-Gaussian identically distributed random variables to their common law, and the convergence of some other related random variables to a constant. Some other minor results concern large deviations for the inverse of the tempered subordinators considered in this paper; actually, in some results, these inverse processes appear as random time-changes of other independent processes.

Large deviations for departures from a shared buffer

1997 ◽  
Vol 34 (03) ◽  
pp. 753-766 ◽  
Author(s):  
Neil O'connell
Keyword(s):  
Large Deviations ◽  
Large Deviation ◽  
The State ◽  
Stationary Case ◽  
Service Capacity ◽  
Equilibrium System ◽  
Large Deviation Principles ◽  
Service Policy ◽  
Shared Buffer

In this paper we describe how the joint large deviation properties of traffic streams are altered when the traffic passes through a shared buffer according to a FCFS service policy with stochastic service capacity. We also consider the stationary case, proving large deviation principles for the state of the system in equilibrium and for departures from an equilibrium system.

scholarly journals A Two-Dimensional Risk Model with Proportional Reinsurance

2011 ◽  
Vol 48 (3) ◽  
pp. 749-765 ◽  
Author(s):  
Andrei L. Badescu ◽  
Eric C. K. Cheung ◽  
Landy Rabehasaina
Keyword(s):  
Laplace Transform ◽  
Risk Model ◽  
Poisson Processes ◽  
General Distribution ◽  
Proportional Reinsurance ◽  
Two Dimensional ◽  
Open Problems ◽  
Compound Poisson ◽  
Compound Poisson Processes ◽  
The Laplace Transform

In this paper we consider an extension of the two-dimensional risk model introduced in Avram, Palmowski and Pistorius (2008a). To this end, we assume that there are two insurers. The first insurer is subject to claims arising from two independent compound Poisson processes. The second insurer, which can be viewed as a different line of business of the same insurer or as a reinsurer, covers a proportion of the claims arising from one of these two compound Poisson processes. We derive the Laplace transform of the time until ruin of at least one insurer when the claim sizes follow a general distribution. The surplus level of the first insurer when the second insurer is ruined first is discussed at the end in connection with some open problems.

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