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).

Large deviations in the supercritical branching process

1993 ◽  
Vol 25 (04) ◽  
pp. 757-772 ◽  
Author(s):  
J. D. Biggins ◽  
N. H. Bingham
Keyword(s):  
Distribution Function ◽  
Large Deviations ◽  
Population Size ◽  
Branching Process ◽  
Large Deviation ◽  
Moment Generating Function ◽  
Large Deviation Theory ◽  
Tail Behaviour ◽  
The Moment ◽  
Deviation Theory

The tail behaviour of the limit of the normalized population size in the simple supercritical branching process, W, is studied. Most of the results concern those cases when a tail of the distribution function of W decays exponentially quickly. In essence, knowledge of the behaviour of transforms can be combined with some ‘large-deviation' theory to get detailed information on the oscillation of the distribution function of W near zero or at infinity. In particular we show how an old result of Harris (1948) on the asymptotics of the moment-generating function of W translates to tail behaviour.

Large deviations in the supercritical branching process

1993 ◽  
Vol 25 (4) ◽  
pp. 757-772 ◽  
Author(s):  
J. D. Biggins ◽  
N. H. Bingham
Keyword(s):  
Distribution Function ◽  
Large Deviations ◽  
Population Size ◽  
Branching Process ◽  
Large Deviation ◽  
Moment Generating Function ◽  
Large Deviation Theory ◽  
Tail Behaviour ◽  
The Moment ◽  
Deviation Theory

The tail behaviour of the limit of the normalized population size in the simple supercritical branching process, W, is studied. Most of the results concern those cases when a tail of the distribution function of W decays exponentially quickly. In essence, knowledge of the behaviour of transforms can be combined with some ‘large-deviation' theory to get detailed information on the oscillation of the distribution function of W near zero or at infinity. In particular we show how an old result of Harris (1948) on the asymptotics of the moment-generating function of W translates to tail behaviour.

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.

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 Dividend Problems in the Diffusion Model with Interest and Exponentially Distributed Observation Time

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Cuilian Wang ◽  
Xiao Liu
Keyword(s):  
Laplace Transform ◽  
Diffusion Model ◽  
Generating Function ◽  
Moment Generating Function ◽  
Observation Time ◽  
Integrodifferential Equations ◽  
Barrier Strategy ◽  
Ruin Time ◽  
The Laplace Transform ◽  
The Moment

Consider dividend problems in the diffusion model with interest and exponentially distributed observation time where dividends are paid according to a barrier strategy. Assume that dividends can only be paid with a certain probability at each point of time; that is, on each observation, if the surplus exceeds the barrier level, the excess is paid as dividend. In this paper, integrodifferential equations for the moment-generating function, thenth moment function, and the Laplace transform of ruin time are derived; explicit expressions for the expected discounted dividends paid until ruin and the Laplace transform of ruin time are also obtained.

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 Large deviations for the leaves in some random trees

2009 ◽  
Vol 41 (3) ◽  
pp. 845-873 ◽  
Author(s):  
Wlodek Bryc ◽  
David Minda ◽  
Sunder Sethuraman
Keyword(s):  
Markov Chains ◽  
Large Deviations ◽  
Random Graphs ◽  
Large Deviation ◽  
Preferential Attachment ◽  
Random Trees ◽  
Explicit Computation ◽  
Large Deviation Principles ◽  
Recursive Trees

Large deviation principles and related results are given for a class of Markov chains associated to the ‘leaves' in random recursive trees and preferential attachment random graphs, as well as the ‘cherries’ in Yule trees. In particular, the method of proof, combining analytic and Dupuis–Ellis-type path arguments, allows for an explicit computation of the large deviation pressure.

scholarly journals Large Deviations in Testing Squared Radial Ornstein-Uhleneck Model

2011 ◽  
Vol 2011 ◽  
pp. 1-11
Author(s):  
Shoujiang Zhao ◽  
Qiaojing Liu
Keyword(s):  
Large Deviations ◽  
Hypothesis Testing ◽  
Likelihood Ratio ◽  
Large Deviation ◽  
Decay Rates ◽  
Moderate Deviations ◽  
Large Deviation Principles ◽  
Log Likelihood ◽  
Log Likelihood Ratio ◽  
Error Probabilities

We study the large deviations and moderate deviations of hypothesis testing for squared radial Ornstein-Uhleneck model. Large deviation principles for the log-likelihood ratio are obtained, by which we give negative regions in testing squared radial Ornstein-Uhleneck model and get the decay rates of the error probabilities.

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