Remarks on a link between the Laplace transform and distribution function of a nonnegative random variable

2009 ◽  
Vol 79 (13) ◽  
pp. 1501-1508 ◽  
Author(s):  
Leonid Rozovsky
Keyword(s):  
Distribution Function ◽  
Laplace Transform ◽  
Random Variable ◽  
Nonnegative Random Variable ◽  
The Laplace Transform

On the limiting distribution of a supercritical branching process in a random environment

1992 ◽  
Vol 29 (03) ◽  
pp. 499-518 ◽  
Author(s):  
Ben Hambly
Keyword(s):  
Distribution Function ◽  
Laplace Transform ◽  
Family Size ◽  
Random Environment ◽  
Branching Process ◽  
Random Variable ◽  
Limiting Distribution ◽  
The Laplace Transform

We consider an increasing supercritical branching process in a random environment and obtain bounds on the Laplace transform and distribution function of the limiting random variable. There are two possibilities that can be distinguished depending on the nature of the component distributions of the environment. If the minimum family size of each is 1, the growth will be as a power depending on a parameter α. If the minimum family sizes of some are greater than 1, it will be exponential, depending on a parameter γ. We obtain bounds on the distribution function analogous to those found for the simple Galton-Watson case. It is not possible to obtain exact estimates and we are only able to obtain bounds to within ε of the parameters.

scholarly journals A Technique for Computing the PDFs and CDFs of Nonnegative Infinitely Divisible Random Variables

2011 ◽  
Vol 48 (01) ◽  
pp. 217-237 ◽  
Author(s):  
Mark S. Veillette ◽  
Murad S. Taqqu
Keyword(s):  
Distribution Function ◽  
Laplace Transform ◽  
Cumulative Distribution Function ◽  
Stable Distribution ◽  
Random Variable ◽  
Cumulative Distribution ◽  
Infinitely Divisible ◽  
The Laplace Transform ◽  
Laplace Transform Inversion ◽  
Probability Density Function Pdf

We present a method for computing the probability density function (PDF) and the cumulative distribution function (CDF) of a nonnegative infinitely divisible random variable X. Our method uses the Lévy-Khintchine representation of the Laplace transform Ee-λX = e-ϕ(λ), where ϕ is the Laplace exponent. We apply the Post-Widder method for Laplace transform inversion combined with a sequence convergence accelerator to obtain accurate results. We demonstrate this technique on several examples, including the stable distribution, mixtures thereof, and integrals with respect to nonnegative Lévy processes.

scholarly journals A Technique for Computing the PDFs and CDFs of Nonnegative Infinitely Divisible Random Variables

2011 ◽  
Vol 48 (1) ◽  
pp. 217-237 ◽  
Author(s):  
Mark S. Veillette ◽  
Murad S. Taqqu
Keyword(s):  
Distribution Function ◽  
Laplace Transform ◽  
Cumulative Distribution Function ◽  
Stable Distribution ◽  
Random Variable ◽  
Cumulative Distribution ◽  
Infinitely Divisible ◽  
The Laplace Transform ◽  
Laplace Transform Inversion ◽  
Probability Density Function Pdf

We present a method for computing the probability density function (PDF) and the cumulative distribution function (CDF) of a nonnegative infinitely divisible random variable X. Our method uses the Lévy-Khintchine representation of the Laplace transform Ee-λX = e-ϕ(λ), where ϕ is the Laplace exponent. We apply the Post-Widder method for Laplace transform inversion combined with a sequence convergence accelerator to obtain accurate results. We demonstrate this technique on several examples, including the stable distribution, mixtures thereof, and integrals with respect to nonnegative Lévy processes.

On the limiting distribution of a supercritical branching process in a random environment

1992 ◽  
Vol 29 (3) ◽  
pp. 499-518 ◽  
Author(s):  
Ben Hambly
Keyword(s):  
Distribution Function ◽  
Laplace Transform ◽  
Family Size ◽  
Random Environment ◽  
Branching Process ◽  
Random Variable ◽  
Limiting Distribution ◽  
The Laplace Transform

We consider an increasing supercritical branching process in a random environment and obtain bounds on the Laplace transform and distribution function of the limiting random variable. There are two possibilities that can be distinguished depending on the nature of the component distributions of the environment. If the minimum family size of each is 1, the growth will be as a power depending on a parameter α. If the minimum family sizes of some are greater than 1, it will be exponential, depending on a parameter γ. We obtain bounds on the distribution function analogous to those found for the simple Galton-Watson case. It is not possible to obtain exact estimates and we are only able to obtain bounds to within ε of the parameters.

scholarly journals On a conjecture of Seneta on regular variation of truncated moments

2021 ◽  
Vol 109 (123) ◽  
pp. 77-82
Author(s):  
Péter Kevei
Keyword(s):  
Distribution Function ◽  
Relative Stability ◽  
Regular Variation ◽  
Domain Of Attraction ◽  
Random Variable ◽  
Nonnegative Random Variable ◽  
Equivalent Characterization ◽  
Regularly Varying ◽  
Normal Law

We prove that h?(x) = ??x0 y??1F?(y)dy is regularly varying with index ? [0, ?) if and only if V?(x) = ?[0,x] y?dF(y) is regularly varying with the same index, where ? > 0, F(x) is a distribution function of a nonnegative random variable, and F?(x) = 1?F(x). This contains at ? = 0, ?= 1 a result of Rogozin [8] on relative stability, and at ? = 0, ? = 2 a new, equivalent characterization of the domain of attraction of the normal law. For ? = 0 and ? > 0 our result implies a recent conjecture by Seneta [9].

Dirichlet Random Walks

2014 ◽  
Vol 51 (04) ◽  
pp. 1081-1099 ◽  
Author(s):  
Gérard Letac ◽  
Mauro Piccioni
Keyword(s):  
Random Walk ◽  
Laplace Transform ◽  
Dirichlet Process ◽  
Hypergeometric Functions ◽  
Bessel Functions ◽  
Random Variable ◽  
Infinite Divisibility ◽  
Stieltjes Transform ◽  
The Laplace Transform ◽  
Infinite Divisible Distributions

This paper provides tools for the study of the Dirichlet random walk inRd. We compute explicitly, for a number of cases, the distribution of the random variableWusing a form of Stieltjes transform ofWinstead of the Laplace transform, replacing the Bessel functions with hypergeometric functions. This enables us to simplify some existing results, in particular, some of the proofs by Le Caër (2010), (2011). We extend our results to the study of the limits of the Dirichlet random walk when the number of added terms goes to ∞, interpreting the results in terms of an integral by a Dirichlet process. We introduce the ideas of Dirichlet semigroups and Dirichlet infinite divisibility and characterize these infinite divisible distributions in the sense of Dirichlet when they are concentrated on the unit sphere ofRd.

scholarly journals The distribution and asympotic behaviour of the negative Wiener–Hopf factor for Lévy processes with rational positive jumps

2019 ◽  
Vol 56 (4) ◽  
pp. 1086-1105
Author(s):  
Ekaterina T. Kolkovska ◽  
Ehyter M. Martín-González
Keyword(s):  
Distribution Function ◽  
Laplace Transform ◽  
Probability Density ◽  
Lévy Processes ◽  
Levy Processes ◽  
Regularity Conditions ◽  
Asymptotic Results ◽  
Lévy Measure ◽  
The Laplace Transform ◽  
Additional Regularity

AbstractWe study the distribution of the negative Wiener–Hopf factor for a class of two-sided jump Lévy processes whose positive jumps have a rational Laplace transform. The positive Wiener–Hopf factor for this class of processes was studied by Lewis and Mordecki (2008). Here we obtain a formula for the Laplace transform of the negative Wiener–Hopf factor, as well as an explicit expression for its probability density in terms of sums of convolutions of known functions. Under additional regularity conditions on the Lévy measure of the studied processes, we also provide asymptotic results as $u\to-\infty$ for the distribution function F(u) of the negative Wiener–Hopf factor. We illustrate our results in some particular examples.

Continued Fraction Analysis of the Duration of an Excursion in an M/M/∞ System

1998 ◽  
Vol 35 (01) ◽  
pp. 165-183
Author(s):  
Fabrice Guillemin ◽  
Didier Pinchon
Keyword(s):  
Laplace Transform ◽  
Hypergeometric Functions ◽  
Continued Fraction ◽  
Complex Analysis ◽  
Markov Property ◽  
Random Variable ◽  
Charlier Polynomials ◽  
Kummer’S Function ◽  
The Laplace Transform ◽  
Strong Markov

We show in this paper how the Laplace transform θ* of the duration θ of an excursion by the occupation process {Λ t } of an M/M/∞ system above a given threshold can be obtained by means of continued fraction analysis. The representation of θ* by a continued fraction is established and the [m−1/m] Padé approximants are computed by means of well known orthogonal polynomials, namely associated Charlier polynomials. It turns out that the continued fraction considered is an S fraction and as a consequence the Stieltjes transform of some spectral measure. Then, using classic asymptotic expansion properties of hypergeometric functions, the representation of the Laplace transform θ* by means of Kummer's function is obtained. This allows us to recover an earlier result obtained via complex analysis and the use of the strong Markov property satisfied by the occupation process {Λ t }. The continued fraction representation enables us to further characterize the distribution of the random variable θ.

A Diffusion Model for a System Subject to Continuous Wear

1987 ◽  
Vol 1 (4) ◽  
pp. 405-416 ◽  
Author(s):  
Laurence A. Baxter ◽  
Eui Yong Lee
Keyword(s):  
Distribution Function ◽  
Brownian Motion ◽  
Laplace Transform ◽  
Poisson Process ◽  
The State ◽  
Stationary Case ◽  
The Laplace Transform ◽  
Negative Drift ◽  
State Changes ◽  
Random Amount

A model for a system whose state changes continuously with time is introduced. It is assumed that the system is modeled by Brownian motion with negative drift and an absorbing barrier at the origin. A repairman arrives according to a Poisson process and increases the state of the system by a random amount if the state is below a threshold α. Explicit expressions are deduced for the distribution function of X(t), the state of the system at time 1, if X(t) ≤ α and for the Laplace transform of the density of X( t). The stationary case is examined in detail.

APLIKASI METODE KERNEL DALAM ESTIMASI FUNGSI HAZARD

2005 ◽  
Vol 6 (2) ◽  
pp. 13
Author(s):  
Bambang Avip Priatna Martadiputra
Keyword(s):  
Distribution Function ◽  
Probability Density Function ◽  
Kernel Methods ◽  
Hazard Function ◽  
Kernel Method ◽  
Random Variable ◽  
Nonnegative Random Variable ◽  
Cumulative Hazard ◽  
Cumulative Hazard Function ◽  
Key Word

Let T be a nonnegative random variable representing the lifetimes of individuals in some population. Let f(t) denote the probability density function of T and F(t) denote the distribution function of T, the hazard function of T defined as  F(t) - 1  S(t)   whereS(t) f(t) h(t)   If equation (1) integrated we have cumulative hazard function H (t).  This paper describes application of kernel method for estimation of hazard function h (.) based censoring data. And then we will show that the hazard estimator is unbiased asymptotically, consistent, and normal asymptotically. Key word: kernel methods, estimation hazard function.

Sign in / Sign up

Export Citation Format

Share Document