scholarly journals Laplace Transforms of Probability Distributions and Their Inversions are Easy on Logarithmic Scales

2008 ◽  
Vol 45 (2) ◽  
pp. 531-541 ◽  
Author(s):  
A. G. Rossberg
Keyword(s):  
Laplace Transform ◽  
High Precision ◽  
Computational Efficiency ◽  
Generating Functions ◽  
Heavy Tails ◽  
Probability Distributions ◽  
Random Variables ◽  
Numerical Examples ◽  
Pareto Distributions ◽  
The Laplace Transform

It is shown that, when expressing arguments in terms of their logarithms, the Laplace transform of a function is related to the antiderivative of this function by a simple convolution. This allows efficient numerical computations of moment generating functions of positive random variables and their inversion. The application of the method is straightforward, apart from the necessity to implement it using high-precision arithmetics. In numerical examples the approach is demonstrated to be particularly useful for distributions with heavy tails, such as lognormal, Weibull, or Pareto distributions, which are otherwise difficult to handle. The computational efficiency compared to other methods is demonstrated for an M/G/1 queueing problem.

scholarly journals Laplace Transforms of Probability Distributions and Their Inversions are Easy on Logarithmic Scales

2008 ◽  
Vol 45 (02) ◽  
pp. 531-541 ◽  
Author(s):  
A. G. Rossberg
Keyword(s):  
Laplace Transform ◽  
High Precision ◽  
Computational Efficiency ◽  
Generating Functions ◽  
Heavy Tails ◽  
Probability Distributions ◽  
Random Variables ◽  
Numerical Examples ◽  
Pareto Distributions ◽  
The Laplace Transform

It is shown that, when expressing arguments in terms of their logarithms, the Laplace transform of a function is related to the antiderivative of this function by a simple convolution. This allows efficient numerical computations of moment generating functions of positive random variables and their inversion. The application of the method is straightforward, apart from the necessity to implement it using high-precision arithmetics. In numerical examples the approach is demonstrated to be particularly useful for distributions with heavy tails, such as lognormal, Weibull, or Pareto distributions, which are otherwise difficult to handle. The computational efficiency compared to other methods is demonstrated for an M/G/1 queueing problem.

scholarly journals Degenerate Stirling Polynomials of the Second Kind and Some Applications

Symmetry ◽  
2019 ◽  
Vol 11 (8) ◽  
pp. 1046 ◽  
Author(s):  
Taekyun Kim ◽  
Dae San Kim ◽  
Han Young Kim ◽  
Jongkyum Kwon
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Probability Distributions ◽  
Random Variables

Recently, the degenerate λ -Stirling polynomials of the second kind were introduced and investigated for their properties and relations. In this paper, we continue to study the degenerate λ -Stirling polynomials as well as the r-truncated degenerate λ -Stirling polynomials of the second kind which are derived from generating functions and Newton’s formula. We derive recurrence relations and various expressions for them. Regarding applications, we show that both the degenerate λ -Stirling polynomials of the second and the r-truncated degenerate λ -Stirling polynomials of the second kind appear in the expressions of the probability distributions of appropriate random variables.

scholarly journals Approximating the Laplace transform of the sum of dependent lognormals

2016 ◽  
Vol 48 (A) ◽  
pp. 203-215 ◽  
Author(s):  
Patrick J. Laub ◽  
Søren Asmussen ◽  
Jens L. Jensen ◽  
Leonardo Rojas-Nandayapa
Keyword(s):  
Monte Carlo ◽  
Laplace Transform ◽  
Taylor Expansion ◽  
Multivariate Normal ◽  
Numerical Examples ◽  
Quasi Monte Carlo ◽  
Error Factor ◽  
The Laplace Transform ◽  
Laplace Transform Inversion ◽  
Mean Vector

AbstractLet (X1,...,Xn) be multivariate normal, with mean vector 𝛍 and covariance matrix 𝚺, and letSn=eX1+⋯+eXn. The Laplace transform ℒ(θ)=𝔼e-θSn∝∫exp{-hθ(𝒙)}d𝒙 is represented as ℒ̃(θ)I(θ), where ℒ̃(θ) is given in closed form andI(θ) is the error factor (≈1). We obtain ℒ̃(θ) by replacinghθ(𝒙) with a second-order Taylor expansion around its minimiser 𝒙*. An algorithm for calculating the asymptotic expansion of 𝒙*is presented, and it is shown thatI(θ)→ 1 as θ→∞. A variety of numerical methods for evaluatingI(θ) is discussed, including Monte Carlo with importance sampling and quasi-Monte Carlo. Numerical examples (including Laplace-transform inversion for the density ofSn) are also given.

scholarly journals Heavy-tailed random matrices

2018 ◽  
Author(s):  
Vladimir Kravtsov
Keyword(s):  
Random Matrices ◽  
Heavy Tails ◽  
Probability Distributions ◽  
Basin Of Attraction ◽  
Random Variables ◽  
Universal Properties ◽  
Random Matrix Ensembles ◽  
Free Random Variables ◽  
Heavy Tailed ◽  
Non Gaussian

This article considers non-Gaussian random matrices consisting of random variables with heavy-tailed probability distributions. In probability theory heavy tails of distributions describe rare but violent events which usually have a dominant influence on the statistics. Furthermore, they completely change the universal properties of eigenvalues and eigenvectors of random matrices. This article focuses on the universal macroscopic properties of Wigner matrices belonging to the Lévy basin of attraction, matrices representing stable free random variables, and a class of heavy-tailed matrices obtained by parametric deformations of standard ensembles. It first examines the properties of heavy-tailed symmetric matrices known as Wigner–Lévy matrices before discussing free random variables and free Lévy matrices as well as heavy-tailed deformations. In particular, it describes random matrix ensembles obtained from standard ensembles by a reweighting of the probability measure. It also analyses several matrix models belonging to heavy-tailed random matrices and presents methods for integrating them.

scholarly journals Descriptor fractional linear systems with regular pencils

2013 ◽  
Vol 23 (2) ◽  
pp. 309-315 ◽  
Author(s):  
Tadeusz Kaczorek
Keyword(s):  
Laplace Transform ◽  
Linear Systems ◽  
Discrete Time ◽  
Continuous Time ◽  
The State ◽  
Transition Matrices ◽  
State Equations ◽  
Numerical Examples ◽  
The Laplace Transform ◽  
Time Linear

Methods for finding solutions of the state equations of descriptor fractional discrete-time and continuous-time linear systems with regular pencils are proposed. The derivation of the solution formulas is based on the application of the Z transform, the Laplace transform and the convolution theorems. Procedures for computation of the transition matrices are proposed. The efficiency of the proposed methods is demonstrated on simple numerical examples.

A Technique for Increasing the Accuracy of the Numerical Inversion of the Laplace Transform With Applications

1973 ◽  
Vol 40 (4) ◽  
pp. 1110-1112 ◽  
Author(s):  
B. S. Berger ◽  
S. Duangudom
Keyword(s):  
Laplace Transform ◽  
Initial Value Problems ◽  
Numerical Inversion ◽  
Initial Value ◽  
Numerical Examples ◽  
Oscillatory Function ◽  
Higher Order Terms ◽  
The Laplace Transform ◽  
Image Function ◽  
Inversion Techniques

A technique is introduced which extends the range of useful approximation of numerical inversion techniques to many cycles of an oscillatory function without requiring either the evaluation of the image function for many values of s or the computation of higher-order terms. The technique consists in reducing a given initial value problem defined over some interval into a sequence of initial value problems defined over a set of subintervals. Several numerical examples demonstrate the utility of the method.

scholarly journals Combination of Laplace transform and residual power series techniques to solve autonomous n-dimensional fractional nonlinear systems

2021 ◽  
Vol 10 (1) ◽  
pp. 282-292
Author(s):  
Marwan Alquran ◽  
Maysa Alsukhour ◽  
Mohammed Ali ◽  
Imad Jaradat
Keyword(s):  
Nonlinear Systems ◽  
Power Series ◽  
Laplace Transform ◽  
Iterative Algorithm ◽  
Autonomous Systems ◽  
Numerical Examples ◽  
The Laplace Transform ◽  
Residual Power Series ◽  
The Impact ◽  
Residual Power

Abstract In this work, a new iterative algorithm is presented to solve autonomous n-dimensional fractional nonlinear systems analytically. The suggested scheme is combination of two methods; the Laplace transform and the residual power series. The methodology of this algorithm is presented in details. For the accuracy and effectiveness purposes, two numerical examples are discussed. Finally, the impact of the fractional order acting on these autonomous systems is investigated using graphs and tables.

On a modified counter with prolonging dead time

1985 ◽  
Vol 22 (3) ◽  
pp. 678-687 ◽  
Author(s):  
A. Dvurečenskij ◽  
G. A. Ososkov
Keyword(s):  
Laplace Transform ◽  
Dead Time ◽  
Random Variables ◽  
The Laplace Transform ◽  
Interarrival Times

Emitted particles arrive at the counter with prolonging dead time so that the interarrival times and the lengths of impulses in any dead time are independent but not necessarily identically distributed random variables, and whenever the counter is idle then the following evolution starts from the beginning. For this class of counters we derive the probability laws of the numbers of particles arriving at the counters during their dead times, and the Laplace transform of the cycle, respectively.

Some probability distributions connected with recording apparatus

1948 ◽  
Vol 44 (3) ◽  
pp. 335-341 ◽  
Author(s):  
C. Domb
Keyword(s):  
Probability Distribution ◽  
Laplace Transform ◽  
Asymptotic Behaviour ◽  
Explicit Expression ◽  
Probability Distributions ◽  
Practical Interest ◽  
Observation Time ◽  
Constant Length ◽  
Recording Apparatus ◽  
The Laplace Transform

1. The problem to be discussed in the present paper arises when the finite resolving time of a recording apparatus is taken into account. Events are divided into two classes, recorded and unrecorded. Any recorded event is followed by a dead interval of a certain length of time, during which any other event which occurs will be unrecorded. A typical example is an α-particle counter; a recorded particle causes the chamber to ionize, and no other particle can be recorded until the chamber has deionized. In the case when the dead interval is of constant length τ, if the observation time t lies between (n − 1)τ and nτ, the number of recorded events must be 0, 1, 2, …, n − 1 or n. We shall assume that the probability of an event occurring in the interval [t, t + dt] is λdt, λ being constant; this is the case of most practical interest. The probability distribution of recorded events for dead intervals of constant length has been determined by Ruark and Devol (2). The explicit expression of this distribution is fairly complicated, and it is therefore difficult to manipulate. The method employed in the present paper leads naturally to the Laplace transform of the distribution with respect to the time t, and this is relatively simple. The method can easily be generalized to deal with the case of dead intervals which are not all equal, but follow a probability distribution u(τ) dτ. Finally the Laplace transform is very convenient for determining the asymptotic behaviour of the distribution of recorded events for large times of observation.

On a modified counter with prolonging dead time

1985 ◽  
Vol 22 (03) ◽  
pp. 678-687
Author(s):  
A. Dvurečenskij ◽  
G. A. Ososkov
Keyword(s):  
Laplace Transform ◽  
Dead Time ◽  
Random Variables ◽  
The Laplace Transform ◽  
Interarrival Times

Emitted particles arrive at the counter with prolonging dead time so that the interarrival times and the lengths of impulses in any dead time are independent but not necessarily identically distributed random variables, and whenever the counter is idle then the following evolution starts from the beginning. For this class of counters we derive the probability laws of the numbers of particles arriving at the counters during their dead times, and the Laplace transform of the cycle, respectively.

Sign in / Sign up

Export Citation Format

Share Document