Sur Une Équation Fonctionnelle Et SES Applications: Une Extension Du Théorème De Kesten-Stigum Concernant Des Processus De Branchement

1997 ◽  
Vol 29 (2) ◽  
pp. 353-373 ◽  
Author(s):  
Quansheng Liu
Keyword(s):  
Invariant Measures ◽  
Random Variables ◽  
Particle Systems ◽  
Independent Random Variables ◽  
Probability Measures ◽  
Optimal Conditions ◽  
Branching Random Walks ◽  
Random Integer ◽  
Watson Process ◽  
Infinite Particle

Given a random integer N ≧ 0 and a sequence of random variables Ai ≧ 0, we define a transformation T on the class of probability measures on [0, ∞) by letting Tμ be the distribution of , where {Zi} are independent random variables with distribution μ, which are independent of N and of {Ai} as well. We obtain the optimal conditions for the functional equation μ = Tμ to have a non-trivial solution of finite mean, and we study the existence of moments of the solutions. The work unifies the corresponding theorems of Kesten-Stigum concerning the Galton-Watson process, of Biggins for branching random walks, of Kahane-Peyrière for a model of turbulence of Yaglom made precise by Mandelbrot, and of Durrett-Liggett for the study of invariant measures for certain infinite particle systems.

Sur Une Équation Fonctionnelle Et SES Applications: Une Extension Du Théorème De Kesten-Stigum Concernant Des Processus De Branchement

1997 ◽  
Vol 29 (02) ◽  
pp. 353-373 ◽  
Author(s):  
Quansheng Liu
Keyword(s):  
Invariant Measures ◽  
Random Variables ◽  
Particle Systems ◽  
Independent Random Variables ◽  
Probability Measures ◽  
Optimal Conditions ◽  
Branching Random Walks ◽  
Random Integer ◽  
Watson Process ◽  
Infinite Particle

Given a random integerN≧ 0 and a sequence of random variablesAi≧ 0, we define a transformation T on the class of probability measures on [0, ∞) by lettingTμ be the distribution of, where {Zi} are independent random variables with distribution μ, which are independent of N and of {Ai} as well. We obtain the optimal conditions for the functional equation μ =Tμ to have a non-trivial solution of finite mean, and we study the existence of moments of the solutions. The work unifies the corresponding theorems of Kesten-Stigum concerning the Galton-Watson process, of Biggins for branching random walks, of Kahane-Peyrière for a model of turbulence of Yaglom made precise by Mandelbrot, and of Durrett-Liggett for the study of invariant measures for certain infinite particle systems.

scholarly journals An infinite particle system with additive interactions

1979 ◽  
Vol 11 (02) ◽  
pp. 355-383 ◽  
Author(s):  
Richard Durrett
Keyword(s):  
Necessary Conditions ◽  
Particle System ◽  
Particle Systems ◽  
Compact Set ◽  
Translation Invariant ◽  
Limiting Behavior ◽  
Branching Random Walks ◽  
Sufficient And Necessary Conditions ◽  
Infinite Particle ◽  
Number Of Particles

The models under consideration are a class of infinite particle systems which can be written as a superposition of branching random walks. This paper gives some results about the limiting behavior of the number of particles in a compact set ast→ ∞ and also gives both sufficient and necessary conditions for the existence of a non-trivial translation-invariant stationary distribution.

Some central limit analogues for supercritical Galton-Watson processes

1971 ◽  
Vol 8 (01) ◽  
pp. 52-59 ◽  
Author(s):  
C. C. Heyde
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Random Variables ◽  
Almost Sure Convergence ◽  
Independent Random Variables ◽  
Strong Law ◽  
Large Numbers ◽  
Watson Process ◽  
Image Position

It is possible to interpret the classical central limit theorem for sums of independent random variables as a convergence rate result for the law of large numbers. For example, ifXi, i= 1, 2, 3, ··· are independent and identically distributed random variables withEXi=μ, varXi= σ2< ∞ andthen the central limit theorem can be written in the formThis provides information on the rate of convergence in the strong lawas. (“a.s.” denotes almost sure convergence.) It is our object in this paper to discuss analogues for the super-critical Galton-Watson process.

Some central limit analogues for supercritical Galton-Watson processes

1971 ◽  
Vol 8 (1) ◽  
pp. 52-59 ◽  
Author(s):  
C. C. Heyde
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Random Variables ◽  
Almost Sure Convergence ◽  
Independent Random Variables ◽  
Strong Law ◽  
Large Numbers ◽  
Watson Process ◽  
Image Position

It is possible to interpret the classical central limit theorem for sums of independent random variables as a convergence rate result for the law of large numbers. For example, if Xi, i = 1, 2, 3, ··· are independent and identically distributed random variables with EXi = μ, var Xi = σ2 < ∞ and then the central limit theorem can be written in the form This provides information on the rate of convergence in the strong law as . (“a.s.” denotes almost sure convergence.) It is our object in this paper to discuss analogues for the super-critical Galton-Watson process.

scholarly journals An infinite particle system with additive interactions

1979 ◽  
Vol 11 (2) ◽  
pp. 355-383 ◽  
Author(s):  
Richard Durrett
Keyword(s):  
Necessary Conditions ◽  
Particle System ◽  
Particle Systems ◽  
Compact Set ◽  
Translation Invariant ◽  
Limiting Behavior ◽  
Branching Random Walks ◽  
Sufficient And Necessary Conditions ◽  
Infinite Particle ◽  
Number Of Particles

The models under consideration are a class of infinite particle systems which can be written as a superposition of branching random walks. This paper gives some results about the limiting behavior of the number of particles in a compact set as t → ∞ and also gives both sufficient and necessary conditions for the existence of a non-trivial translation-invariant stationary distribution.

Modified algorithm for fast bandwidth selection for kernel estimates of multidimensional probability densities

2020 ◽  
pp. 9-13
Author(s):  
A. V. Lapko ◽  
V. A. Lapko
Keyword(s):  
Probability Density ◽  
Optimal Parameter ◽  
Random Variables ◽  
Kernel Functions ◽  
Independent Random Variables ◽  
Functional Dependencies ◽  
Approximation Properties ◽  
Probability Density Estimation ◽  
Multidimensional Probability ◽  
Selection Of

An original technique has been justified for the fast bandwidths selection of kernel functions in a nonparametric estimate of the multidimensional probability density of the Rosenblatt–Parzen type. The proposed method makes it possible to significantly increase the computational efficiency of the optimization procedure for kernel probability density estimates in the conditions of large-volume statistical data in comparison with traditional approaches. The basis of the proposed approach is the analysis of the optimal parameter formula for the bandwidths of a multidimensional kernel probability density estimate. Dependencies between the nonlinear functional on the probability density and its derivatives up to the second order inclusive of the antikurtosis coefficients of random variables are found. The bandwidths for each random variable are represented as the product of an undefined parameter and their mean square deviation. The influence of the error in restoring the established functional dependencies on the approximation properties of the kernel probability density estimation is determined. The obtained results are implemented as a method of synthesis and analysis of a fast bandwidths selection of the kernel estimation of the two-dimensional probability density of independent random variables. This method uses data on the quantitative characteristics of a family of lognormal distribution laws.

Methodology of Calculation the Terminal Settling Velocity Distribution of Irregular Particles for High Values of the Reynold’s Number/Metodologia Wyliczania Rozkładu Granicznej Prędkości Opadania Ziaren Nieregularnych Dla Wysokich Wartości Liczb Reynoldsa

2014 ◽  
Vol 59 (2) ◽  
pp. 553-562 ◽  
Author(s):  
Agnieszka Surowiak ◽  
Marian Brożek
Keyword(s):  
Velocity Distribution ◽  
Settling Velocity ◽  
Random Variables ◽  
Independent Random Variables ◽  
Calculation Algorithm ◽  
Shape Coefficient ◽  
Geometrical Properties ◽  
Size And Shape ◽  
Physical Density ◽  
Settling Velocity Distribution

Abstract Settling velocity of particles, which is the main parameter of jig separation, is affected by physical (density) and the geometrical properties (size and shape) of particles. The authors worked out a calculation algorithm of particles settling velocity distribution for irregular particles assuming that the density of particles, their size and shape constitute independent random variables of fixed distributions. Applying theorems of probability, concerning distributions function of random variables, the authors present general formula of probability density function of settling velocity irregular particles for the turbulent motion. The distributions of settling velocity of irregular particles were calculated utilizing industrial sample. The measurements were executed and the histograms of distributions of volume and dynamic shape coefficient, were drawn. The separation accuracy was measured by the change of process imperfection of irregular particles in relation to spherical ones, resulting from the distribution of particles settling velocity.

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