Maximal branching processes and ‘long-range percolation’

1970 ◽  
Vol 7 (01) ◽  
pp. 89-98
Author(s):  
John Lamperti
Keyword(s):  
Probability Distribution Function ◽  
Branching Process ◽  
Branching Processes ◽  
Random Variables ◽  
Random Variable ◽  
Independent Random Variables ◽  
Transition Matrices ◽  
A Chain ◽  
Image Position ◽  
The Right

In the first part of this paper, we will consider a class of Markov chains on the non-negative integers which resemble the Galton-Watson branching process, but with one major difference. If there are k individuals in the nth “generation”, and are independent random variables representing their respective numbers of offspring, then the (n + 1)th generation will contain max individuals rather than as in the branching case. Equivalently, the transition matrices Pij of the chains we will study are to be of the form where F(.) is the probability distribution function of a non-negative, integervalued random variable. The right-hand side of (1) is thus the probability that the maximum of i independent random variables distributed by F has the value j. Such a chain will be called a “maximal branching process”.

Maximal branching processes and ‘long-range percolation’

1970 ◽  
Vol 7 (1) ◽  
pp. 89-98 ◽  
Author(s):  
John Lamperti
Keyword(s):  
Probability Distribution Function ◽  
Branching Process ◽  
Branching Processes ◽  
Random Variables ◽  
Random Variable ◽  
Independent Random Variables ◽  
Transition Matrices ◽  
A Chain ◽  
Image Position ◽  
The Right

In the first part of this paper, we will consider a class of Markov chains on the non-negative integers which resemble the Galton-Watson branching process, but with one major difference. If there are k individuals in the nth “generation”, and are independent random variables representing their respective numbers of offspring, then the (n + 1)th generation will contain max individuals rather than as in the branching case. Equivalently, the transition matrices Pij of the chains we will study are to be of the form where F(.) is the probability distribution function of a non-negative, integervalued random variable. The right-hand side of (1) is thus the probability that the maximum of i independent random variables distributed by F has the value j. Such a chain will be called a “maximal branching process”.

Continuity for multi-type branching processes with varying environments

1999 ◽  
Vol 36 (01) ◽  
pp. 139-145 ◽  
Author(s):  
Owen Dafydd Jones
Keyword(s):  
Linear Combination ◽  
Branching Process ◽  
Branching Processes ◽  
Random Variables ◽  
Main Tool ◽  
Concentration Function ◽  
Independent Random Variables ◽  
Varying Environments

Conditions are derived for the components of the normed limit of a multi-type branching process with varying environments, to be continuous on (0, ∞). The main tool is an inequality for the concentration function of sums of independent random variables, due originally to Petrov. Using this, we show that if there is a discontinuity present, then a particular linear combination of the population types must converge to a non-random constant (Equation (1)). Ensuring this can not happen provides the desired continuity conditions.

Convergence rates in the law of large numbers. II

1968 ◽  
Vol 64 (2) ◽  
pp. 485-488 ◽  
Author(s):  
V. K. Rohatgi
Keyword(s):  
Convergence Rates ◽  
Random Variables ◽  
Almost Sure Convergence ◽  
Random Variable ◽  
Rates Of Convergence ◽  
Independent Random Variables ◽  
Large Numbers ◽  
Present Context ◽  
Image Position ◽  
Uniformly Bounded

Let {Xn: n ≥ 1} be a sequence of independent random variables and write Suppose that the random vairables Xn are uniformly bounded by a random variable X in the sense thatSet qn(x) = Pr(|Xn| > x) and q(x) = Pr(|Xn| > x). If qn ≤ q and E|X|r < ∞ with 0 < r < 2 then we have (see Loève(4), 242)where ak = 0, if 0 < r < 1, and = EXk if 1 ≤ r < 2 and ‘a.s.’ stands for almost sure convergence. the purpose of this paper is to study the rates of convergence ofto zero for arbitrary ε > 0. We shall extend to the present context, results of (3) where the case of identically distributed random variables was treated. The techniques used here are strongly related to those of (3).

scholarly journals Probabalistic modelling of the results of economic activity as a function of random values

2020 ◽  
pp. 102-112
Author(s):  
Olesya Martyniuk ◽  
Stepan Popina ◽  
Serhii Martyniuk
Keyword(s):  
Mathematical Modeling ◽  
Distribution Function ◽  
Probability Distribution ◽  
Economic Activity ◽  
Probability Distribution Function ◽  
Random Variables ◽  
Economic Structure ◽  
Random Variable ◽  
Independent Random Variables ◽  
Research Results

Introduction. Mathematical modeling of economic processes is necessary for the unambiguous formulation and solution of the problem. In the economic sphere this is the most important aspect of the activity of any enterprise, for which economic-mathematical modeling is the tool that allows to make adequate decisions. However, economic indicators that are factors of a model are usually random variables. An economic-mathematical model is proposed for calculating the probability distribution function of the result of economic activity on the basis of the known dependence of this result on factors influencing it and density of probability distribution of these factors. Methods. The formula was used to calculate the random variable probability distribution function, which is a function of other independent random variables. The method of estimation of basic numerical characteristics of the investigated functions of random variables is proposed: mathematical expectation that in the probabilistic sense is the average value of the result of functioning of the economic structure, as well as its variance. The upper bound of the variation of the effective feature is indicated. Results. The cases of linear and power functions of two independent variables are investigated. Different cases of two-dimensional domain of possible values of indicators, which are continuous random variables, are considered. The application of research results to production functions is considered. Examples of estimating the probability distribution function of a random variable are offered. Conclusions. The research results allow in the probabilistic sense to estimate the result of the economic structure activity on the basis of the probabilistic distributions of the values of the dependent variables. The prospect of further research is to apply indirect control over economic performance based on economic and mathematical modeling.

On Fibonacci (or lagged Bienaymé-Galton-Watson) branching processes

1981 ◽  
Vol 18 (03) ◽  
pp. 583-591
Author(s):  
C. C. Heyde
Keyword(s):  
Rate Of Convergence ◽  
Discrete Time ◽  
Population Model ◽  
Branching Processes ◽  
Random Variables ◽  
Random Variable ◽  
Independent Random Variables ◽  
Lines Of Descent ◽  
Number Of Individuals ◽  
First Time

This paper is concerned with a discrete-time population model in which a new individual entering the population at time t can produce offspring for the first time at time t + 2 and then subsequently at times t + 3, t + 4, ···. The numbers of offspring produced on each occasion are independent random variables each with the distribution of Z for which EZ = m &lt;∞, and individuals have independent lines of descent. This model is contrasted with the corresponding Bienaymé-Galton-Watson one. If Xn denotes the number of individuals in the population at time n, it is shown that z–nXn almost surely converges to a random variable W, as n→∞, where Various properties of W are obtained, in particular W &gt; 0 a.s. if and only if EZ | log Z | &lt; ∞ Results are also given on the rate of convergence of to z when Var Z &lt; ∞ and these display a surprising dependence on the size of z.

Stochastic majorization of random variables by proportional equilibrium rates

1987 ◽  
Vol 19 (04) ◽  
pp. 854-872 ◽  
Author(s):  
J. George Shanthikumar
Keyword(s):  
Random Variables ◽  
Random Variable ◽  
Independent Random Variables ◽  
Concave Functions ◽  
Schur Convex ◽  
Image Position ◽  
Stochastic Majorization

The equilibrium rate rY of a random variable Y with support on non-negative integers is defined by rY (0) = 0 and rY (n) = P[Y = n – 1]/P[Y – n], Let (j = 1, …, m; i = 1,2) be 2m independent random variables that have proportional equilibrium rates with (j = 1, …, m; i = 1, 2) as the constant of proportionality. When the equilibrium rate is increasing and concave [convex] it is shown that , …, ) majorizes implies , …, for all increasing Schur-convex [concave] functions whenever the expectations exist. In addition if , (i = 1, 2), then

A theorem on the cumulative product of independent random variables

1959 ◽  
Vol 55 (4) ◽  
pp. 333-337 ◽  
Author(s):  
Harold Ruben
Keyword(s):  
Sufficient Conditions ◽  
Random Variables ◽  
Random Variable ◽  
Necessary And Sufficient Conditions ◽  
Independent Random Variables ◽  
Product Theorem ◽  
Image Position ◽  
Introductory Discussion ◽  
Necessary And Sufficient ◽  
Complex Valued

1. Introductory discussion and summary. Consider a sequence {ui} of independent real or complex-valued random variables such that E(ui) = 1, and a sequence of mutually exclusive events S1, S2,…, such that Si depends only on u1, u2, …,ui, with ΣP(Sj) = 1. Define the random variable n = n(u1, u2,…) = m when Sm occurs. We shall obtain the necessary and sufficient conditions under whichreferred to as the product theorem.

scholarly journals Conditional limit theorems for branching processes

1991 ◽  
Vol 4 (4) ◽  
pp. 263-292 ◽  
Author(s):  
Lajos Takács
Keyword(s):  
Asymptotic Behavior ◽  
Limit Theorems ◽  
Branching Process ◽  
Branching Processes ◽  
Random Variables ◽  
Random Variable ◽  
Conditional Limit Theorems ◽  
Total Progeny ◽  
Number Of Individuals ◽  
Conditional Limit

Let [ξ(m),m=0,1,2,…] be a branching process in which each individual reproduces independently of the others and has probability pj(j=0,1,2,…) of giving rise to j descendants in the following generation. The random variable ξ(m) is the number of individuals in the mth generation. It is assumed that P{ξ(0)=1}=1. Denote by ρ the total progeny, μ, the time of extinction, and τ, the total number of ancestors of all the individuals in the process. This paper deals with the distributions of the random variables ξ(m), μ and τ under the condition that ρ=n and determines the asymptotic behavior of these distributions in the case where n→∞ and m→∞ in such a way that m/n tends to a finite positive limit.

A uniform limit theorem and exponential limit law for critical multitype age-dependent branching processes

1978 ◽  
Vol 15 (2) ◽  
pp. 235-242 ◽  
Author(s):  
Martin I. Goldstein
Keyword(s):  
Limit Theorem ◽  
Generating Function ◽  
Branching Process ◽  
Branching Processes ◽  
Uniform Limit ◽  
Second Moment ◽  
Age Dependent ◽  
The Mean ◽  
Image Position ◽  
The Right

Let Z(t) ··· (Z1(t), …, Zk (t)) be an indecomposable critical k-type age-dependent branching process with generating function F(s, t). Denote the right and left eigenvalues of the mean matrix M by u and v respectively and suppose μ is the vector of mean lifetimes, i.e. Mu = u, vM = v.It is shown that, under second moment assumptions, uniformly for s ∈ ([0, 1]k of the form s = 1 – cu, c a constant. Here vμ is the componentwise product of the vectors and Q[u] is a constant.This result is then used to give a new proof of the exponential limit law.

scholarly journals Attainable bounds for expectations

1982 ◽  
Vol 33 (3) ◽  
pp. 411-420 ◽  
Author(s):  
E. Seneta ◽  
N. C. Weber
Keyword(s):  
Lower Bound ◽  
Generating Functions ◽  
Branching Process ◽  
Simple Technique ◽  
Branching Processes ◽  
Random Variables ◽  
Random Variable ◽  
Malthusian Parameter ◽  
Age Dependent ◽  
Coefficient Of Skewness

AbstractA simple technique for obtaining bounds in terms of means and variances for the expectations of certain functions of random variables in a given class is examined. The bounds given are sharp in the sense that they are attainable by at least one random variable in the class. This technique is applied to obtain bounds for moment generating functions, the coefficient of skewness and parameters associated with branching processes. In particular an improved lower bound for the Malthusian parameter in an age-dependent branching process is derived.

Sign in / Sign up

Export Citation Format

Share Document