Series expansions of probability generating functions and bounds for the extinction probability of a branching process

1980 ◽  
Vol 17 (04) ◽  
pp. 939-947 ◽  
Author(s):  
D. J. Daley ◽  
Prakash Narayan
Keyword(s):  
Remainder Term ◽  
Branching Process ◽  
Factorial Moment ◽  
Probability Generating Function ◽  
Taylor Series Expansion ◽  
Random Variable ◽  
Extinction Probability ◽  
Series Expansions ◽  
Power Series Expansions ◽  
New Inequalities

In the Taylor series expansion about s = 1 of the probability generating function f(s) of a non-negative integer-valued random variable with finite nth factorial moment the remainder term is proportional to another p.g.f. This leads to simple proofs of other power series expansions for p.g.f.'s, including an inversion formula giving the distribution in terms of the moments (when this can be done). Old and new inequalities for the extinction probability of a branching process are established.

Series expansions of probability generating functions and bounds for the extinction probability of a branching process

1980 ◽  
Vol 17 (4) ◽  
pp. 939-947 ◽  
Author(s):  
D. J. Daley ◽  
Prakash Narayan
Keyword(s):  
Remainder Term ◽  
Branching Process ◽  
Factorial Moment ◽  
Probability Generating Function ◽  
Taylor Series Expansion ◽  
Random Variable ◽  
Extinction Probability ◽  
Series Expansions ◽  
Power Series Expansions ◽  
New Inequalities

In the Taylor series expansion about s = 1 of the probability generating function f(s) of a non-negative integer-valued random variable with finite nth factorial moment the remainder term is proportional to another p.g.f. This leads to simple proofs of other power series expansions for p.g.f.'s, including an inversion formula giving the distribution in terms of the moments (when this can be done). Old and new inequalities for the extinction probability of a branching process are established.

Inequalities for branching processes

1966 ◽  
Vol 3 (01) ◽  
pp. 261-267 ◽  
Author(s):  
C. R. Heathcote ◽  
E. Seneta
Keyword(s):  
Generating Function ◽  
Conditional Distribution ◽  
Branching Process ◽  
Branching Processes ◽  
Probability Generating Function ◽  
Random Variable ◽  
Expected Time ◽  
Time To Extinction ◽  
Explicit Formulae

Summary If F(s) is the probability generating function of a non-negative random variable, the nth functional iterate Fn (s) = Fn– 1 (F(s)) generates the distribution of the size of the nth generation of a simple branching process. In general it is not possible to obtain explicit formulae for many quantities involving Fn (s), and this paper considers certain bounds and approximations. Bounds are found for the Koenigs-type iterates lim n→∞ m −n {1−Fn (s)}, 0 ≦ s ≦ 1 where m = F′ (1) < 1 and F′′ (1) < ∞; for the expected time to extinction and for the limiting conditional-distribution generating function limn→∞{Fn (s) − Fn (0)} [1 – Fn (0)]–1. Particular attention is paid to the case F(s) = exp {m(s − 1)}.

An age dependent branching process with variable lifetime distribution

1972 ◽  
Vol 4 (3) ◽  
pp. 453-474 ◽  
Author(s):  
Robert Fildes
Keyword(s):  
Integral Equations ◽  
Branching Process ◽  
Probability Generating Function ◽  
Random Variable ◽  
Lifetime Distribution ◽  
Distribution Functions ◽  
Mean Square ◽  
Initial Particle ◽  
Age Dependent ◽  
Number Of Particles

A branching process with variable lifetime distribution is defined by a sequence of distribution functions {Gi(t)}, together with a probability generating function, h(s) = Σk∞= 0pksk. An ith generation particle lives a random length of time, determined by Gi(t). At the end of a particle's life it produces children, the number being determined by h(s). These offspring behave like the initial particle except they are (i + 1)th generation particles and have lifetime distribution Gi + 1 (t).Let Zi(t) be the number of particles alive at time t, the initial particle being born into the ith generation. Integral equations are derived for the moments of Zi(t) and it is shown that for some constants Ni, γ, a, Zi (t)/(Nitγ-1eαt) converges in mean square to a proper random variable.

scholarly journals New Inequalities of Cusa–Huygens Type

Mathematics ◽  
2021 ◽  
Vol 9 (17) ◽  
pp. 2101
Author(s):  
Ling Zhu
Keyword(s):  
Power Series ◽  
Bernoulli Numbers ◽  
Series Expansions ◽  
Power Series Expansions ◽  
New Inequalities ◽  
Better Than

Using the power series expansions of the functions cotx,1/sinx and 1/sin2x, and the estimate of the ratio of two adjacent even-indexed Bernoulli numbers, we improve Cusa–Huygens inequality in two directions on 0,π/2. Our results are much better than those in the existing literature.

The fractional linear probability generating function in the random environment branching process

1994 ◽  
Vol 31 (01) ◽  
pp. 38-47 ◽  
Author(s):  
D. R. Grey ◽  
Lu Zhunwei
Keyword(s):  
Random Environment ◽  
Generating Functions ◽  
Branching Process ◽  
Probability Generating Function ◽  
Random Variable ◽  
Random Environments ◽  
Distributed Environments ◽  
Absolutely Continuous ◽  
Tail Behaviour ◽  
Linear Probability

In a branching process with random environments, the probability of ultimate extinction is a function of the environment sequence, and is therefore a random variable. Explicit results about the distribution of this random variable are difficult to obtain in general. Here we assume independent and identically distributed environments and use the special properties of fractional linear generating functions to derive some explicit distributions, which may be singular or absolutely continuous, depending on the values of certain parameters. We also consider briefly tail behaviour close to 1, and provide an extension to cases where probability generating functions are not fractional linear.

An unexpected connection between branching processes and optimal stopping

2000 ◽  
Vol 37 (3) ◽  
pp. 613-626 ◽  
Author(s):  
David Assaf ◽  
Larry Goldstein ◽  
Ester Samuel-Cahn
Keyword(s):  
Optimal Stopping ◽  
Branching Process ◽  
Branching Processes ◽  
Infinite Horizon ◽  
Random Variable ◽  
Rates Of Convergence ◽  
Extinction Probability ◽  
Independent Random Variables ◽  
Rule Method ◽  
Watson Process

A curious connection exists between the theory of optimal stopping for independent random variables, and branching processes. In particular, for the branching process Zn with offspring distribution Y, there exists a random variable X such that the probability P(Zn = 0) of extinction of the nth generation in the branching process equals the value obtained by optimally stopping the sequence X1,…, Xn, where these variables are i.i.d. distributed as X. Generalizations to the inhomogeneous and infinite horizon cases are also considered. This correspondence furnishes a simple ‘stopping rule’ method for computing various characteristics of branching processes, including rates of convergence of the nth generation's extinction probability to the eventual extinction probability, for the supercritical, critical and subcritical Galton-Watson process. Examples, bounds, further generalizations and a connection to classical prophet inequalities are presented. Throughout, the aim is to show how this unexpected connection can be used to translate methods from one area of applied probability to another, rather than to provide the most general results.

An age dependent branching process with variable lifetime distribution

1972 ◽  
Vol 4 (03) ◽  
pp. 453-474 ◽  
Author(s):  
Robert Fildes
Keyword(s):  
Integral Equations ◽  
Branching Process ◽  
Probability Generating Function ◽  
Random Variable ◽  
Lifetime Distribution ◽  
Distribution Functions ◽  
Mean Square ◽  
Initial Particle ◽  
Age Dependent ◽  
Number Of Particles

A branching process with variable lifetime distribution is defined by a sequence of distribution functions {G i (t)}, together with a probability generating function, h(s) = Σ k ∞= 0pks k . An ith generation particle lives a random length of time, determined by G i (t). At the end of a particle's life it produces children, the number being determined by h(s). These offspring behave like the initial particle except they are (i + 1)th generation particles and have lifetime distribution G i + 1 (t). Let Z i (t) be the number of particles alive at time t, the initial particle being born into the ith generation. Integral equations are derived for the moments of Z i (t) and it is shown that for some constants N i , γ, a, Z i (t)/(N i t γ-1 e αt ) converges in mean square to a proper random variable.

scholarly journals An unexpected connection between branching processes and optimal stopping

2000 ◽  
Vol 37 (03) ◽  
pp. 613-626 ◽  
Author(s):  
David Assaf ◽  
Larry Goldstein ◽  
Ester Samuel-Cahn
Keyword(s):  
Optimal Stopping ◽  
Branching Process ◽  
Branching Processes ◽  
Infinite Horizon ◽  
Random Variable ◽  
Rates Of Convergence ◽  
Extinction Probability ◽  
Independent Random Variables ◽  
Rule Method ◽  
Watson Process

A curious connection exists between the theory of optimal stopping for independent random variables, and branching processes. In particular, for the branching process Z n with offspring distribution Y, there exists a random variable X such that the probability P(Z n = 0) of extinction of the nth generation in the branching process equals the value obtained by optimally stopping the sequence X 1,…, X n , where these variables are i.i.d. distributed as X. Generalizations to the inhomogeneous and infinite horizon cases are also considered. This correspondence furnishes a simple ‘stopping rule’ method for computing various characteristics of branching processes, including rates of convergence of the nth generation's extinction probability to the eventual extinction probability, for the supercritical, critical and subcritical Galton-Watson process. Examples, bounds, further generalizations and a connection to classical prophet inequalities are presented. Throughout, the aim is to show how this unexpected connection can be used to translate methods from one area of applied probability to another, rather than to provide the most general results.

The fractional linear probability generating function in the random environment branching process

1994 ◽  
Vol 31 (1) ◽  
pp. 38-47 ◽  
Author(s):  
D. R. Grey ◽  
Lu Zhunwei
Keyword(s):  
Random Environment ◽  
Generating Functions ◽  
Branching Process ◽  
Probability Generating Function ◽  
Random Variable ◽  
Random Environments ◽  
Distributed Environments ◽  
Absolutely Continuous ◽  
Tail Behaviour ◽  
Linear Probability

In a branching process with random environments, the probability of ultimate extinction is a function of the environment sequence, and is therefore a random variable. Explicit results about the distribution of this random variable are difficult to obtain in general. Here we assume independent and identically distributed environments and use the special properties of fractional linear generating functions to derive some explicit distributions, which may be singular or absolutely continuous, depending on the values of certain parameters. We also consider briefly tail behaviour close to 1, and provide an extension to cases where probability generating functions are not fractional linear.

Inequalities for branching processes

1966 ◽  
Vol 3 (1) ◽  
pp. 261-267 ◽  
Author(s):  
C. R. Heathcote ◽  
E. Seneta
Keyword(s):  
Generating Function ◽  
Conditional Distribution ◽  
Branching Process ◽  
Branching Processes ◽  
Probability Generating Function ◽  
Random Variable ◽  
Expected Time ◽  
Time To Extinction ◽  
Explicit Formulae

SummaryIf F(s) is the probability generating function of a non-negative random variable, the nth functional iterate Fn(s) = Fn–1 (F(s)) generates the distribution of the size of the nth generation of a simple branching process. In general it is not possible to obtain explicit formulae for many quantities involving Fn(s), and this paper considers certain bounds and approximations. Bounds are found for the Koenigs-type iterates limn→∞m−n {1−Fn(s)}, 0 ≦ s ≦ 1 where m = F′(1) < 1 and F′′(1) < ∞; for the expected time to extinction and for the limiting conditional-distribution generating function limn→∞{Fn(s) − Fn(0)} [1 – Fn(0)]–1. Particular attention is paid to the case F(s) = exp {m(s − 1)}.

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