On a result of Rubin and Vere-Jones concerning subcritical branching processes

1976 ◽  
Vol 13 (4) ◽  
pp. 804-808 ◽  
Author(s):  
Fred M. Hoppe
Keyword(s):  
Mass Distribution ◽  
Generating Function ◽  
Branching Processes ◽  
Probability Generating Function ◽  
Single Parameter ◽  
Limit Distributions ◽  
Arbitrary Mass ◽  
One To One ◽  
Watson Process ◽  
Conditional Limit

If a subcritical Galton-Watson process is initiated with an arbitrary mass distribution, then it is known that under certain conditions proper conditional limit distributions exist, depending on a single parameter. It is shown here that there is a one-to-one correspondence between these distributions and those arising from the process with a linear offspring probability generating function.

On a result of Rubin and Vere-Jones concerning subcritical branching processes

1976 ◽  
Vol 13 (04) ◽  
pp. 804-808
Author(s):  
Fred M. Hoppe
Keyword(s):  
Mass Distribution ◽  
Generating Function ◽  
Branching Processes ◽  
Probability Generating Function ◽  
Single Parameter ◽  
Limit Distributions ◽  
Arbitrary Mass ◽  
One To One ◽  
Watson Process ◽  
Conditional Limit

If a subcritical Galton-Watson process is initiated with an arbitrary mass distribution, then it is known that under certain conditions proper conditional limit distributions exist, depending on a single parameter. It is shown here that there is a one-to-one correspondence between these distributions and those arising from the process with a linear offspring probability generating function.

A note on a functional equation arising in Galton-Watson branching processes

1971 ◽  
Vol 8 (3) ◽  
pp. 589-598 ◽  
Author(s):  
Krishna B. Athreya
Keyword(s):  
Functional Equation ◽  
Continuous Function ◽  
Generating Function ◽  
Branching Process ◽  
Branching Processes ◽  
Probability Generating Function ◽  
Age Dependent ◽  
Watson Process ◽  
Natural Way

The functional equation ϕ(mu) = h(ϕ(u)) where is a probability generating function with 1 < m = h'(1 –) < ∞ and where F(t) is a non-decreasing right continuous function with F(0 –) = 0, F(0 +) < 1 and F(+ ∞) = 1 arises in a Galton-Watson process in a natural way. We prove here that for any if and only if This unifies several results in the literature on the supercritical Galton-Watson process. We generalize this to an age dependent branching process case as well.

A note on a functional equation arising in Galton-Watson branching processes

1971 ◽  
Vol 8 (03) ◽  
pp. 589-598 ◽  
Author(s):  
Krishna B. Athreya
Keyword(s):  
Functional Equation ◽  
Continuous Function ◽  
Generating Function ◽  
Branching Process ◽  
Branching Processes ◽  
Probability Generating Function ◽  
Age Dependent ◽  
Watson Process ◽  
Natural Way

The functional equation ϕ(mu) = h(ϕ(u)) where is a probability generating function with 1 &lt; m = h'(1 –) &lt; ∞ and where F(t) is a non-decreasing right continuous function with F(0 –) = 0, F(0 +) &lt; 1 and F(+ ∞) = 1 arises in a Galton-Watson process in a natural way. We prove here that for any if and only if This unifies several results in the literature on the supercritical Galton-Watson process. We generalize this to an age dependent branching process case as well.

scholarly journals The Critical Galton-Watson Process Without Further Power Moments

2007 ◽  
Vol 44 (03) ◽  
pp. 753-769 ◽  
Author(s):  
S. V. Nagaev ◽  
V. Wachtel
Keyword(s):  
Asymptotic Behavior ◽  
Limit Theorem ◽  
Generating Function ◽  
Branching Process ◽  
Alternative Proof ◽  
Conditional Limit Theorem ◽  
Power Moments ◽  
Watson Process ◽  
Conditional Limit

In this paper we prove a conditional limit theorem for a critical Galton-Watson branching process {Z n ; n ≥ 0} with offspring generating function s + (1 − s)L((1 − s)−1), where L(x) is slowly varying. In contrast to a well-known theorem of Slack (1968), (1972) we use a functional normalization, which gives an exponential limit. We also give an alternative proof of Sze's (1976) result on the asymptotic behavior of the nonextinction probability.

Moments for stationary and quasi-stationary distributions of markov chains

1985 ◽  
Vol 22 (01) ◽  
pp. 148-155 ◽  
Author(s):  
E. Seneta ◽  
R. L. Tweedie
Keyword(s):  
Invariant Measure ◽  
Markov Chains ◽  
Stationary Distributions ◽  
Limit Distributions ◽  
Sufficient Set ◽  
Watson Process ◽  
Necessary And Sufficient ◽  
Conditional Limit ◽  
Finite Moments

A necessary and sufficient set of conditions is given for the finiteness of a general moment of the R -invariant measure of an R -recurrent substochastic matrix. The conditions are conceptually related to Foster's theorem. The result extends that of [8], and is illustratively applied to the single and multitype subcritical Galton–Watson process to find conditions for Yaglom-type conditional limit distributions to have finite moments.

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) &lt; 1 and F′′ (1) &lt; ∞; 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)}.

A theorem on discrete time branching processes allowing immigration

1970 ◽  
Vol 7 (02) ◽  
pp. 446-450 ◽  
Author(s):  
John F. Reynolds
Keyword(s):  
Population Size ◽  
Generating Function ◽  
Discrete Time ◽  
Branching Processes ◽  
Probability Generating Function ◽  
Image Position

We consider a population which evolves at discrete points in time by branching and immigration, and in which each member reproduces independently of all others. Let Fn (x) denote the probability generating function (P.G.F.) of the number of offspring produced by each member of the nth generation, Bn– 1(x) the P.G.F. of the number of immigrants joining the nth generation and Zn the population size in the nth generation. We write

Functional equations and the Galton-Watson process

1969 ◽  
Vol 1 (1) ◽  
pp. 1-42 ◽  
Author(s):  
E. Seneta
Keyword(s):  
Generating Function ◽  
Functional Equations ◽  
Probability Generating Function ◽  
Single Individual ◽  
Fundamental Assumption ◽  
Offspring Distribution ◽  
Watson Process

In the present exposition we are concerned only with the simple Galton-Watson process, initiated by a single ancestor (Harris (1963), Chapter I). Let denote the probability generating function of the offspring distribution of a single individual. Our fundamental assumption, which holds throughout the sequel, is that fj ≠ 1, j = 0,1,2, …; in particular circumstances it shall be necessary to strengthen this to 0 < f0 ≡ F(0) < 1, which is the relevant assumption when extinction behaviour is to be considered. (Even so, our assumptions will always differ slightly from those of Harris (1963), p. 5.)

Moments for stationary and quasi-stationary distributions of markov chains

1985 ◽  
Vol 22 (1) ◽  
pp. 148-155 ◽  
Author(s):  
E. Seneta ◽  
R. L. Tweedie
Keyword(s):  
Invariant Measure ◽  
Markov Chains ◽  
Stationary Distributions ◽  
Limit Distributions ◽  
Sufficient Set ◽  
Watson Process ◽  
Necessary And Sufficient ◽  
Conditional Limit ◽  
Finite Moments

A necessary and sufficient set of conditions is given for the finiteness of a general moment of the R -invariant measure of an R -recurrent substochastic matrix. The conditions are conceptually related to Foster's theorem. The result extends that of [8], and is illustratively applied to the single and multitype subcritical Galton–Watson process to find conditions for Yaglom-type conditional limit distributions to have finite moments.

scholarly journals Branching processes in random environment die slowly

2008 ◽  
Vol DMTCS Proceedings vol. AI,... (Proceedings) ◽  
Author(s):  
Vladimir Vatutin ◽  
Andreas Kyprianou
Keyword(s):  
Random Walk ◽  
Random Environment ◽  
Generating Functions ◽  
Limit Theorems ◽  
Branching Process ◽  
Branching Processes ◽  
Limit Distributions ◽  
Conditional Limit Theorems ◽  
International Audience ◽  
Conditional Limit

International audience Let $Z_n,n=0,1,\ldots,$ be a branching process evolving in the random environment generated by a sequence of iid generating functions $f_0(s),f_1(s),\ldots,$ and let $S_0=0$, $S_k=X_1+ \ldots +X_k,k \geq 1$, be the associated random walk with $X_i=\log f_{i-1}^{\prime}(1), \tau (m,n)$ be the left-most point of minimum of $\{S_k,k \geq 0 \}$ on the interval $[m,n]$, and $T=\min \{ k:Z_k=0\}$. Assuming that the associated random walk satisfies the Doney condition $P(S_n > 0) \to \rho \in (0,1), n \to \infty$, we prove (under the quenched approach) conditional limit theorems, as $n \to \infty$, for the distribution of $Z_{nt}, Z_{\tau (0,nt)}$, and $Z_{\tau (nt,n)}, t \in (0,1)$, given $T=n$. It is shown that the form of the limit distributions essentially depends on the location of $\tau (0,n)$ with respect to the point $nt$.

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