scholarly journals Asymptotics of fluctuations in Crump‒Mode‒Jagers processes: the lattice case

2018 ◽  
Vol 50 (A) ◽  
pp. 141-171
Author(s):  
Svante Janson
Keyword(s):  
Generating Function ◽  
Branching Processes ◽  
Age Distribution ◽  
Symbolic Calculus ◽  
Asymptotic Results ◽  
Absolute Value ◽  
Malthusian Parameter ◽  
Asymptotically Normal ◽  
Complex Roots ◽  
Degenerate Cases

Abstract Consider a supercritical Crump‒Jagers process in which all births are at integer times (the lattice case). Let μ̂(z) be the generating function of the intensity of the offspring process, and consider the complex roots of μ̂(z)=1. The root of smallest absolute value is e-α=1∕m, where α>0 is the Malthusian parameter; let γ* be the root of second smallest absolute value. Subject to some technical conditions, the second-order fluctuations of the age distribution exhibit one of three types of behaviour: (i) when γ*>e-α∕2=m-1∕2, they are asymptotically normal; (ii) when γ*=e-α∕2, they are still asymptotically normal, but with a larger variance; and (iii) when γ*<e-α∕2, the fluctuations are in general oscillatory and (degenerate cases excluded) do not converge in distribution. This trichotomy is similar to what has been observed in related situations, such as some other branching processes and for Pólya urns. The results lead to a symbolic calculus describing the limits. The asymptotic results also apply to the total of other (random) characteristics of the population.

Mean-square and almost-sure convergence of supercritical age-dependent branching processes

1974 ◽  
Vol 11 (04) ◽  
pp. 678-686
Author(s):  
Edgar Z. Ganuza ◽  
S. D. Durham
Keyword(s):  
Branching Process ◽  
Branching Processes ◽  
Age Distribution ◽  
Almost Sure Convergence ◽  
Mean Square ◽  
Malthusian Parameter ◽  
Age Dependent ◽  
Mean Square Convergence ◽  
Watson Process ◽  
Successive Generations

Letting Z(t) be the number of objects alive at time t in a general supercritical age-dependent branching process generated by a single ancestor born at time 0, one achieves (Theorem 1) mean-square convergence of Z(t)/E[Z(t)] provided and , where N(t) is the number of offspring of the initial ancestor born by time t and α is the (positive) Malthusian parameter defined by . If the stronger conditions that (Theorem 2) and hold also, then the convergence is almost-sure. It is of interest that the embedded Galton-Watson process of successive generations need not have a finite mean for the conditions of the above theorems to hold. Similar results are obtained for the age-distribution as well.

Mean-square and almost-sure convergence of supercritical age-dependent branching processes

1974 ◽  
Vol 11 (4) ◽  
pp. 678-686 ◽  
Author(s):  
Edgar Z. Ganuza ◽  
S. D. Durham
Keyword(s):  
Branching Process ◽  
Branching Processes ◽  
Age Distribution ◽  
Almost Sure Convergence ◽  
Mean Square ◽  
Malthusian Parameter ◽  
Age Dependent ◽  
Mean Square Convergence ◽  
Watson Process ◽  
Successive Generations

Letting Z(t) be the number of objects alive at time t in a general supercritical age-dependent branching process generated by a single ancestor born at time 0, one achieves (Theorem 1) mean-square convergence of Z(t)/E[Z(t)] provided and , where N(t) is the number of offspring of the initial ancestor born by time t and α is the (positive) Malthusian parameter defined by . If the stronger conditions that (Theorem 2) and hold also, then the convergence is almost-sure. It is of interest that the embedded Galton-Watson process of successive generations need not have a finite mean for the conditions of the above theorems to hold. Similar results are obtained for the age-distribution as well.

Parametric inference in Markov branching processes with time-dependent random immigration rate

1985 ◽  
Vol 22 (03) ◽  
pp. 503-517
Author(s):  
Helmut Pruscha
Keyword(s):  
Point Processes ◽  
Traditional Approach ◽  
Branching Processes ◽  
External Source ◽  
Time Dependent ◽  
Parametric Inference ◽  
Asymptotic Results ◽  
Subcritical Case ◽  
Immigration Rate ◽  
Pearson Type

The present paper deals with continuous-time Markov branching processes allowing immigration. The immigration rate is allowed to be random and time-dependent where randomness may stem from an external source or from state-dependence. Unlike the traditional approach, we base the analysis of these processes on the theory of multivariate point processes. Using the tools of this theory, asymptotic results on parametric inference are derived for the subcritical case. In particular, the limit distributions of some parametric estimators and of Pearson-type statistics for testing simple and composite hypotheses are established.

scholarly journals Sorting using complete subintervals and the maximum number of runs in a randomly evolving sequence: Extended abstract.

2007 ◽  
Vol DMTCS Proceedings vol. AH,... (Proceedings) ◽  
Author(s):  
Svante Janson
Keyword(s):  
Order Term ◽  
Combinatorial Problem ◽  
Random Order ◽  
Priority Queues ◽  
Sorting Algorithm ◽  
Asymptotic Results ◽  
First Order ◽  
Asymptotically Normal ◽  
International Audience ◽  
Space Requirements

International audience We study the space requirements of a sorting algorithm where only items that at the end will be adjacent are kept together. This is equivalent to the following combinatorial problem: Consider a string of fixed length n that starts as a string of 0's, and then evolves by changing each 0 to 1, with the n changes done in random order. What is the maximal number of runs of 1's? We give asymptotic results for the distribution and mean. It turns out that, as in many problems involving a maximum, the maximum is asymptotically normal, with fluctuations of order $n^{1/2}$, and to the first order well approximated by the number of runs at the instance when the expectation is maximized, in this case when half the elements have changed to 1; there is also a second order term of order $n^{1/3}$. We also treat some variations, including priority queues and sock-sorting.

Malthusian behaviour of the critical and subcritical age-dependent branching processes with arbitrary state space

1976 ◽  
Vol 13 (3) ◽  
pp. 455-465
Author(s):  
D. I. Saunders
Keyword(s):  
State Space ◽  
Branching Process ◽  
Branching Processes ◽  
Rates Of Convergence ◽  
Expected Number ◽  
Subcritical Case ◽  
Arbitrary State ◽  
Malthusian Parameter ◽  
Age Dependent ◽  
Number Of Individuals

For the age-dependent branching process with arbitrary state space let M(x, t, A) be the expected number of individuals alive at time t with states in A given an initial individual at x. Subject to various conditions it is shown that M(x, t, A)e–at converges to a non-trivial limit where α is the Malthusian parameter (α = 0 for the critical case, and is negative in the subcritical case). The method of proof also yields rates of convergence.

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.

Strict supercritical generation-dependent Crump–Mode–Jagers branching processes

1978 ◽  
Vol 10 (04) ◽  
pp. 744-763 ◽  
Author(s):  
L. Edler
Keyword(s):  
Generating Functions ◽  
Positive Real Number ◽  
Branching Processes ◽  
Positive Real ◽  
Quadratic Mean ◽  
Malthusian Parameter ◽  
Renewal Equations ◽  
Branching Model ◽  
Age Dependent ◽  
Number Of Individuals

The general age-dependent branching model of Crump, Mode and Jagers will be generalized towards generation-dependent varying lifespan and reproduction distributions. A system of integral and renewal equations is established for the generating functions and the first two moments of Zi (t) (the number of individuals alive at time t), if the population was initiated at time 0 by one ancestor of age 0 from generation i. Convergence in quadratic mean of Zi (t)/EZi (t) as t tends to infinity is obtained if the generation-dependent reproduction functions converge to a supercritical one. In particular, if this convergence is slow enough t γ exp (αt) is the asymptotic behavior of EZi (t) for t tending to infinity, where γ is a positive real number and α the Malthusian parameter of growth of the limiting reproduction function.

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.

Sur une procédure de branchement déterministe et ses dérivées aléatoires

1994 ◽  
Vol 31 (02) ◽  
pp. 333-347
Author(s):  
Thierry Huillet ◽  
Andrzej Kłopotowski
Keyword(s):  
Branching Processes ◽  
Asymptotic Results ◽  
Renewal Equations ◽  
Age Dependent ◽  
Probabilistic Version

This paper is concerned with the description of both a deterministic and stochastic branching procedure. The renewal equations for the deterministic branching population are first derived which allow for asymptotic results on the ‘number' and ‘generation' processes. A probabilistic version of these processes is then studied which presents some discrepancy with the standard Harris age-dependent branching processes.

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