On Largest Offspring in a Critical Branching Process with Finite Variance

2013 ◽  
Vol 50 (03) ◽  
pp. 791-800 ◽  
Author(s):  
Jean Bertoin
Keyword(s):  
Limit Theorem ◽  
Branching Process ◽  
Shape Parameter ◽  
Weak Limit ◽  
Finite Variance ◽  
Poisson Measure ◽  
Doubly Stochastic ◽  
Weak Limit Theorem ◽  
Watson Process ◽  
Critical Branching Process

Continuing the work in Bertoin (2011) we study the distribution of the maximal number X * k of offspring amongst all individuals in a critical Galton‒Watson process started with k ancestors, treating the case when the reproduction law has a regularly varying tail F̅ with index −α for α > 2 (and, hence, finite variance). We show that X * k suitably normalized converges in distribution to a Fréchet law with shape parameter α/2; this contrasts sharply with the case 1< α<2 when the variance is infinite. More generally, we obtain a weak limit theorem for the offspring sequence ranked in decreasing order, in terms of atoms of a certain doubly stochastic Poisson measure.

scholarly journals On Largest Offspring in a Critical Branching Process with Finite Variance

2013 ◽  
Vol 50 (3) ◽  
pp. 791-800 ◽  
Author(s):  
Jean Bertoin
Keyword(s):  
Limit Theorem ◽  
Branching Process ◽  
Shape Parameter ◽  
Weak Limit ◽  
Finite Variance ◽  
Poisson Measure ◽  
Doubly Stochastic ◽  
Weak Limit Theorem ◽  
Watson Process ◽  
Critical Branching Process

Continuing the work in Bertoin (2011) we study the distribution of the maximal number X*k of offspring amongst all individuals in a critical Galton‒Watson process started with k ancestors, treating the case when the reproduction law has a regularly varying tail F̅ with index −α for α > 2 (and, hence, finite variance). We show that X*k suitably normalized converges in distribution to a Fréchet law with shape parameter α/2; this contrasts sharply with the case 1< α<2 when the variance is infinite. More generally, we obtain a weak limit theorem for the offspring sequence ranked in decreasing order, in terms of atoms of a certain doubly stochastic Poisson measure.

scholarly journals On the Maximal Offspring in a Critical Branching Process with Infinite Variance

2011 ◽  
Vol 48 (02) ◽  
pp. 576-582 ◽  
Author(s):  
Jean Bertoin
Keyword(s):  
Branching Process ◽  
Shape Parameter ◽  
Scale Parameter ◽  
Infinite Variance ◽  
Regularly Varying ◽  
Watson Process ◽  
Critical Branching Process

We investigate the maximal number M k of offspring amongst all individuals in a critical Galton-Watson process started with k ancestors. We show that when the reproduction law has a regularly varying tail with index -α for 1 &lt; α &lt; 2, then k -1 M k converges in distribution to a Frechet law with shape parameter 1 and scale parameter depending only on α.

scholarly journals On the Maximal Offspring in a Critical Branching Process with Infinite Variance

2011 ◽  
Vol 48 (2) ◽  
pp. 576-582 ◽  
Author(s):  
Jean Bertoin
Keyword(s):  
Branching Process ◽  
Shape Parameter ◽  
Scale Parameter ◽  
Infinite Variance ◽  
Regularly Varying ◽  
Watson Process ◽  
Critical Branching Process

We investigate the maximal number Mk of offspring amongst all individuals in a critical Galton-Watson process started with k ancestors. We show that when the reproduction law has a regularly varying tail with index -α for 1 < α < 2, then k-1Mk converges in distribution to a Frechet law with shape parameter 1 and scale parameter depending only on α.

scholarly journals A Weak Limit Theorem for Galton-Watson Processes in Varying Environments

2014 ◽  
Vol 2014 ◽  
pp. 1-4
Author(s):  
Zhenlong Gao ◽  
Yanhua Zhang
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Weak Limit ◽  
The Central Limit Theorem ◽  
Weak Limit Theorem ◽  
Varying Environments ◽  
Watson Process ◽  
Varying Environment

We extend Donsker’s theorem and the central limit theorem of classical Galton-Watson process to the Galton-Watson processes in varying environment.

scholarly journals A Weak Limit Theorem for Generalized Jiřina Processes

2009 ◽  
Vol 46 (2) ◽  
pp. 453-462 ◽  
Author(s):  
Yuqiang Li
Keyword(s):  
Limit Theorem ◽  
Diffusion Process ◽  
Nonlinear Diffusion ◽  
Weak Limit ◽  
Weak Limit Theorem ◽  
Lévy Jumps

In this paper we prove that a sequence of scaled generalized Jiřina processes can converge weakly to a nonlinear diffusion process with Lévy jumps under certain conditions.

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.

scholarly journals Weak limit theorem for a nonlinear quantum walk

2018 ◽  
Vol 17 (9) ◽  
Author(s):  
Masaya Maeda ◽  
Hironobu Sasaki ◽  
Etsuo Segawa ◽  
Akito Suzuki ◽  
Kanako Suzuki
Keyword(s):  
Limit Theorem ◽  
Quantum Walk ◽  
Weak Limit ◽  
Nonlinear Quantum ◽  
Weak Limit Theorem

A representation for the limiting random variable of a branching process with infinite mean and some related problems

1978 ◽  
Vol 15 (02) ◽  
pp. 225-234 ◽  
Author(s):  
Harry Cohn ◽  
Anthony G. Pakes
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Convergence Rate ◽  
Branching Process ◽  
Extreme Value Distribution ◽  
Random Variable ◽  
Random Sum ◽  
Strong Limit ◽  
Strong Limit Theorem ◽  
Watson Process

It is known that for a Bienaymé– Galton–Watson process {Zn } whose mean m satisfies 1 &lt; m &lt; ∞, the limiting random variable in the strong limit theorem can be represented as a random sum of i.i.d. random variables and hence that convergence rate results follow from a random sum central limit theorem. This paper develops an analogous theory for the case m = ∞ which replaces ‘sum' by ‘maximum'. In particular we obtain convergence rate results involving a limiting extreme value distribution. An associated estimation problem is considered.

scholarly journals A Weak Limit Theorem for Generalized Jiřina Processes

2009 ◽  
Vol 46 (02) ◽  
pp. 453-462 ◽  
Author(s):  
Yuqiang Li
Keyword(s):  
Limit Theorem ◽  
Diffusion Process ◽  
Nonlinear Diffusion ◽  
Weak Limit ◽  
Weak Limit Theorem ◽  
Lévy Jumps

In this paper we prove that a sequence of scaled generalized Jiřina processes can converge weakly to a nonlinear diffusion process with Lévy jumps under certain conditions.

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