scholarly journals Individuals at the origin in the critical catalytic branching random walk

2003 ◽  
Vol DMTCS Proceedings vol. AC,... (Proceedings) ◽  
Author(s):  
Valentin Topchii ◽  
Vladimir Vatutin
Keyword(s):  
Random Walk ◽  
Limit Theorem ◽  
Continuous Time ◽  
Branching Random Walk ◽  
Conditional Limit Theorem ◽  
Catalytic Branching ◽  
International Audience ◽  
Number Of Individuals ◽  
Conditional Limit

International audience A continuous time branching random walk on the lattice $\mathbb{Z}$ is considered in which individuals may produce children at the origin only. Assuming that the underlying random walk is symmetric and the offspring reproduction law is critical we prove a conditional limit theorem for the number of individuals at the origin.

The maximum and time to absorption of a left-continuous random walk

1976 ◽  
Vol 13 (3) ◽  
pp. 444-454 ◽  
Author(s):  
P. J. Green
Keyword(s):  
Random Walk ◽  
Limit Theorem ◽  
Joint Distribution ◽  
Distribution Of The Maximum ◽  
Conditional Limit Theorem ◽  
Continuous Random Walk ◽  
Conditional Limit

For a left-continuous random walk, absorbing at 0, the joint distribution of the maximum and time to absorption is derived. A description of the tails of the distributions and a conditional limit theorem are obtained for the cases where absorption is certain.

The maximum and time to absorption of a left-continuous random walk

1976 ◽  
Vol 13 (03) ◽  
pp. 444-454 ◽  
Author(s):  
P. J. Green
Keyword(s):  
Random Walk ◽  
Limit Theorem ◽  
Joint Distribution ◽  
Distribution Of The Maximum ◽  
Conditional Limit Theorem ◽  
Continuous Random Walk ◽  
Conditional Limit

For a left-continuous random walk, absorbing at 0, the joint distribution of the maximum and time to absorption is derived. A description of the tails of the distributions and a conditional limit theorem are obtained for the cases where absorption is certain.

Efficiency, Sufficiency, and Optimality

2017 ◽  
Author(s):  
Amos Golan
Keyword(s):  
Limit Theorem ◽  
Computational Efficiency ◽  
The Other ◽  
Information Compression ◽  
Information Theoretic ◽  
Conditional Limit Theorem ◽  
Concept Of Information ◽  
Conditional Limit

In this chapter I provide additional rationalization for using the info-metrics framework. This time the justifications are in terms of the statistical, mathematical, and information-theoretic properties of the formalism. Specifically, in this chapter I discuss optimality, statistical and computational efficiency, sufficiency, the concentration theorem, the conditional limit theorem, and the concept of information compression. These properties, together with the other properties and measures developed in earlier chapters, provide logical, mathematical, and statistical justifications for employing the info-metrics framework.

scholarly journals Strong Local Survival of Branching Random Walks is Not Monotone

2014 ◽  
Vol 46 (02) ◽  
pp. 400-421 ◽  
Author(s):  
Daniela Bertacchi ◽  
Fabio Zucca
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Generating Function ◽  
Continuous Time ◽  
Local Property ◽  
Branching Random Walk ◽  
Infinite Dimensional ◽  
Branching Random Walks ◽  
Local Survival ◽  
Branching Law

In this paper we study the strong local survival property for discrete-time and continuous-time branching random walks. We study this property by means of an infinite-dimensional generating functionGand a maximum principle which, we prove, is satisfied by every fixed point ofG. We give results for the existence of a strong local survival regime and we prove that, unlike local and global survival, in continuous time, strong local survival is not a monotone property in the general case (though it is monotone if the branching random walk is quasitransitive). We provide an example of an irreducible branching random walk where the strong local property depends on the starting site of the process. By means of other counterexamples, we show that the existence of a pure global phase is not equivalent to nonamenability of the process, and that even an irreducible branching random walk with the same branching law at each site may exhibit nonstrong local survival. Finally, we show that the generating function of an irreducible branching random walk can have more than two fixed points; this disproves a previously known result.

Sign in / Sign up

Export Citation Format

Share Document