On a conditional invariance principle for a critical Galton–Watson branching process

2005 ◽  
Vol 15 (1) ◽  
Author(s):  
V. I. Afanasyev
Keyword(s):  
Invariance Principle ◽  
Branching Process ◽  
Conditional Invariance

Branching processes in a Markov random environment

2014 ◽  
Vol 24 (6) ◽  
Author(s):  
Elena E. Dyakonova
Keyword(s):  
Limit Theorem ◽  
Invariance Principle ◽  
Random Environment ◽  
Branching Processes ◽  
Functional Limit Theorem ◽  
Functional Limit ◽  
Markov Random ◽  
Tail Behaviour ◽  
Conditional Invariance ◽  
Number Of Particles

AbstractThe paper is concerned with critical branching processes in a Markov random environment. A conditional functional limit theorem for the number of particles in a process and a conditional invariance principle are proved. The asymptotic tail behaviour for the distributions of the maximum and the total number of particles in a process is found.

Some Contributions to the Theory of Near-Critical Bisexual Branching Processes

2007 ◽  
Vol 44 (02) ◽  
pp. 492-505
Author(s):  
M. Molina ◽  
M. Mota ◽  
A. Ramos
Keyword(s):  
Branching Process ◽  
Branching Processes ◽  
Sufficient Conditions ◽  
Positive Probability ◽  
Limit Laws ◽  
Bisexual Branching Processes ◽  
Probabilistic Evolution

We investigate the probabilistic evolution of a near-critical bisexual branching process with mating depending on the number of couples in the population. We determine sufficient conditions which guarantee either the almost sure extinction of such a process or its survival with positive probability. We also establish some limiting results concerning the sequences of couples, females, and males, suitably normalized. In particular, gamma, normal, and degenerate distributions are proved to be limit laws. The results also hold for bisexual Bienaymé–Galton–Watson processes, and can be adapted to other classes of near-critical bisexual branching processes.

scholarly journals Mixing rates for potentials of non-summable variations

2021 ◽  
pp. 1-18
Author(s):  
CHRISTOPHE GALLESCO ◽  
DANIEL Y. TAKAHASHI
Keyword(s):  
Invariance Principle ◽  
Type Inequality ◽  
Upper Bounds ◽  
Decay Of Correlations ◽  
Relaxation Rates ◽  
Weak Invariance Principle ◽  
Weak Invariance ◽  
Coupling Inequality ◽  
Mixing Rates

Abstract Mixing rates, relaxation rates, and decay of correlations for dynamics defined by potentials with summable variations are well understood, but little is known for non-summable variations. This paper exhibits upper bounds for these quantities for dynamics defined by potentials with square-summable variations. We obtain these bounds as corollaries of a new block coupling inequality between pairs of dynamics starting with different histories. As applications of our results, we prove a new weak invariance principle and a Hoeffding-type inequality.

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