scholarly journals Functional Limit Laws of Strassen and Wichura type for multiple generations of branching processes

2009 ◽  
pp. 131-152
Author(s):  
Jim Kuelbs ◽  
Anand N. Vidyashankar
Keyword(s):  
Branching Processes ◽  
Functional Limit ◽  
Limit Laws ◽  
Multiple Generations

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.

Diffusion limits of critical branching processes conditioned on extinction in the near future

1976 ◽  
Vol 13 (02) ◽  
pp. 247-254
Author(s):  
Warren W. Esty
Keyword(s):  
Branching Processes ◽  
Limit Laws ◽  
Age Dependent ◽  
Diffusion Limits ◽  
Near Future

Limit laws and limiting diffusions are obtained for critical branching processes, either Galton-Watson or age-dependent, conditioned on extinction in the interval (T, cT], 1 < c, as T→∞, and also as T→∞ and c ↓ 1.

Functional limit theorems for high-level subcritical branching processes in random environment

2014 ◽  
Vol 24 (5) ◽  
Author(s):  
Valeriy I. Afanasyev
Keyword(s):  
Random Environment ◽  
Limit Theorems ◽  
Branching Process ◽  
Branching Processes ◽  
Moment Generating Function ◽  
Functional Limit ◽  
Functional Limit Theorems ◽  
The Moment ◽  
High Level ◽  
Passage Times

AbstractThe paper is concerned with subcritical branching process in random environment. It is assumed that the moment-generating function of steps of the associated random walk is equal to 1 for some positive value of the argument. Functional limit theorems for sizes of various generations and passage times to various levels are put forward.

Functional limit theorems for stochastic processes based on embedded processes

1975 ◽  
Vol 7 (01) ◽  
pp. 123-139 ◽  
Author(s):  
Richard F. Serfozo
Keyword(s):  
Stochastic Process ◽  
Stochastic Processes ◽  
Limit Theorems ◽  
Functional Limit ◽  
Functional Limit Theorems ◽  
Limit Laws ◽  
Iterated Logarithm ◽  
Large Numbers ◽  
Subordinated Processes ◽  
Analogous Functional

The techniques used by Doeblin and Chung to obtain ordinary limit laws (central limit laws, weak and strong laws of large numbers, and laws of the iterated logarithm) for Markov chains, are extended to obtain analogous functional limit laws for stochastic processes which have embedded processes satisfying these laws. More generally, it is shown how functional limit laws of a stochastic process are related to those of a process embedded in it. The results herein unify and extend many existing limit laws for Markov, semi-Markov, queueing, regenerative, semi-stationary, and subordinated processes.

scholarly journals Asymptotic Behaviour of Continuous Time and State Branching Processes

2000 ◽  
Vol 68 (1) ◽  
pp. 68-84 ◽  
Author(s):  
Zeng-Hu Li
Keyword(s):  
Asymptotic Behaviour ◽  
Continuous Time ◽  
Limit Theorems ◽  
Branching Processes ◽  
Limit Laws ◽  
Classical Setting

AbstractWe prove some limit theorems for contiunous time and state branching processes. The non-degenerate limit laws are obtained in critical and non-critical cases by conditioning or introducing immigration processes. The limit laws in non-critical cases are characterized in terms of the cononical measure of the cumulant semigroup. The proofs are based on estimates of the cumulant semigroup derived from the forward and backward equations, which are easier than the proffs in the classical setting.

Regularly varying functions in the theory of simple branching processes

1974 ◽  
Vol 6 (3) ◽  
pp. 408-420 ◽  
Author(s):  
E. Seneta
Keyword(s):  
Invariant Measures ◽  
Function Theory ◽  
Branching Processes ◽  
Regularly Varying Function ◽  
Asymptotic Structure ◽  
Regularly Varying Functions ◽  
Limit Laws ◽  
Intimate Connection ◽  
Similar Fact ◽  
Regularly Varying

It is demonstrated for the non-critical and the explosive cases of the simple Bienaymé-Galton-Watson (B. G. W.) process (with and without immigration) that there exists a natural and intimate connection between regularly varying function theory and the asymptotic structure of the limit laws and corresponding norming constants. A similar fact had been demonstrated in connection with their invariant measures in [22]. This earlier study is complemented here by a similar analysis of the process where immigration occurs only at points of “emptiness” of the B. G. W. process.

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