Robust Estimation in Multitype Branching Processes Based on their Asymptotic Properties

2000 Mathematics Subject Classification: 60J80.

Controlled Multitype Branching Models: Geometric Growth

2000 Mathematics Subject Classi cation: 60J80, 60F25.

A Note on the Extinction Problem for Controlled Multitype Branching Processes

2000 Mathematics Subject Classification: 60J80, 60J10.

A Note on the Asymptotic Behaviour of a Periodic Multitype Galton-Watson Branching Process

2000 Mathematics Subject Classification: 60J80.

Using Inside-Outside Algorithm for Estimation of the Offspring Distribution in Multitype Branching Processes

Multitype branching processes (MTBP) model branching structures, where the nodes of the resulting tree are particles of different types. Usually such a process is not observable in the sense of the whole tree, but only as the “generation” at a given moment in time, which consists of the number of particles of every type. This requires an EM-type algorithm to obtain a maximum likelihood (ML) estimate of the parameters of the branching process. Using a version of the inside-outside algorithm for stochastic context-free grammars (SCFG), such an estimate could be obtained for the offspring distribution of the process.

The collision branching process

We consider a branching model, which we call the collision branching process (CBP), that accounts for the effect of collisions, or interactions, between particles or individuals. We establish that there is a unique CBP, and derive necessary and sufficient conditions for it to be nonexplosive. We review results on extinction probabilities, and obtain explicit expressions for the probability of explosion and the expected hitting times. The upwardly skip-free case is studied in some detail.

On the Maximum of a Branching Process Conditioned on the Total Progeny

The maximum M of a critical Bienaymé-Galton-Watson process conditioned on the total progeny N is studied. Imbedding of the process in a random walk is used. A limit theorem for the distribution of M as N → ∞ is proved. The result is trasferred to the non-critical processes. A corollary for the maximal strata of a random rooted labeled tree is obtained.

Some Theoretical Results on the Progeny of a Bisexual Galton-Watson Branching Process

A Superadditive Bisexual Galton-Watson Branching Process is considered and the total number of mating units, females and males, until the n-th generation, are studied. In particular some results about the stochastic monotony, probability generating functions and moments are obtained. Finally, the limit behaviour of those variables suitably normed is investigated.

The Maximal Number of Particles in a Branching Process with State-Dependent Immigration

AMS subject classification: 60J80, 60J15.

Bootstrap for Critical Branching Process with Non-Stationary Immigration

2000 Mathematics Subject Classification: Primary 60J80, Secondary 62F12, 60G99.

Option Pricing by Branching Process

2000 Mathematics Subject Classification: 60J80, 62P05.

Some Probabilistic Results in a Bisexual Branching Process with Immigration

2000 Mathematics Subject Classification: 60J80.

Estimation of the Offspring Mean in a General Single-Type Size-Dependent Branching Process

2000 Mathematics Subject Classification: 60J80, 62F12, 62P10

Total Progeny in a Subcritical Branching Process with two Types of Immigration

2000 Mathematics Subject Classification: 60J80, 60F05

Crump-Mode-Jagers Branching Process: Modelling and Application for Human Population

2010 Mathematics Subject Classification: 60J85, 92D25.