Proofs of L log L Criteria for Mean Behavior of Branching Processes
نویسنده
چکیده
The Kesten-Stigum Theorem is a fundamental criterion for the rate of growth of a supercritical branching process, showing that an L log L condition is decisive. In critical and subcritical cases, results of Kolmogorov and later authors give the rate of decay of the probability that the process survives at least n generations. We give conceptual proofs of these theorems based on comparisons of Galton-Watson measure to another measure on the space of trees. This approach also explains Yaglom’s exponential limit law for conditioned critical branching processes via a simple characterization of the exponential distribution.
منابع مشابه
When Does a Branching Process Grow Like its Mean ? Conceptual Proofs of L log L Criteria
The Kesten-Stigum Theorem is a fundamental criterion for the rate of growth of a supercritical branching process, showing that an L logL condition is decisive. In critical and subcritical cases, results of Kolmogorov and later authors give the rate of decay of the probability that the process survives at least n generations. Through a uni ed approach, we give conceptual proofs of these theorems...
متن کاملWhen Does a Branching Process Grow Like its Mean ?
The Kesten-Stigum Theorem is a fundamental criterion for the rate of growth of a supercritical branching process, showing that an L log L condition is decisive. In critical and subcritical cases, results of Kolmogorov and later authors give the rate of decay of the probability that the process survives at least n generations. Through a uniied approach, we give conceptual proofs of these theorem...
متن کاملCentral Limit Theorem in Multitype Branching Random Walk
A discrete time multitype (p-type) branching random walk on the real line R is considered. The positions of the j-type individuals in the n-th generation form a point process. The asymptotic behavior of these point processes, when the generation size tends to infinity, is studied. The central limit theorem is proved.
متن کاملEntropies and combinatorics of random branching processes and context-free languages
AWract-The entropies and combinatorics of trees that branch according to fixed but finite numbers of rules are studied. Context-free grammars are used to categorize the ways in which nodes branch to yield daughter nodes, thus providing an organized setting to examine the entropies for random branching processes whose realizations are trees and whose probabilities are determined by probabilities...
متن کاملPharmacokinetic Behavior of Theophylline following PEEP in Critically Ill Patients with Acute Lung Injury
The effect of Positive End Expiratory Pressure (PEEP) on the hepatic elimination of low to moderate extraction ratio drugs has not been clearly defined. We prospectively investigated the effect of PEEP on the clearance of theophylline in 30 (20 males and 10 females) intubated critically ill adult patients with acute lung injury/acute respiratory distress syndrome (ALI/ARDS). The Mean (±SD) age ...
متن کامل