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Limit theorems for a moderately subcritical branching process in a random environment

Limit theorems for a moderately subcritical branching process in a random environment -- Let {,,} be a moderately subcritical branching process in a random environment with linear-fractional generating functions, m,, be the conditional expectation of ,, with respect to the random environment. We prove theorems on convergence of the sequence of random processes {,,,/m,,,,,, / e (0,1)| ,, >0} as -> °° in distribution, and of the initial and final segments of the random sequence /mo, / m i , . . . , ,,/m,, considered under the condition that {,, > 0}. 1. Let sequences of random variables /=0 be identically distributed and independent for different n. We introduce the generating functions A sequence of non-negative integer-valued random variables {, 0} is called a branching process in the random environment {, E N0}, if E(/"' |§,,,,...., ,...,7 ) = (^(^, No (for simplicity we assume that = 1). In [1], the classification of branching processes in a random environment depending on the sign of Eln/ 0 '(l) is given: if Eln/ 0 ; (l) > 0, then the process is supercritical, if ln/ 0 '(l) = 0, the process is critical, and if ln/ 0 '(l) < 0, the process is subcritical. In its turn, in [2] it is http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Discrete Mathematics and Applications de Gruyter

Limit theorems for a moderately subcritical branching process in a random environment

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Publisher
de Gruyter
Copyright
Copyright © 2009 Walter de Gruyter
ISSN
0924-9265
eISSN
1569-3929
DOI
10.1515/dma.1998.8.1.35
Publisher site
See Article on Publisher Site

Abstract

-- Let {,,} be a moderately subcritical branching process in a random environment with linear-fractional generating functions, m,, be the conditional expectation of ,, with respect to the random environment. We prove theorems on convergence of the sequence of random processes {,,,/m,,,,,, / e (0,1)| ,, >0} as -> °° in distribution, and of the initial and final segments of the random sequence /mo, / m i , . . . , ,,/m,, considered under the condition that {,, > 0}. 1. Let sequences of random variables /=0 be identically distributed and independent for different n. We introduce the generating functions A sequence of non-negative integer-valued random variables {, 0} is called a branching process in the random environment {, E N0}, if E(/"' |§,,,,...., ,...,7 ) = (^(^, No (for simplicity we assume that = 1). In [1], the classification of branching processes in a random environment depending on the sign of Eln/ 0 '(l) is given: if Eln/ 0 ; (l) > 0, then the process is supercritical, if ln/ 0 '(l) = 0, the process is critical, and if ln/ 0 '(l) < 0, the process is subcritical. In its turn, in [2] it is

Journal

Discrete Mathematics and Applications – de Gruyter

Published: Jan 1, 1998

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