Growth rates for pure birth Markov chains

A pure birth Markov chain is a continuous time Markov chain { Z ( t ) : t ≥ 0 } with state space S ≡ { 0 , 1 , 2 , … } such that for each i ≥ 0 the chain stays in state i for a random length of time that is exponentially distributed with mean λ i − 1 and then jumps to ( i + 1 ) . Suppose b ( ⋅ ) is...

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Bibliographic Details
Published in:Statistics & probability letters 2008-09, Vol.78 (12), p.1534-1540
Main Author: Athreya, K.B.
Format: Article
Language:English
Subjects:
Exact sciences and technology
General topics
Markov processes
Mathematics
Probability and statistics
Probability theory and stochastic processes
Sciences and techniques of general use
Statistics
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Summary:A pure birth Markov chain is a continuous time Markov chain { Z ( t ) : t ≥ 0 } with state space S ≡ { 0 , 1 , 2 , … } such that for each i ≥ 0 the chain stays in state i for a random length of time that is exponentially distributed with mean λ i − 1 and then jumps to ( i + 1 ) . Suppose b ( ⋅ ) is a function from ( 0 , ∞ ) → ( 0 , ∞ ) that is nondecreasing and ↑ ∞ . This paper addresses the two questions: (1) Given { λ i } i ≥ 0 what is the growth rate of Z ( t ) ? (2) Given b ( ⋅ ) does there exist { λ i } such that Z ( t ) grows at rate b ( t ) ?
ISSN:0167-7152
1879-2103
DOI:10.1016/j.spl.2008.01.016