Continuity for multi-type branching processes with varying environments

Owen Dafydd Jones

doi:10.1239/jap/1032374236

Continuity for multi-type branching processes with varying environments

Owen Dafydd Jones^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

2 Citations (Scopus)

Abstract

Conditions are derived for the components of the normed limit of a multi-type branching process with varying environments, to be continuous on (0,∞). The main tool is an inequality for the concentration function of sums of independent random variables, due originally to Petrov. Using this, we show that if there is a discontinuity present, then a particular linear combination of the population types must converge to a non-random constant (Equation (1)). Ensuring this can not happen provides the desired continuity conditions.

Original language	English
Pages (from-to)	139-145
Number of pages	7
Journal	Journal of Applied Probability
Volume	36
Issue number	1
DOIs	https://doi.org/10.1239/jap/1032374236
Publication status	Published - 1 Jan 1999

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AB - Conditions are derived for the components of the normed limit of a multi-type branching process with varying environments, to be continuous on (0,∞). The main tool is an inequality for the concentration function of sums of independent random variables, due originally to Petrov. Using this, we show that if there is a discontinuity present, then a particular linear combination of the population types must converge to a non-random constant (Equation (1)). Ensuring this can not happen provides the desired continuity conditions.

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KW - Continuity

KW - Multi-type

KW - Varying environment

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Continuity for multi-type branching processes with varying environments

Abstract

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