### Abstract

Original language | English |
---|---|

Pages (from-to) | 1563-1581 |

Number of pages | 18 |

Journal | Proceedings A: Mathematical, Physical and Engineering Sciences |

Volume | 461 |

Issue number | 2057 |

DOIs | |

Publication status | Published - 2005 |

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### Keywords

- immigration
- stochastic population
- quantum optics
- schoenberg distribution
- statistics

### Cite this

*Proceedings A: Mathematical, Physical and Engineering Sciences*,

*461*(2057), 1563-1581. https://doi.org/10.1098/rspa.2004.1414

}

*Proceedings A: Mathematical, Physical and Engineering Sciences*, vol. 461, no. 2057, pp. 1563-1581. https://doi.org/10.1098/rspa.2004.1414

**The evolution of a batch-immigration death process subject to counts.** / Gillespie, C.; Renshaw, E.

Research output: Contribution to journal › Article

TY - JOUR

T1 - The evolution of a batch-immigration death process subject to counts

AU - Gillespie, C.

AU - Renshaw, E.

PY - 2005

Y1 - 2005

N2 - A bivariate batch immigration-death process is developed to study the degree to which the fundamental structure of a hidden stochastic process can be inferred purely from counts of escaping individuals. This question is of immense importance in fields such as quantum optics, where externally based radiation elucidates the nature of the underlying electromagnetic radiation process. Batches of i immigrants enter the population at rate αqi, and each individual dies independently at rate μ. General expressions are developed for the population size cumulants and probabilities, together with those for the associated counting process. The strong link between these two structures is highlighted through two specific examples, involving k-batch immigration for i=k, and Schoenberg-batch immigration over i=2m (m =0, 1, 2, ...), and shows that high quality inferences on the hidden population process can be inferred purely from externally counted observations.

AB - A bivariate batch immigration-death process is developed to study the degree to which the fundamental structure of a hidden stochastic process can be inferred purely from counts of escaping individuals. This question is of immense importance in fields such as quantum optics, where externally based radiation elucidates the nature of the underlying electromagnetic radiation process. Batches of i immigrants enter the population at rate αqi, and each individual dies independently at rate μ. General expressions are developed for the population size cumulants and probabilities, together with those for the associated counting process. The strong link between these two structures is highlighted through two specific examples, involving k-batch immigration for i=k, and Schoenberg-batch immigration over i=2m (m =0, 1, 2, ...), and shows that high quality inferences on the hidden population process can be inferred purely from externally counted observations.

KW - immigration

KW - stochastic population

KW - quantum optics

KW - schoenberg distribution

KW - statistics

UR - http://dx.doi.org/10.1098/rspa.2004.1414

U2 - 10.1098/rspa.2004.1414

DO - 10.1098/rspa.2004.1414

M3 - Article

VL - 461

SP - 1563

EP - 1581

JO - Proceedings A: Mathematical, Physical and Engineering Sciences

JF - Proceedings A: Mathematical, Physical and Engineering Sciences

SN - 1364-5021

IS - 2057

ER -