Central limit theorem for a critical multitype branching process in random environments

Abstract Let {Zn, n = 0, 1, 2, . . .} be a supercritical branching process, {Nt, t ≥ 0} be a Poisson process independent of {Zn, n = 0, 1, 2, . . .}, then {ZNt, t ≥ 0} is a supercritical Poisson random indexed branching process. We show a law of large numbers, central limit theorem, and large and moderate deviation principles for log ZNt.

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Central limit theorem and recurrence for random walks in bistochastic random environments

Journal of Mathematical Physics ◽

10.1063/1.3005226 ◽

2008 ◽

Vol 49 (12) ◽

pp. 125213 ◽

Cited By ~ 5

Author(s):

Marco Lenci

Keyword(s):

Central Limit Theorem ◽

Limit Theorem ◽

Random Walks ◽

Central Limit ◽

Random Environments

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Berry-Esseen bound for nearly critical branching processes with immigration

Bulletin of Taras Shevchenko National University of Kyiv. Series: Physics and Mathematics ◽

10.17721/1812-5409.2019/4.5 ◽

2019 ◽

pp. 42-49

Author(s):

Ya. Khusanbaev ◽

S. Sharipov ◽

V. Golomoziy

Keyword(s):

Central Limit Theorem ◽

Limit Theorem ◽

Rate Of Convergence ◽

Central Limit ◽

Branching Process ◽

Branching Processes ◽

Branching Process With Immigration ◽

Critical Branching Process

In this paper, we consider a nearly critical branching process with immigration. We obtain the rate of convergence in central limit theorem for nearly critical branching processes with immigration.

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Stochastic multi-type SIR epidemics among a population partitioned into households

Advances in Applied Probability ◽

10.1017/s000186780001065x ◽

2001 ◽

Vol 33 (1) ◽

pp. 99-123 ◽

Cited By ~ 22

Author(s):

Frank Ball ◽

Owen D. Lyne

Keyword(s):

Central Limit Theorem ◽

Limit Theorem ◽

Stochastic Model ◽

Central Limit ◽

Epidemic Model ◽

Branching Process ◽

Finite Population ◽

Exact Distribution ◽

Final Outcome ◽

Process Approximation

We consider a stochastic model for the spread of an SIR (susceptible → infective → removed) epidemic among a closed, finite population that contains several types of individuals and is partitioned into households. The infection rate between two individuals depends on the types of the transmitting and receiving individuals and also on whether the infection is local (i.e., within a household) or global (i.e., between households). The exact distribution of the final outcome of the epidemic is outlined. A branching process approximation for the early stages of the epidemic is described and made fully rigorous, by considering a sequence of epidemics in which the number of households tends to infinity and using a coupling argument. This leads to a threshold theorem for the epidemic model. A central limit theorem for the final outcome of epidemics which take off is derived, by exploiting an embedding representation.

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