The Galton–Watson process is the simplest example of a branching process. The relationship between the offspring distribution, and, when the extinction occurs almost surely, the distribution of the total progeny is well known. In this paper, we illustrate the relationship between these two distributions when we consider the large deviation rate function (provided by Cramér’s theorem) for empirical means of i.i.d. random variables. We also consider the case with a random initial population. In the final part, we present large deviation results for sequences of estimators of the offspring mean based on i.i.d. replications of total progeny.

Macci, C., Pacchiarotti, B. (2017). Large deviations for i.i.d. replications of the total progeny of a Galton–Watson process. MODERN STOCHASTICS: THEORY AND APPLICATIONS, 4(1), 1-13 [10.15559/16-VMSTA72].

### Large deviations for i.i.d. replications of the total progeny of a Galton–Watson process

#### Abstract

The Galton–Watson process is the simplest example of a branching process. The relationship between the offspring distribution, and, when the extinction occurs almost surely, the distribution of the total progeny is well known. In this paper, we illustrate the relationship between these two distributions when we consider the large deviation rate function (provided by Cramér’s theorem) for empirical means of i.i.d. random variables. We also consider the case with a random initial population. In the final part, we present large deviation results for sequences of estimators of the offspring mean based on i.i.d. replications of total progeny.
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Esperti anonimi
Settore MAT/06 - Probabilita' e Statistica Matematica
English
Macci, C., Pacchiarotti, B. (2017). Large deviations for i.i.d. replications of the total progeny of a Galton–Watson process. MODERN STOCHASTICS: THEORY AND APPLICATIONS, 4(1), 1-13 [10.15559/16-VMSTA72].
Macci, C; Pacchiarotti, B
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/2108/179311`