Abstract
Branching Processes in Random Environment (BPREs) $(Z_n:n\geq0)$ are the generalization of Galton-Watson processes where 'in each generation' the reproduction law is picked randomly in an i.i.d. manner. The associated random walk of the environment has increments distributed like the logarithmic mean of the offspring distributions. This random walk plays a key role in the asymptotic behavior. In this paper, we study the upper large deviations of the BPRE $Z$ when the reproduction law may have heavy tails. More precisely, we obtain an expression for the limit of $-\log \mathbb{P}(Z_n\geq \exp(\theta n))/n$ when $n\rightarrow \infty$. It depends on the rate function of the associated random walk of the environment, the logarithmic cost of survival $\gamma:=-\lim_{n\rightarrow\infty} \log \mathbb{P}(Z_n \gt 0)/n$ and the polynomial rate of decay $\beta$ of the tail distribution of $Z_1$. This rate function can be interpreted as the optimal way to reach a given "large" value. We then compute the rate function when the reproduction law does not have heavy tails. Our results generalize the results of Böinghoff & Kersting (2009) and Bansaye & Berestycki (2008) for upper large deviations. Finally, we derive the upper large deviations for the Galton-Watson processes with heavy tails.
Highlights
Branching processes in random environment have been introduced in [3] and [21]
Branching Processes in Random Environment (BPREs) have originally been studied under the assumption of i.i.d. geometric -or more generally linear fractional- offspring distributions [1, 17]
In [5], the authors present a general upper bound for the rate function and compute it in the special case of each individual leaving at least one offspring, i.e. P(Z1 = 0) = 0
Summary
Branching processes in random environment have been introduced in [3] and [21]. In such processes, for each generation, an offspring distribution is chosen at random, independently of other generations. Several results about the large deviations of branching processes in random environment for offspring distributions with weak tails have been proved. In [5], the authors present a general upper bound for the rate function and compute it in the special case of each individual leaving at least one offspring, i.e. P(Z1 = 0) = 0. In [9] an expression of the upper rate function is derived when the reproduction laws have geometrically bounded tails. We refer to [12] for more precise asymptotic results on the survival probability of subcritical BPREs. For large deviations without heavy tails, it has been already observed in [9] that γ is of importance in the limit theorems only in the strongly subcritical case.
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