Discrete Time Branching Processes in Random Environment

3Large Deviations for BPREs

3.1. Introduction

In large deviation theory, we consider events with probabilities that are asymptotically exponentially small. For BPREs, events of two different types have been studied: upper large deviation events of the form {Z_n ≥ e^xn} with x ≥ 0 and lower large deviation events such as {Z_n ≤ e^xn} or {Z_n = z} with z > 0. In Chapter 2, Theorem 2.5, we already encountered a large deviation result on BPREs which then offered a motivation to introduce different subcritical regimes.

In this chapter, we derive an upper large deviation result which extends the claim of Theorem 2.5. We do not refer to this theorem but start anew from the observation that, in any case, the limit

[3.1]

exists with 0 ≤ η < ∞; moreover,

[3.2]

This follows from the properties of subadditive sequences (see [DEM 93, DEN 00]). Namely, we have the inequality

meaning that the sequence − ln (Z_n > 0), n ≥ 0 is subadditive. Theorem 2.5 provides a formula for the value of η.

Apart from the existence of the limit η, we shall apply Cramer’s classical large deviation theorem on sums of i.i.d. ...