Skip to Main Content
Discrete Time Branching Processes in Random Environment
book

Discrete Time Branching Processes in Random Environment

by Götz Kersting, Vladimir Vatutin
November 2017
Intermediate to advanced content levelIntermediate to advanced
306 pages
6h 24m
English
Wiley-ISTE
Content preview from 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 {Znexn} with x ≥ 0 and lower large deviation events such as {Zn exn} or {Zn = 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

exists with 0 η < ∞; moreover,

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

Image

meaning that the sequence − ln images(Zn > 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. ...

Become an O’Reilly member and get unlimited access to this title plus top books and audiobooks from O’Reilly and nearly 200 top publishers, thousands of courses curated by job role, 150+ live events each month,
and much more.
Start your free trial

You might also like

Reinventing the Organization for GenAI and LLMs

Reinventing the Organization for GenAI and LLMs

Ethan Mollick
Delayed and Network Queues

Delayed and Network Queues

Aliakbar Montazer Haghighi, Dimitar P. Mishev

Publisher Resources

ISBN: 9781786302526Purchase book