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Large deviations for the branching random walk with heavy-tailed associated random walk - a principle of one big jump

Jakob Stonner

Abstract

We prove a version of Nagaev's theorem for the branching random walk with heavy-tailed associated random walk. For a branching random walk on $\mathbb{R}$ we consider the random measure $Z_n = \sum_{|u|=n} e^{-V_u} δ_{V_u}$ where $V_u$, $|u| = n$ denote the positions of the particles in the $n$-th generation. Under the assumption that $\mathbb{E}[Z_1(\cdot)]$ is a probability distribution with regularly varying tail, we prove that $Z_n((n\mathbb{E}[X] + t_n , \infty)) = W n\mathbb{P}(X > t_n )(1 + o(1))$ in $L^1$ as $n \to \infty$ where $W$ is a non-zero random variable, $t_n \uparrow \infty$ grows suitably fast, and $X$ has law $\mathbb{E}[Z_1(\cdot)]$. The result is explained probabilistically by a principle of one big jump for the branching random walk.

Large deviations for the branching random walk with heavy-tailed associated random walk - a principle of one big jump

Abstract

We prove a version of Nagaev's theorem for the branching random walk with heavy-tailed associated random walk. For a branching random walk on we consider the random measure where , denote the positions of the particles in the -th generation. Under the assumption that is a probability distribution with regularly varying tail, we prove that in as where is a non-zero random variable, grows suitably fast, and has law . The result is explained probabilistically by a principle of one big jump for the branching random walk.
Paper Structure (17 sections, 12 theorems, 187 equations)

This paper contains 17 sections, 12 theorems, 187 equations.

Table of Contents

  1. Introduction
  2. Notation and overview
  3. Notation and overview
  4. Model setup and main results
  5. Branching Random Walk
  6. Assumptions, examples and result statement
  7. Proof of Theorem \ref{['thm:brw-large-deviations']}
  8. Preliminaries
  9. Overview
  10. Proof of Theorem \ref{['thm:brw-large-deviations']} -- Part 1: Preparation
  11. Proof of Theorem \ref{['thm:brw-large-deviations']} -- Part 2: A weak law of large numbers
  12. Proof of Lemma \ref{['lemma:Zhat-wlln']} in case \ref{['item:case2']}
  13. Proof of Lemma \ref{['lemma:Zhat-wlln']} in case \ref{['item:case1']}
  14. Proof of Theorem \ref{['thm:brw-large-deviations']} -- Part 3: Conclusion
  15. Proof of Proposition \ref{['prop:stochdom']}
  16. ...and 2 more sections

Key Result

Theorem 2.5

Assume eq:heavy-tailed through eq:mu-reg-var with $p > 2$ and fix $a > \sqrt{p-2}$ and a sequence $(t_n)_{n \in \mathbb{N}}$ with $t_n \geq a \sigma \sqrt{n \log n}$ for all $n \in \mathbb{N}$. Further suppose that one of the following conditions is satisfied: Then we have where $W$ is the limit of Biggins' martingale. Moreover, in the case item:case2 the convergence holds in $L^q$ for any $q <

Theorems & Definitions (27)

  • Remark 2.1
  • Remark 2.2
  • Example 2.3
  • Remark 2.4
  • Theorem 2.5
  • Remark 2.6
  • Lemma 3.1: Many-to-one formula,Biggins1997aShi2015
  • Theorem 3.2: Kyprianou2000
  • Theorem 3.3: Nagaev1979
  • Definition 3.4
  • ...and 17 more