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The extremal point process for branching random walk with stretched exponential displacements

Piotr Dyszewski, Nina Gantert

Abstract

We investigate a branching random walk where the displacements are independent from the branching mechanism and have a stretched exponential distribution. We describe the positions of the particles in the vicinity of the rightmost particle in terms of point process convergence. As a consequence we give a~new limit theorem for the position of the rightmost particle. Our methods rely on providing precise large deviations for sums of i.i.d. random variables with stretched exponential distributions outside the so-called one big jump regime.

The extremal point process for branching random walk with stretched exponential displacements

Abstract

We investigate a branching random walk where the displacements are independent from the branching mechanism and have a stretched exponential distribution. We describe the positions of the particles in the vicinity of the rightmost particle in terms of point process convergence. As a consequence we give a~new limit theorem for the position of the rightmost particle. Our methods rely on providing precise large deviations for sums of i.i.d. random variables with stretched exponential distributions outside the so-called one big jump regime.
Paper Structure (8 sections, 19 theorems, 208 equations, 2 figures)

This paper contains 8 sections, 19 theorems, 208 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main result
  4. Trimming lemmas
  5. Decoupling
  6. Convergence of the point processes
  7. Large deviations for stretched exponential tails
  8. Appendix

Key Result

Theorem 3.1

Suppose that Assumptions as:BP and as:RW are in force. Let $d_n$ and $a_n$ be given by eq:3:dnDef and eq:3:anDef respectively and take $b_n = d_n +\tau_n$, where $\tau_n$ is given in eq:3:tau. Then with $\Lambda$ given by eq:3:LambdaLimit.

Figures (2)

  • Figure 1: BRW with steps distributed uniformly on the interval (-1/2, 1/2) (left) and regularly varying steps with index $\gamma = 2$ (right). The red lines indicate the displacements of the rightmost particles from its place of birth.
  • Figure 2: BRW with stretched exponential steps with $r=3/4$ (left) and $r=1/3$ (right). The red lines indicate the displacements of the rightmost particles from its place of birth.

Theorems & Definitions (38)

  • Definition 2.1
  • Definition 2.2
  • Theorem 3.1
  • Corollary 3.2
  • Remark 3.3
  • Remark 3.4
  • Remark 3.5
  • Lemma 4.1
  • Lemma 4.2
  • proof
  • ...and 28 more