# Edgeworth expansions for profiles of lattice branching random walks

@article{Grubel2015EdgeworthEF,
title={Edgeworth expansions for profiles of lattice branching random walks},
author={Rudolf Grubel and Zakhar Kabluchko},
journal={arXiv: Probability},
year={2015},
pages={2103-2134}
}
• Published 2015
• Mathematics
• arXiv: Probability
• Consider a branching random walk on $\mathbb Z$ in discrete time. Denote by $L_n(k)$ the number of particles at site $k\in\mathbb Z$ at time $n\in\mathbb N_0$. By the profile of the branching random walk (at time $n$) we mean the function $k\mapsto L_n(k)$. We establish the following asymptotic expansion of $L_n(k)$, as $n\to\infty$:  e^{-\varphi(0)n} L_n(k) = \frac{e^{-\frac 12 x_n^2(k)}}{\sqrt {2\pi \varphi''(0) n}} \sum_{j=0}^r \frac{F_j(x_n(k))}{n^{j/2}} + o\left(n^{-\frac{r+1}{2}}\right… CONTINUE READING

#### Citations

##### Publications citing this paper.
SHOWING 1-2 OF 2 CITATIONS

## General Edgeworth expansions with applications to profiles of random trees

• Mathematics
• 2016
VIEW 7 EXCERPTS
CITES BACKGROUND, METHODS & RESULTS

## Mode and Edgeworth Expansion for the Ewens Distribution and the Stirling Numbers

• Mathematics, Computer Science
• J. Integer Seq.
• 2016

#### References

##### Publications referenced by this paper.
SHOWING 1-10 OF 39 REFERENCES

## Exact convergence rates in central limit theorems for a branching random walk with a random environment in time

• Mathematics
• 2015
VIEW 5 EXCERPTS
HIGHLY INFLUENTIAL

## Width and mode of the profile for some random trees of logarithmic height

• Mathematics
• 2006
VIEW 4 EXCERPTS
HIGHLY INFLUENTIAL

## Profiles of random trees: correlation and width of random recursive trees and binary search trees

VIEW 4 EXCERPTS
HIGHLY INFLUENTIAL

## Exact Convergence Rates For The Distribution of Particles in Branching Random Walks

VIEW 3 EXCERPTS
HIGHLY INFLUENTIAL

## Uniform Convergence of Martingales in the Branching Random Walk

VIEW 8 EXCERPTS
HIGHLY INFLUENTIAL

## Spatial growth of a branching process of particles living in R

VIEW 4 EXCERPTS
HIGHLY INFLUENTIAL

## Growth rates in the branching random walk

VIEW 14 EXCERPTS
HIGHLY INFLUENTIAL

## MARTINGALE CONVERGENCE IN THE BRANCHING RANDOM WALK

VIEW 13 EXCERPTS
HIGHLY INFLUENTIAL

## OF RANDOM VARIABLES

• Mathematics
• 2016
VIEW 2 EXCERPTS

## Mod-φ Convergence

• Mathematics
• 2015
VIEW 1 EXCERPT