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}
}
  • Rudolf Grubel, Zakhar Kabluchko
  • 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

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