格点分支随机游动剖面的 Edgeworth 展开
概率论
2016-06-14 v2
摘要
考虑离散时间下 Z \mathbb Z Z L n ( k ) L_n(k) L n ( k ) n ∈ N 0 n\in\mathbb N_0 n ∈ N 0 k ∈ Z k\in\mathbb Z k ∈ Z k ↦ L n ( k ) k\mapsto L_n(k) k ↦ L n ( k ) n n n n → ∞ n\to\infty n → ∞ L n ( k ) L_n(k) L n ( k ) e − φ ( 0 ) n L n ( k ) = e − 1 2 x n 2 ( k ) 2 π φ ′ ′ ( 0 ) n ∑ j = 0 r F j ( x n ( k ) ) n j / 2 + o ( n − r + 1 2 ) a . s . , 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) \quad a.s., e − φ ( 0 ) n L n ( k ) = 2 π φ ′′ ( 0 ) n e − 2 1 x n 2 ( k ) j = 0 ∑ r n j /2 F j ( x n ( k )) + o ( n − 2 r + 1 ) a . s . , r ∈ N 0 r\in\mathbb N_0 r ∈ N 0 φ ( β ) = log ∑ k ∈ Z e β k E L 1 ( k ) \varphi(\beta)=\log \sum_{k\in\mathbb Z} e^{\beta k} \mathbb E L_1(k) φ ( β ) = log ∑ k ∈ Z e β k E L 1 ( k ) x n ( k ) = k − φ ′ ( 0 ) n φ ′ ′ ( 0 ) n . x_n(k) = \frac{k-\varphi'(0) n}{\sqrt{\varphi''(0)n}}. x n ( k ) = φ ′′ ( 0 ) n k − φ ′ ( 0 ) n . k ∈ Z k\in\mathbb Z k ∈ Z F j F_j F j φ \varphi φ r = 0 , 1 , 2 r=0,1,2 r = 0 , 1 , 2 L n ( k n ) L_n(k_n) L n ( k n ) k n ∈ Z k_n\in\mathbb Z k n ∈ Z n n n arg max k ∈ Z L n ( k ) \arg \max_{k\in\mathbb Z} L_n(k) arg max k ∈ Z L n ( k ) max k ∈ Z L n ( k ) \max_{k\in\mathbb Z} L_n(k) max k ∈ Z L n ( k ) φ ′ ( 0 ) \varphi'(0) φ ′ ( 0 )
引用
@article{arxiv.1503.04616,
title = {Edgeworth expansions for profiles of lattice branching random walks},
author = {Rudolf Grübel and Zakhar Kabluchko},
journal= {arXiv preprint arXiv:1503.04616},
year = {2016}
}
备注
34 pages, 5 figures