arXiv 1503.046162015-03math.PR

格点分支随机游动轮廓的Edgeworth展开

Edgeworth expansions for profiles of lattice branching random walks

Rudolf Grübel, Zakhar Kabluchko

备注：: 34 pages, 5 figures

摘要：

考虑$\mathbb Z$上离散时间的分支随机游动。用$L_n(k)$表示在时间$n\in\mathbb N_0$时在位点$k\in\mathbb Z$处的颗粒数。分支随机游走的轮廓（在时间$n$）是指函数$k\mapsto L_n(k)$。我们建立了$L_n(k)$的以下渐近展开式，作为$n\to\infty$：$$ e^{-φ(0)n} L_n(k) = \frac{e^{-\frac 12 x_n^2(k)}}{\sqrt {2πφ''(0) n}} \sum_{j=0}^r \frac{F_j(x_n(k))}{n^{j/2}} + o\left(n^{-\frac{r+1}{2}}\right) \quad a.s., $$其中$r\in\mathbb N_0$是任意的，$φ(β)=\log \sum_{k\in\mathbb Z} e^{βk} \mathbb E L_1(k)$是分支随机游动的强度的累积生成函数，并且$$ x_n(k) = \frac{k-φ'(0) n}{\sqrt{φ''(0)n}}. $$该展开式在具有概率$1$的$k\in\mathbb Z$中一致有效，并且$F_j$是其随机系数可以通过$φ$的导数和$φ$的极限的导数来表示的多项式。比金斯鞅在$0$。使用指数倾斜，我们还建立了更一般的扩展，覆盖整个范围内的分支随机游动除了它的极值。作为这种扩展的$r=0,1,2$的应用，我们恢复在一个统一的方式一些已知的结果，并建立几个新的极限定理。特别是，我们研究了A. S。个人职业号码$L_n(k_n)$的行为，其中$k_n\in\mathbb Z$以某种常规方式依赖于$n$。我们也证明了A. S。极限定理的模式$\arg \max_{k\in\mathbb Z} L_n(k)$和高度$\max_{k\in\mathbb Z} L_n(k)$的轮廓。这些量的渐近行为取决于漂移参数$φ'(0)$是整数、非整数有理数还是无理数。

英文摘要：

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 $n\to\infty$: $$ e^{-φ(0)n} L_n(k) = \frac{e^{-\frac 12 x_n^2(k)}}{\sqrt {2πφ''(0) n}} \sum_{j=0}^r \frac{F_j(x_n(k))}{n^{j/2}} + o\left(n^{-\frac{r+1}{2}}\right) \quad a.s., $$ where $r\in\mathbb N_0$ is arbitrary, $φ(β)=\log \sum_{k\in\mathbb Z} e^{βk} \mathbb E L_1(k)$ is the cumulant generating function of the intensity of the branching random walk and $$ x_n(k) = \frac{k-φ'(0) n}{\sqrt{φ''(0)n}}. $$ The expansion is valid uniformly in $k\in\mathbb Z$ with probability $1$ and the $F_j$'s are polynomials whose random coefficients can be expressed through the derivatives of $φ$ and the derivatives of the limit of the Biggins martingale at $0$. Using exponential tilting, we also establish more general expansions covering the whole range of the branching random walk except its extreme values. As an application of this expansion for $r=0,1,2$ we recover in a unified way a number of known results and establish several new limit theorems. In particular, we study the a.s. behavior of the individual occupation numbers $L_n(k_n)$, where $k_n\in\mathbb Z$ depends on $n$ in some regular way. We also prove a.s. limit theorems for the mode $\arg \max_{k\in\mathbb Z} L_n(k)$ and the height $\max_{k\in\mathbb Z} L_n(k)$ of the profile. The asymptotic behavior of these quantities depends on whether the drift parameter $φ'(0)$ is integer, non-integer rational, or irrational.

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