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## 统计代写|离散时间鞅理论代写martingale代考|Random Walks in Mixing Environments

In this section, we show that the local drift $V$ satisfies the assumptions (3.21), (3.22) if the random field $\left{p_{z} \circ \tau_{x}: x \in \mathbb{Z}^{d}\right}$ is sufficiently strongly mixing for each $z \in \Lambda$.
Denote by $d$ the distance on $\mathbb{Z}^{d}$ defined by $d(x, y)=\sum_{1 \leq i \leq d}\left|x_{i}-y_{i}\right|, x$, $y \in \mathbb{Z}^{d}$. We extend this notion to sets in the natural way. For $x \in \mathbb{Z}^{d}$ and subsets $\Gamma, \Gamma_{1}, \Gamma_{2}$ of $\mathbb{Z}^{d}, d(x, \Gamma)$ stands for the distance from $x$ to $\Gamma$ defined by $d(x, \Gamma)=\min {y \in \Gamma} d(x, y)$, and $d\left(\Gamma{1}, \Gamma_{2}\right)$ for the distance between $\Gamma_{1}$ and $\Gamma_{2}$, defined by $d\left(I_{1}, I_{2}\right)=\min {x \in \Gamma{1}}, y \in \Gamma_{2} d(x, y)$.

For an arbitrary finite subset $\Gamma$ of $\mathbb{Z}^{d}$ and a positive integer $m$, denote by $\Gamma_{m}^{c}$ the set of sites which are a distance at least $m$ from $\Gamma: \Gamma_{m}^{c}=\left{x \in \mathbb{Z}^{d}: d(x, \Gamma) \geq m\right}$. For a subset $A$ of $\mathbb{Z}^{d}$ and $z \in \Lambda$, denote by $\mathscr{F}{z}(A)$ the $\sigma$-algebra generated by $p{z} \circ \tau_{x}$ for $x \in A$. For $m \geq 1$, a finite subset $\Gamma$ of $\mathbb{Z}^{d}$ and $z \in \Lambda$, let
$$\alpha_{m}\left(\Gamma ; p_{z}\right):=\sup \left{|\mathbb{Q}(A \cap B)-\mathbb{Q}(A) \mathbb{Q}(B)|: A \in \mathscr{F}{z}(\Gamma), B \in \mathscr{F}{z}\left(\Gamma_{m}^{c}\right)\right}$$
The $\alpha$-mixing coefficients of the random field $\left{p_{z} \circ \tau_{x}: x \in \mathbb{Z}^{d}\right}$ are defined as $\alpha_{m}\left(p_{z}\right):=\sup {\Gamma} \alpha{m}\left(\Gamma ; p_{z}\right)$, where the supremum is carried over all finite subsets $\Gamma$ of $\mathbb{Z}^{d}$. The following result which allows to control the covariance of random variables in terms of the mixing coefficient is Theorem 17.2.1 of Ibragimov and Linnik (1971), p. 306 and Lemma 3 in p. 10 of Doukhan (1994).

## 统计代写|离散时间鞅理论代写martingale代考|Doubly Stochastic Random Walks in Dimension d = 1

As the title suggests, in this section, we examine, doubly stochastic random walks in dimension $d=1$ satisfying assumptions (H1)-(H4). The main result states that the generator of the environment process associated to such randoms walks satisfies a sector condition provided the local drift has zero mean with respect to the ergodic measure $\mathbb{Q}$ and the random rates satisfy an elliptic condition (3.20). This statement follows from the fact, presented in Theorem $3.17$ below, that in dimension 1 the generator of the environment process associated to a doubly stochastic random walk can be written as the sum of three pieces: its symmetric part; an operator in divergence form with bounded coefficients; and an operator which vanishes if the expectation of the local drift vanishes.
We start with the decomposition of the generator of the environment process.
Theorem $3.17$ Suppose that the field $\left{p_{z}: z \in \mathbb{Z}^{d}\right}$ satisfies conditions (H1)-(H4). Then, there exist a non-random finite set $\Lambda_{0}$ and a random field $\left{a_{y, z}: y, z \in \Lambda_{0}\right}$, $a_{y, z}$ in $B(\Omega)$, such that the generator $L$ of the process ${\eta(t): t \geq 0}$ satisfies
$$L f=S f+\langle V\rangle_{\mathbb{Q}} D_{1} f+\sum_{y, z \in \Lambda_{0}} D_{y}^{}\left(a_{y, z} D_{z} f\right)$$ for all $f$ in $L^{2}(\mathbb{Q})$, where $S$ is the symmetric part of the generator $L$ given by (3.10) and $D_{z}, D_{y}^{}$ are the operators defined by (3.9).

## 统计代写|离散时间鞅理论代写martingale代考|Symmetric Random Walks

In this section, we present two results on the transition probability of symmetric rãndom walks used in thé chāpter. By a symmetric, simplẽ random wallk on $\mathbb{Z}^{d}$ starting at $x$ we understand a random sequence $\left{X_{n}, n \geq 0\right}$, defined over a probability space $(\Sigma, \mathscr{W}, \mathbb{Q})$, taking values on a $d$-dimensional integer lattice and such that $\mathbb{Q}\left[X_{0}=x\right]=1$, and
$$\mathbb{Q}\left[X_{n+1}=x_{n+1} \mid X_{n}=x_{n}, \ldots, X_{0}=x_{0}\right]= \begin{cases}(2 d)^{-1} & \text { if }\left|x_{n}-x_{n+1}\right|_{\infty}=1, \ 0 & \text { otherwise. }\end{cases}$$

In the particular case $d=1$ we have
$$\mathbb{Q}\left[X_{2 n}=2 y\right]=\frac{1}{2^{2 n}}\left(\begin{array}{c} 2 n \ n-y \end{array}\right), \quad \mathbb{Q}\left[X_{2 n+1}=2 y+1\right]=\frac{1}{2^{2 n+1}}\left(\begin{array}{c} 2 n+1 \ n-y \end{array}\right),$$
for $y \in \mathbb{Z}$, where by convention $\left(\begin{array}{c}n \ m\end{array}\right)=0$ whenever $m>n$, or $m<0$.
Suppose that $\left{a_{n}, n \geq 1\right},\left{b_{n}, n \geq 1\right}$ are two sequences of positive numbers. We write $a_{n} \asymp b_{n}$ if
$$0<\liminf {n \rightarrow+\infty} a{n} / b_{n} \leq \limsup {n \rightarrow+\infty} a{n} / b_{n}<+\infty$$
By Stirling’s formula for each $n \geq 1$ there exists $\theta \in(0,1)$ such that $n !=\sqrt{2 \pi n}$ $\left(n e^{-1}\right)^{n} e^{\theta /(12 n)}$. Therefore for any $y$ such that $|y|_{\infty}<n$
\begin{aligned} &\mathbb{Q}\left[X_{2 n}=2 y\right] \ &\because n^{-1 / 2}\left[1-\left(\frac{y}{n}\right)^{2}\right]^{-1 / 2} \exp \left{-n \log \left[1-\left(\frac{y}{n}\right)^{2}\right]-y \log \frac{1+y / n}{1-y / n}\right} \ &=n^{-1 / 2}\left[1-\left(\frac{y}{n}\right)^{2}\right]^{-1 / 2} \exp \left{-n \sum_{p=1}^{+\infty} \frac{1}{p(2 p-1)}\left(\frac{y}{n}\right)^{2 p}\right} \end{aligned}
A similar asymptotic formula holds for $\mathbb{Q}\left[X_{2 n+1}=2 y+1\right]$. As a result for any $\delta \in(0,1)$ there exist $0<c_{U}<C_{U}$, depending on $\delta$, such that
$$\frac{c_{U}}{n^{1 / 2}} \exp \left{-\frac{C_{U} x^{2}}{n}\right} \leq \mathbb{Q}\left[X_{n}=x\right] \leq \frac{C_{U}}{n^{1 / 2}} \exp \left{-\frac{c_{U} x^{2}}{n}\right}$$

## 统计代写|离散时间鞅理论代写martingale代考|Random Walks in Mixing Environments

$\backslash 1$ eft 的分隔符缺失或无法识别

## 统计代写|离散时间鞅理论代写martingale代考|Doubly Stochastic Random Walks in Dimension d = 1

left 的分隔符缺失或无法识别
\begin{aligned} &a_{y, z} \text { 在 } B(\Omega) \text { ，使得生成器 } L \text { 过程的 } \eta(t): t \geq 0 \ &=S f+\langle V\rangle_{\mathbb{Q}} D_{1} f+\sum_{y, z \in \Lambda_{0}} D_{y}\left(a_{y, z} D_{z} f\right) \end{aligned}

## 统计代写|离散时间鞅理论代写martingale代考|Symmetric Random Walks

$$\mathbb{Q}\left[X_{n+1}=x_{n+1} \mid X_{n}=x_{n}, \ldots, X_{0}=x_{0}\right]=\left{(2 d)^{-1} \quad \text { if }\left|x_{n}-x_{n+1}\right|{\infty}=1,0 \quad\right. \text { otherwise }$$ 在特定情况下 $d=1$ 我们有 $$\mathbb{Q}\left[X{2 n}=2 y\right]=\frac{1}{2^{2 n}}(2 n n-y), \quad \mathbb{Q}\left[X_{2 n+1}=2 y+1\right]=\frac{1}{2^{2 n+1}}(2 n+1 n-y),$$

$$0<\liminf n \rightarrow+\infty a n / b_{n} \leq \lim \sup n \rightarrow+\infty a n / b_{n}<+\infty$$

$\backslash 1 \mathrm{eft}$ 的分隔符缺失或无法识别

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