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

In the previous section we proved a central limit theorem, in $L^{1}$ with respect to the environment, for random walks with random conductances. The approach, derived from the theory developed in Chap. 2, applies to a large class of processes. In this section, we present a general framework, which may appear abstract at a first reading, in which similar arguments can be used. The reader is invited to keep in mind the example of the first section as a particular case of the class of processes introduced below.

On a probability space $(\Omega, \mathscr{F}, \mathbb{Q})$ we are given a group of transformations $\tau_{x}$ : $\Omega \rightarrow \Omega, x \in \mathbb{Z}^{d}$, which preserves the measure $\mathbb{Q}: \tau_{0}$ is the identity, $\tau_{x} \circ \tau_{y}=\tau_{x+y}$ and $\mathbb{Q}\left[\tau_{x} A\right]=\mathbb{Q}[A]$ for all $x, y \in \mathbb{Z}^{d}, A \in \mathscr{F}$. We assume that the action of the group is ergodic: if $A$ is such that $\mathbb{Q}\left[\left(\tau_{x} A\right) \Delta A\right]=0$ for all $x \in \mathbb{Z}^{d}$ then $\mathbb{Q}[A]$ is either 0 or 1 . Here $\Delta$ stands for the symmetric difference: $A \Delta B=(A \backslash B) \cup(B \backslash A)$.
Suppose that we are given a family of non-negative functions $p_{x}: \Omega \rightarrow[0, \infty)$, $x \in \mathbb{Z}^{d}$, which represent the rate at which a random walk in random environment jumps from the origin to $x$ if the environment is $\omega$. In the example of the previous section, $\Omega=[a, b]^{\mathbb{B}{d}}$ and $p{x}(\xi)=\xi\left(0, e_{j}\right)$ if $x=e_{j}$ for some $1 \leq j \leq 2 d, p_{x}(\xi)=$ 0 otherwise. We shall assume that the functions $\left{p_{x}: x \in \mathbb{Z}^{d}\right}$ satisfy the following conditions.
(H1) Boundedness: $p_{z}$ belongs to $B(\Omega)$ for all $z \in \mathbb{Z}^{d}$;
(H2) Finite range: There exists a deterministic $R>0$ such that $p_{z}=0$ if $|z| \geq R$, where $|\cdot|$ stands for the Euclidean norm of $\mathbb{R}^{d}$;
(H3) Irreducibility: For $\mathbb{Q}$-a.e. $\omega$ there exists a random set $\Lambda(\omega)$ generating $\mathbb{Z}^{d}$ such that $p_{z}(\omega)>0, z \in \Lambda(\omega)$
(H4) Double stochasticity: $\sum_{z \in \mathbb{Z}^{d}} p_{-z}\left(\tau_{z} \omega\right)=\sum_{z \in \mathbb{Z}^{d}} p_{z}(\omega)$ for $\mathbb{Q}$-a.e. $\omega$.
As before, $B(\Omega)$ stands for the space of bounded measurable functions $f: \Omega \rightarrow$ $\mathbb{R}$ endowed with the sup norm. Moreover, a subset $\Lambda$ of $\mathbb{Z}^{d}$ is said to generate $\mathbb{Z}^{d}$ if for any $x$ in $\mathbb{Z}^{d}$ one can find $n \geq 1$ and $z_{1}, \ldots, z_{n} \in \Lambda$ for which $x=\sum_{i=1}^{n} z_{i}$. One can easily check that the random walks with random conductances examined in the previous section satisfy all the above conditions.

For each fixed $\omega$, we define a random walk $\left{X_{t}^{\omega}: t \geq 0\right}$ over a probability space $(\Sigma, \mathscr{A}, \mathbb{P})$ whose random jump rate from $y$ to $z$, denoted by $p(y, z ; \omega)$, is given by
$$p(y, z ; \omega):=p_{z-y}\left(\tau_{y} \omega\right)$$

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

In this section we give an example of a non-reversible random walk satisfying hypotheses (H1)-(H4), (3.14), (3.15), (3.17). We start with the definition of a cycle.
A cycle $C$ of length $n$ is a sequence of $n$ sites of $\mathbb{Z}^{d}$ starting and ending at the same point: $\left(y_{0}, y_{1}, \ldots, y_{n-1}, y_{n}=y_{0}\right), y_{i} \neq y_{i+1}, 0 \leq i \leq n-1$. To a cycle $C$ of length $n$, we associate a zero-mean probability measure $p_{C}$ on $\mathbb{Z}^{d}$ which does not charge the origin defined by
$$p_{C}(x)=\frac{1}{n} \sum_{j=0}^{n-1} \mathbf{1}\left{x=y_{j+1}-y_{j}\right}$$
$p_{C}$ has mean zero since
$$\sum_{x \in \mathbb{Z}^{d}} x p_{C}(x)=\frac{1}{n} \sum_{x \in \mathbb{Z}^{d}} \sum_{j=0}^{n-1} x \mathbf{1}\left{x=y_{j+1}-y_{j}\right}$$

$$=\frac{1}{n} \sum_{j=0}^{n-1}\left{y_{j+1}-y_{j}\right}=\frac{y_{n}-y_{0}}{n}=0 .$$
A probability measure associated to a cycle $C$ is called a cyclic probability measure. The measure associated to a cycle $C=\left(y_{0}, y_{1}, \ldots, y_{n-1}, y_{0}\right)$ translated by $x$ coincides with the one associated to $C:$ If $C+x=\left(y_{0}+x, y_{1}+x, \ldots, y_{n-1}+x, y_{0}+\right.$ $x), p_{C+x}(\cdot)=p_{C}(\cdot)$. The same observation holds for the probability measure associated to the cycle $C^{\prime}=\left(y_{1}, y_{2}, \ldots, y_{n-1}, y_{0}, y_{1}\right)$ obtained by shifting the cycle $C$. We may, in particular, assume that $y_{0}=0$.

A cycle $C=\left(y_{0}, y_{1}, \ldots, y_{n-1}, y_{0}\right)$ is said to be irreducible if $y_{i} \neq y_{j}$ for $0 \leq$ $i \neq j \leq n-1$. Clearly, a cycle $C$ can always be decomposed in a finite number of irreducible cycles. Moreover, if $C$ is decomposed in irreducible cycles $C_{1}, \ldots, C_{k}$, $p_{C}$ is a rational convex combination of $p_{C_{1}}, \ldots, p_{C_{k}}: p_{C}(\cdot)=\sum_{1 \leq j \leq k} w_{j} p_{C_{j}}(\cdot)$ for some strictly positive rationals $w_{j}$ such that $\sum_{1 \leq j \leq k} w_{j}=1$.

The cyclic probability measures are the finite-range zero-mean probability measures on $\mathbb{Z}^{d}$ taking rational values which do not charge the origin. Indeed, fix such a probability measure $p$ and denote its support by $S=\left{x_{1}, \ldots, x_{n}\right}$. There exists a sufficiently large positive integer $M$ for which $p(x)=m(x) / M$, where $m(x)$ are positive integers. Consider the cycle $C=\left(0, x_{1}, 2 x_{1}, \ldots, m\left(x_{1}\right) x_{1}, m\left(x_{1}\right) x_{1}+\right.$ $\left.x_{2}, \ldots, m\left(x_{1}\right) x_{1}+m\left(x_{2}\right) x_{2}, \ldots, m\left(x_{1}\right) x_{1}+\cdots+m\left(x_{n-1}\right) x_{n-1}+\left(m\left(x_{n}\right)-1\right) x_{n}, 0\right)$. It is easy to check that the probability measure associated to this cycle is $p$.

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

Consider a doubly stochastic random walk $\left{X^{\omega}(t): t \geq 0\right}$ as defined in Sect. 3.2. In this section, we show that the sector condition (3.17) is not needed to prove the central limit theorem provided the assumptions on the drift are strengthen.

We shall suppose that the random rates satisfy an ellipticity condition, i.e., that there exists a deterministic constant $\kappa>0$ and a finite deterministic subset $\Lambda \subset \mathbb{Z}^{d}$, generating the lattice, such that $\mathbb{Q}$ almost surely
$$\min {z \in \Lambda} p{z}(\omega) \geq \kappa$$
To simplify the exposition, although not needed, we shall also assume that $p_{z}(\omega)=$ 0 for $z \notin \Lambda$. Denote by $R$ a positive integer such that $p_{z}=0$ for $|z|_{\infty}>R$, where $|x|_{\infty}$ stands for the max norm, $\left|\left(x_{1}, \ldots, x_{d}\right)\right|_{\infty}=\max \left{\left|x_{1}\right|, \ldots,\left|x_{d}\right|\right}$.

To present the extra assumptions needed on the local drift, we first show in Lemma $3.5$ below that any function in the space $L^{2}(\mathbb{Q}) \cap \mathscr{H}{-1}$ can be represented as $\sum{z \in \Lambda} D_{z}^{*}\left(p_{z} \Psi_{z}\right)$ for some $\Psi_{z} \in L^{2}(\mathbb{Q}), z \in \Lambda$.

Let $\mathscr{H}$ be the Hilbert space consisting of all random vectors $F:=\left{F_{z}: z \in \Lambda\right}$, $F_{z}: \Omega \rightarrow \mathbb{R}$, that satisfy
$$|F|_{\mathscr{H}}^{2}:=\frac{1}{2} \sum_{z \in \Lambda}\left\langle p_{z} F_{z}, F_{z}\right\rangle_{\mathbb{Q}}<\infty$$
Denote by $|\cdot|_{\mathscr{H}}$ the respective norm.
Recall from (3.9) the definition of the operator $D_{z}: L^{2}(\mathbb{Q}) \rightarrow L^{2}(\mathbb{Q}), z \in \mathbb{Z}^{d}$. Let $L_{0}^{2}(\mathbb{Q})$ represent the subspace of $L^{2}(\mathbb{Q})$ consisting of all zero mean elements and let $\mathscr{D}: L_{0}^{2}(\mathbb{Q}) \rightarrow \mathscr{H}$ be given by $\mathscr{D} g:=\left{D_{z} g, z \in \Lambda\right}$. The closure of $\mathscr{D}\left(L_{0}^{2}(\mathbb{Q})\right)$ shall be denoted by $\mathscr{H}_{\nabla} \subset \mathscr{H}$. It represents the space of gradients.

It follows from the explicit expression of the Dirichlet form that $\mathscr{H}{\nabla}$ is isomorphic to $\mathscr{H}{1}$, where the isomorphism $\bar{D}: \mathscr{H}{1} \rightarrow \mathscr{H}{\nabla}$ is given by the continuous extension of the operator $\mathscr{D}$.

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

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(H1) 有界性: $p_{z}$ 属于 $B(\Omega)$ 对所有人 $z \in \mathbb{Z}^{d}$;
(H2) 有限范围：存在确定性 $R>0$ 这样 $p_{z}=0$ 如果 $|z| \geq R$ ， 在哪里 $|\cdot|$ 代表欧几里得范数 $\mathbb{R}^{d}$;
(H3) 不可约性: 对于 $\mathbb{Q}$-ae $\omega$ 存在一个随机集 $\Lambda(\omega)$ 生成 $\mathbb{Z}^{d}$ 这样 $p_{z}(\omega)>0, z \in \Lambda(\omega)$
(H4) 双随机性: $\sum_{z \in \mathbb{Z}^{d}} p_{-z}\left(\tau_{z} \omega\right)=\sum_{z \in \mathbb{Z}^{d}} p_{z}(\omega)$ 为了 $\mathbb{Q}$-ae $\omega$.

$$p(y, z ; \omega):=p_{z-y}\left(\tau_{y} \omega\right)$$

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

\left 的分隔符缺失或无法识别
$p_{C}$ 平均为零，因为
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$\backslash$ left 的分隔符缺失或无法识别

$\backslash$ left 的分隔符缺失或无法识别 ．存在一个足够大的正整数 $M$ 为此 $p(x)=m(x) / M$ ，在哪里 $m(x)$ 是正整数。考虑循环 $C=\left(0, x_{1}, 2 x_{1}, \ldots, m\left(x_{1}\right) x_{1}, m\left(x_{1}\right) x_{1}+\right.$
$\left.x_{2}, \ldots, m\left(x_{1}\right) x_{1}+m\left(x_{2}\right) x_{2}, \ldots, m\left(x_{1}\right) x_{1}+\cdots+m\left(x_{n-1}\right) x_{n-1}+\left(m\left(x_{n}\right)-1\right) x_{n}, 0\right)$. 很容易检查与此循环相关的概率测度是否 为 $p$.

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

$$\min z \in \Lambda p z(\omega) \geq \kappa$$

$$|F|{\mathscr{H}}^{2}:=\frac{1}{2} \sum{z \in \Lambda}\left\langle p_{z} F_{z}, F_{z}\right\rangle_{\mathbb{Q}}<\infty$$

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