### 统计代写|离散时间鞅理论代写martingale代考|MAST90019

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## 统计代写|离散时间鞅理论代写martingale代考|Comments and References

Central Limit Theorems for martingales can be found in many textbooks, Billingsley (1995); Durrett (1996); Ethier and Kurtz (1986); Varadhan (2001), for instance. We refer to Whitt (2007) for a recent account.

To our knowledge, the first central limit theorem for Markov chains goes back to Doeblin (1938) who reduced the problem to the case of independent identically distributed random variables. We refer to Nagaev (1957) for a proof along the line of Doeblin’s idea. Gordin (1969) and Gordin and Lifšic (1978) showed that
$$\frac{1}{\sqrt{N}} \sum_{j=0}^{N-1} V\left(X_{j}\right)$$
converges to a mean zero Gaussian random variable if $V$ belongs to the range of the operator $I-P$ in $L^{2}(\pi)$. Lawler (1982) proved an invariance principle for a Markov chain in random environment.

Kozlov (1985) and Kipnis and Varadhan (1986) proposed independently a general method to prove central limit theorems for additive functionals of Markov chains from martingale central limit theorems. The approach presented here follows Kipnis and Varadhan (1986). This seminal paper has been the starting point of much research on asymptotic normality of additive functionals of ergodic Markov chains which is reviewed in the following chapters. De Masi et al. (1989) and Goldstein (1995) considered anti-symmetric additive functionals of reversible Markov chains. Maxwell and Woodroofe $(2000)$ proved that the sequence (1.27) is asymptotically normal for stationary ergodic Markov chains $\left{X_{j}: j \geq 0\right}$ provided $V$ has mean zero with respect to the stationary measure $\pi$ and
$$\sum_{n \geq 1} n^{-3 / 2}\left|\sum_{j=0}^{n-1} P^{j} V\right|<\infty$$

## 统计代写|离散时间鞅理论代写martingale代考|Central Limit Theorem for Continuous Time Martingales

On a probability space $(\Omega, \mathbb{P}, \mathscr{F})$ consider a right continuous, square-integrable martingale $\left{M_{t}: t \geq 0\right}$ with respect to a given filtration $\left{\mathscr{F}{t}: t \geq 0\right}$ satisfying the usual conditions. We refer to Jacod and Shiryaev (1987) for the terminology adopted and some elementary properties of martingales used without further comments. Assume that $M{0}=0$ and denote by $\langle M, M\rangle_{t}$ its predictable quadratic variation. Denote by $\mathbb{E}$ expectation with respect to $\mathbb{P}$.

Theorem 2.1 Assume that the increments of the martingale $M_{t}$ are stationary: for every $t \geq 0, n \geq 1$ and $0 \leq s_{0}<\cdots<s_{n}$, the random vectors $\left(M_{s_{1}}-M_{s_{0}}, \ldots, M_{s_{n}}-\right.$ $\left.M_{s_{n-1}}\right),\left(M_{t+s_{1}}-M_{t+s_{0}}, \ldots, M_{t+s_{n}}-M_{t+s_{n-1}}\right)$ have the same distribution. Assume also that the predictable quadratic variation converges in $L^{1}(\mathbb{P})$ to $\sigma^{2}=\mathbb{E} M_{1}^{2}$ :
$$\lim {n \rightarrow \infty} \mathbb{E}\left|\frac{\langle M, M\rangle{n}}{n}-\sigma^{2}\right|=0 .$$
Then, the distribution of $M_{t} / \sqrt{t}$ conditioned on $\mathscr{F}{0}$ converges in probability, as $t \uparrow \infty$, to a mean zero Gaussian law with variance $\sigma^{2}$ : $$\lim {t \rightarrow \infty} \mathbb{E}\left[\left|\mathbb{E}\left[e^{i \theta M_{t} / \sqrt{t}} \mid \mathscr{F}{0}\right]-e^{-\sigma^{2} \theta^{2} / 2}\right|\right]=0$$ for all $\theta$ in $\mathbb{R}$. The proof of this theorem relies on the next lemma which reduces the problem to proving the central limit theorem for integer times. Lemma 2.2 Under the assumptions of Theorem 2.1, $$\lim {n \rightarrow \infty} \mathbb{E}\left[\sup {n \leq t \leq n+1}\left|\mathbb{E}\left[e^{i \theta M{t} / \sqrt{t}} \mid \mathscr{F}{0}\right]-\mathbb{E}\left[e^{i \theta M{n} / \sqrt{n}} \mid \mathscr{F}{0}\right]\right|\right]=0$$ Proof The difference of conditional expectations appearing in the statement of the lemma equals $$\mathbb{E}\left[\left(\exp \left{i \theta\left[M{t} / \sqrt{t}-M_{n} / \sqrt{n}\right]\right}-1\right) e^{i \theta M_{n} / \sqrt{n}} \mid \mathscr{F}_{0}\right] .$$

## 统计代写|离散时间鞅理论代写martingale代考|The Resolvent Equation

Fix a function $V$ in $L^{2}(\pi) \cap \mathscr{H}{-1}, \lambda>0$ and consider the resolvent equation $$\lambda f{\lambda}-L f_{\lambda}-V$$
Note that $f_{\lambda}=(\lambda-L)^{-1} V$ belongs to the domain of the generator $L$. Taking the scalar product with respect to $f_{\lambda}$ on both sides of this equation we get that
$$\lambda\left\langle f_{\lambda}, f_{\lambda}\right\rangle_{\pi}+\left|f_{\lambda}\right|_{1}^{2}=\left\langle V, f_{\lambda}\right\rangle_{\pi}$$

Hence, by Schwarz inequality ( $2.9)$,
$$\lambda\left\langle f_{\lambda}, f_{\lambda}\right\rangle_{\pi}+\left|f_{\lambda}\right|_{1}^{2} \leq\left|f_{\lambda}\right|_{1}|V|_{-1}$$
so that $\left|f_{\lambda}\right|_{1} \leq|V|_{-1}$. Combining the two previous bounds we easily obtain the stronger estimate
$$\lambda\left\langle f_{\lambda}, f_{\lambda}\right\rangle_{\pi}+\left|f_{\lambda}\right|_{1}^{2} \leq|V|_{-1}^{2} .$$
From the above estimate we conclude that $\lambda f_{\lambda}$ vanishes in $L^{2}(\pi)$ as $\lambda \downarrow 0$ and that $\left{f_{\lambda}: 0<\lambda \leq 1\right}$ forms a bounded sequence in $\mathscr{H}_{1}$ and is therefore weakly precompact.

Another simple consequence of $(2.15)$ is that $(\lambda-L)^{-1}$ extends to a bounded mapping from $\mathscr{H}{-1}$ to $\mathscr{H}{1}$ :

Lemma 2.3 The operator $(\lambda-L)^{-1}$ extends from $L^{2}(\pi)$ to a bounded mapping from $\mathscr{H}{-1}$ to $\mathscr{H}{1}$. Moreover, for any $V \in \mathscr{H}{-1}$ we have $$\left|(\lambda-L)^{-1} V\right|{1} \leq|V|_{-1}$$
We wish to formulate sufficient conditions for the central limit theorem of $t^{-1 / 2} \int_{0}^{t} V\left(X_{s}\right) d s$ in terms of the asymptotic behavior, as $\lambda \downarrow 0$, of the solutions $f_{\lambda}$ of the resolvent equation (2.13). We first observe in Sect. $2.5$ that the condition $V \in \mathscr{H}{-1}$ guarantees that the $L^{2}\left(\mathbb{P}{\pi}\right)$ norm of $t^{-1 / 2} \int_{0}^{t} V\left(X_{s}\right) d s$ remains bounded for large $t$. Next, in Theorem 2.7, we show that a central limit theorem is valid, provided the following two conditions are satisfied:
$$\lim {\lambda \rightarrow 0} \lambda\left|f{\lambda}\right|_{\pi}^{2}=0 \quad \text { and } \quad \lim {\lambda \rightarrow 0}\left|f{\lambda}-f\right|_{1}=0$$
for some $f$ in $\mathscr{H}{1}$. In Theorem $2.14$, we prove that the bound $\sup {0<\lambda \leq 1}\left|L f_{\lambda}\right|_{-1}<\infty$ implies the previous two conditions. Therefore, a central limit theorem holds if $\sup {0<\lambda \leq 1}\left|L f{\lambda}\right|_{-1}<+\infty$.

## 统计代写|离散时间鞅理论代写martingale代考|Comments and References

$$\frac{1}{\sqrt{N}} \sum_{j=0}^{N-1} V\left(X_{j}\right)$$

Kozlov (1985) 和 Kipnis 和 Varadhan (1986) 分别提出了一种通用方法，用于从鞅中心极限定理证明马尔可夫链的 加性泛函的中心极限定理。这里介绍的方法遒循 Kipnis 和 Varadhan (1986)。这篇开创性的论文是对遍历马尔可夫 链的加性泛函的渐近正态性进行大量研究的起点，后续章节将对此进行回顾。德马西等人。(1989) 和 Goldstein (1995) 考虑了可逆马尔可夫链的反对称加性泛函。麦克斯韦和伍德屋顶(2000)证明了序列 (1.27) 对于静止遍历马 尔可夫链是渐近正态的 Veft {X_{j}: \geq O\right } 假如 $V$ 相对于静止测量的平均值为零 $\pi$ 和
$$\sum_{n \geq 1} n^{-3 / 2}\left|\sum_{j=0}^{n-1} P^{j} V\right|<\infty$$

## 统计代写|离散时间鞅理论代写martingale代考|Central Limit Theorem for Continuous Time Martingales

$\mathrm{~ V e f t { \ m a t h s c r { F } { t : ~ t ~ \ g e q ~ O \ r i g h t } ~ 满 足 一 般 条 件 。 我 们 参 考 了 J a c o d ~ 和}$ 鞅的一些基本性质，没有进一步的评论。假使，假设 $M 0=0$ 并表示为 $\langle M, M\rangle_{t}$ 其可预测的二次变化。表示为 $\mathbb{E}$ 关于期望 $\mathbb{P}$.

$$\lim n \rightarrow \infty \mathbb{E}\left|\frac{\langle M, M\rangle n}{n}-\sigma^{2}\right|=0$$

$$\lim t \rightarrow \infty \mathbb{E}\left[\left|\mathbb{E}\left[e^{i \theta M_{t} / \sqrt{t}} \mid \mathscr{F} 0\right]-e^{-\sigma^{2} \theta^{2} / 2}\right|\right]=0$$

$$\lim n \rightarrow \infty \mathbb{E}\left[\sup n \leq t \leq n+1\left|\mathbb{E}\left[e^{i \theta M t / \sqrt{t}} \mid \mathscr{F} 0\right]-\mathbb{E}\left[e^{i \theta M n / \sqrt{n}} \mid \mathscr{F} 0\right]\right|\right]=0$$

\mathbb ${$ E $} \backslash$ left[Veft(\exp \left{i \theta $\backslash \mathrm{~ e f t}$

## 统计代写|离散时间鞅理论代写martingale代考|The Resolvent Equation

$$\lambda f \lambda-L f_{\lambda}-V$$

$$\lambda\left\langle f_{\lambda}, f_{\lambda}\right\rangle_{\pi}+\left|f_{\lambda}\right|{1}^{2}=\left\langle V, f{\lambda}\right\rangle_{\pi}$$

$$\lambda\left\langle f_{\lambda}, f_{\lambda}\right\rangle_{\pi}+\left|f_{\lambda}\right|{1}^{2} \leq\left|f{\lambda}\right|{1}|V|{-1}$$

$$\lambda\left\langle f_{\lambda}, f_{\lambda}\right\rangle_{\pi}+\left|f_{\lambda}\right|{1}^{2} \leq|V|{-1}^{2} .$$

$$\lim \lambda \rightarrow 0 \lambda|f \lambda|{\pi}^{2}=0 \quad \text { and } \quad \lim \lambda \rightarrow 0|f \lambda-f|{1}=0$$

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