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assignmentutor-lab™ 为您的留学生涯保驾护航 在代写随机过程统计Stochastic process statistics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写随机过程统计Stochastic process statistics代写方面经验极为丰富，各种代写随机过程统计Stochastic process statistics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|随机过程统计代写Stochastic process statistics代考|Proof of the Sufficiency

In this subsection, we prove the “if” part of Theorem $2.55$.
Let $G: \mathcal{M} \rightarrow H^{}$ be a $\mu_{1} \times \mu_{2}$-continuous vector measure of bounded variation. We shall show that there exists a $\tilde{g} \in L_{\mathcal{M}}^{1}\left(X_{1} \times X_{2} ; H^{}\right)$ such that
$$G(E)=\int_{E} \tilde{g} d \mu_{1} d \mu_{2}, \quad \forall E \in \mathcal{M} .$$
By Theorem 2.43, the variation $|G|$ of $G$ is a finite measure on $\left(X_{1} \times\right.$ $\left.X_{2}, \mathcal{M}\right)$. It is easy to see that $|G|$ is a $\mu_{1} \times \mu_{2}$-continuous $\left(\mathbb{R}^{+}\right.$-valued) measure. Applying the Radon-Nikodým Theorem, i.e. Theorem $2.33$ (to $|G|$ and $\mu_{1} \times$ $\left.\mu_{2}\right)$, we can find a nonnegative function $k(\cdot) \in L_{\mathcal{M}}^{1}\left(X_{1} \times X_{2}\right)$ such that
$$|G|(A)=\int_{A} k\left(x_{1}, x_{2}\right) d \mu_{1} d \mu_{2}, \quad \forall A \in \mathcal{M} .$$
Clearly, there exists a sequence $\left{A_{j}\right}_{j=1}^{\infty}$ of mutually disjoint members from $\mathcal{M}$ so that $\bigcup_{j=1}^{\infty} A_{j}=X_{1} \times X_{2}$, and $k(\cdot)$ is bounded by a constant $k_{j}>0$ on each $A_{j}$. It follows from (2.51) that
$$|G|(A) \leq k_{j}\left(\mu_{1} \times \mu_{2}\right)(A), \quad \forall A \in \mathcal{M} \text { with } A \subset A_{j} .$$
For each $j \in \mathbb{N}$, define a linear functional $\ell_{j}$ on the subspace $\mathcal{S}$ of $\mathcal{M}$-simple functions in $L_{\mathcal{M}}^{p}\left(X_{1} ; L^{q}\left(X_{2} ; H\right)\right)$ as follows:
$$\ell_{j}(f)=\sum_{i=1}^{i_{0}} G\left(E_{i} \cap A_{j}\right)\left(x_{i}\right),$$
where $i_{0} \in \mathbb{N}$, and $f=\sum_{i=1}^{i_{0}} x_{i} \chi_{E_{i}}$ for some $x_{i} \in H$ and mutually disjoint sets $E_{1}, \cdots E_{i_{0}}$ from $\mathcal{M}$ so that $\bigcup_{i=1}^{i_{0}} E_{i}=X_{1} \times X_{2}$.

## 数学代写|随机过程统计代写Stochastic process statistics代考|A Sequential Banach-Alaoglu-Type

The classical Banach-Alaoglu Theorem (e.g. [60, p. 130]) states that the closed unit ball of the dual space of a normed vector space is compact in the weak* topology. This theorem has an important special (sequential) version, asserting that the closed unit ball of the dual space of a separable normed vector space (resp., the closed unit ball of a reflexive Banach space) is sequentially compact in the weak* topology (resp., the weak topology). In this section, we shall present a sequential Banach-Alaoglu-type theorem for uniformly bounded linear operators (between suitable Banach spaces). As we shall see later, this result will play crucial roles in the study of the optimal control problems for stochastic evolution equations.

Let $X$ and $Y$ be two Banach spaces. Let $\left{y_{n}\right}_{n=1}^{\infty} \subset Y, y \in Y,\left{z_{n}\right}_{n=1}^{\infty} \subset$ $Y^{}$ and $z \in Y^{}$. In the sequel, we denote by
$$\text { (w)- } \lim {n \rightarrow \infty} y{n}=y \text { in } Y$$
when $\left{y_{n}\right}_{n=1}^{\infty}$ weakly converges to $y$ in $Y$; and by
$$\text { (w)- } \lim {n \rightarrow \infty} z{n}=z \text { in } Y^{}$$
when $\left{z_{n}\right}_{n=1}^{\infty}$ weakly* converges to $z$ in $Y^{*}$. Let us show first the following simple result.

# 随机过程统计代考

## 数学代写|随机过程统计代写Stochastic process statistics代考|Proof of the Sufficiency

$$G(E)=\int_{E} \tilde{g} d \mu_{1} d \mu_{2}, \quad \forall E \in \mathcal{M} .$$

$$|G|(A)=\int_{A} k\left(x_{1}, x_{2}\right) d \mu_{1} d \mu_{2}, \quad \forall A \in \mathcal{M} .$$

$$|G|(A) \leq k_{j}\left(\mu_{1} \times \mu_{2}\right)(A), \quad \forall A \in \mathcal{M} \text { with } A \subset A_{j} .$$

$$\ell_{j}(f)=\sum_{i=1}^{i_{0}} G\left(E_{i} \cap A_{j}\right)\left(x_{i}\right),$$

## 数学代写|随机过程统计代写Stochastic process statistics代考|A Sequential Banach-Alaoglu-Type

(w)- $\lim n \rightarrow \infty y n=y$ in $Y$

$(\mathrm{w})-\lim n \rightarrow \infty z n=z$ in $Y$

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assignmentutor™您的专属作业导师
assignmentutor™您的专属作业导师