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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (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 Necessity for a Special Case

In this subsection, we prove the “only if” part of Theorem $2.55$ for the case that $H=\mathbb{R}$.
For any $g \in L_{\mathcal{M}}^{p^{\prime}}\left(X_{1} ; L^{q^{\prime}}\left(X_{2}\right)\right)$, define $F_{g}: L_{\mathcal{M}}^{p}\left(X_{1} ; L^{q}\left(X_{2}\right)\right) \rightarrow \mathbb{R}$ by $F_{g}(f)=\int_{X_{1} \times X_{2}} f\left(x_{1}, x_{2}\right) g\left(x_{1}, x_{2}\right) d \mu_{1} d \mu_{2}, \quad \forall f \in L_{\mathcal{M}}^{p}\left(X_{1} ; L^{q}\left(X_{2}\right)\right) .$
Clearly, $g \mapsto F_{g}$ is a linear map. It follows from Hölder’s inequality that
$$\left|F_{g}(f)\right| \leq|f|{L{\mathcal{M}}^{p}\left(X_{1} ; L^{q}\left(X_{2}\right)\right)}|g|{L{\mathcal{M}}^{p^{\prime}}\left(X_{1} ; L^{q^{\prime}}\left(X_{2}\right)\right)}, \forall f \in L_{\mathcal{M}}^{p}\left(X_{1} ; L^{q}\left(X_{2}\right)\right) .$$
Hence $F_{g} \in L_{\mathcal{M}}^{p}\left(X_{1} ; L^{q}\left(X_{2}\right)\right)^{}$ and $$\left|F_{g}\right|{L{\mathcal{M}}^{p}\left(X_{1} ; L^{q}\left(X_{2}\right)\right)^{}} \leq|g|{p^{\prime}, q^{\prime}} .$$ Therefore, $g \mapsto F{g}$ is a linear non-expanding map. Now, we show that this map is surjective and is an isometry.

To show the surjectivity of $g \mapsto F_{g}$, take any $F \in L_{\mathcal{M}}^{p}\left(X_{1} ; L^{q}\left(X_{2}\right)\right)^{*}$. Since $\chi_{A} \in L_{\mathcal{M}}^{p}\left(X_{1} ; L^{4}\left(X_{2}\right)\right)$ for any $A \in \mathcal{M}$, we may define
$$\nu(A)=F\left(\chi_{A}\right), \quad \forall A \in \mathcal{M} .$$
Then $\nu$ is a finite signed measure on $\left(X_{1} \times X_{2}, \mathcal{M}\right)$, and $\nu \ll \mu_{1} \times \mu_{2}$. By Theorem 2.33, there is a $g \in L_{\mathcal{M}}^{1}\left(X_{1} \times X_{2}\right)$ such that
$$\nu(A)=\int_{A} g d \mu_{1} d \mu_{2}, \quad \forall A \in \mathcal{M},$$

## 数学代写|随机过程统计代写Stochastic process statistics代考|Proof of the Necessity for the General Case

In this subsection, we prove the “only if” part of Theorem $2.55$ for the general case. The proof is divided into two steps.

Step 1. We show that $L_{\mathcal{M}}^{p^{\prime}}\left(X_{1} ; L^{q^{\prime}}\left(X_{2} ; H^{}\right)\right)$ is isometrically isomorphic to a subspace $\mathcal{H}$ of $L_{\mathcal{M}}^{p}\left(X_{1} ; L^{q}\left(X_{2} ; H\right)\right)^{}$.

For any given $g \in L_{\mathcal{M}}^{p^{\prime}}\left(X_{1} ; L^{q^{\prime}}\left(X_{2} ; H^{}\right)\right)$, define a linear functional $F_{g}$ on $L_{\mathcal{M}}^{p}\left(X_{1} ; L^{q}\left(X_{2} ; H\right)\right)$ as follows: $$\begin{gathered} F_{g}(f)=\int_{X_{1} \times X_{2}}\left\langle f\left(x_{1}, x_{2}\right), g\left(x_{1}, x_{2}\right)\right\rangle_{H, H^{}} d \mu_{1} d \mu_{2}, \ \forall f \in L_{\mathcal{M}}^{p}\left(X_{1} ; L^{q}\left(X_{2} ; H\right)\right) . \end{gathered}$$
Then, by means of the Hölder inequality and similarly to (2.34), we conclude that $F_{g}$ belongs to $L_{\mathcal{M}}^{p}\left(X_{1} ; L^{q}\left(X_{2} ; H\right)\right)^{}$, and $$\left|F_{g}\right|{L{\mathcal{M}}^{p}\left(X_{1} ; L^{q}\left(X_{2} ; H\right)\right)^{}} \leq|g|{p^{\prime}, q^{\prime} ; H^{}} .$$ Therefore the norm of $F{g}$ is not greater than $|g|{p^{\prime}, q^{\prime} ; H^{}}$. Define
$$\mathcal{H} \equiv\left{F{g} \mid g \in L_{\mathcal{M}}^{p^{\prime}}\left(X_{1} ; L^{q^{\prime}}\left(X_{2} ; H^{*}\right)\right)\right} .$$

# 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 Necessity for a Special Case

$$\left|F_{g}(f)\right| \leq|f| L \mathcal{M}^{p}\left(X_{1} ; L^{q}\left(X_{2}\right)\right)|g| L \mathcal{M}^{p^{\prime}}\left(X_{1} ; L^{q^{\prime}}\left(X_{2}\right)\right), \forall f \in L_{\mathcal{M}}^{p}\left(X_{1} ; L^{q}\left(X_{2}\right)\right) .$$

$$\left|F_{g}\right| L \mathcal{M}^{p}\left(X_{1} ; L^{q}\left(X_{2}\right)\right) \leq|g| p^{\prime}, q^{\prime} .$$

$$\nu(A)=F\left(\chi_{A}\right), \quad \forall A \in \mathcal{M} .$$

$$\nu(A)=\int_{A} g d \mu_{1} d \mu_{2}, \quad \forall A \in \mathcal{M},$$

## 数学代写|随机过程统计代写Stochastic process statistics代考|Proof of the Necessity for the General Case

$$F_{g}(f)=\int_{X_{1 \times X_{2}}}\left\langle f\left(x_{1}, x_{2}\right), g\left(x_{1}, x_{2}\right)\right\rangle_{H, H} d \mu_{1} d \mu_{2}, \forall f \in L_{\mathcal{M}}^{p}\left(X_{1} ; L^{q}\left(X_{2} ; H\right)\right) .$$

$$\left|F_{g}\right| L \mathcal{M}^{p}\left(X_{1} ; L^{q}\left(X_{2} ; H\right)\right) \leq|g| p^{\prime}, q^{\prime} ; H .$$

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