### 统计代写|随机过程代写stochastic process代考|STAT7004

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• Foundations of Data Science 数据科学基础

## 统计代写|随机过程代写stochastic process代考|Signed Measures

Let us begin with the following notion.
Definition 2.28. A function $\mu: \mathcal{F} \rightarrow[-\infty,+\infty]$ is called a signed measure on $(\Omega, \mathcal{F})$ if
1) $\mu(\emptyset)=0$;
2) $\mu\left(\bigcup_{j=1}^{\infty} A_j\right)=\sum_{j=1}^{\infty} \mu\left(A_j\right)$ for any sequence $\left{A_j\right}$ of mutually disjoint sets from $\mathcal{F}$; and
3) $\mu$ assumes at most one of the values $+\infty$ and $-\infty$.
Example 2.29. Let $\nu$ be a measure on $(\Omega, \mathcal{F})$ and $f$ be a real valued integrable function defined on $(\Omega, \mathcal{F}, \nu)$. Then
$$\mu(A)=\int_A f d \nu, \quad \forall A \in \mathcal{F},$$
defines a signed measure in $(\Omega, \mathcal{F})$. More generally, the above $\mu$ is still a signed measure if $f$ is a measurable function on $(\Omega, \mathcal{F})$, and one of $f^{+}$and $f^{-}$, the positive and negative parts of $f$, is integrable on $(\Omega, \mathcal{F}, \nu)$.

If $\mu$ is a signed measure on $(\Omega, \mathcal{F})$, we call a set $E \subset \Omega$ positive (resp. negative) (w.r.t. $\mu$ ) if for every $F \in \mathcal{F}, E \cap F$ is measurable, and $\mu(E \cap F) \geq 0$ (resp. $\mu(E \cap F) \leq 0)$.

Theorem 2.30. If $\mu$ is a signed measure on $(\Omega, \mathcal{F})$, then there exist two disjoint sets $A$ and $B$, whose union is $\Omega$, such that $A$ is positive and $B$ is negative w.r.t. $\mu$.

The sets $A$ and $B$ in Theorem $2.30$ are said a Hahn decomposition of $\Omega$ w.r.t. $\mu$. It is not difficult to construct examples to show that Hahn decomposition is not unique. However, if
$$\Omega=A_1 \cup B_1 \quad \text { and } \quad \Omega=A_2 \cup B_2$$
are two Hahn decomposition of $\Omega$, then it is easy to show that, for every measurable set $E$, it holds
$$\mu\left(E \cap A_1\right)=\mu\left(E \cap A_2\right) \quad \text { and } \quad \mu\left(E \cap B_1\right)=\mu\left(E \cap B_2\right) .$$

## 统计代写|随机过程代写stochastic process代考|Distribution, Density and Characteristic Functions

Let $\mathbb{P}$ be a probability measure on $(\Omega, \mathcal{F})$, and $X: \Omega \rightarrow H$ be a strongly measurable random variable. Then, as a special case of (2.11), $X$ induces a probability measure $\mathbb{P}_X$ on $(H, \mathcal{B}(H))$ via
$$\mathbb{P}_X(A) \triangleq \mathbb{P}\left(X^{-1}(A)\right), \quad \forall A \in \mathcal{B}(H) .$$
We call $\mathbb{P}_X$ the distribution of $X$. If $X$ is Bochner integrable w.r.t. $\mathbb{P}$, then by (2.5) and using (2.12) in Theorem $2.27$, we see that
$$\mathbb{E} X=\int_H x d \mathbb{P}_X(x) .$$
In the case of $H=\mathbb{R}^m$ (for some $m \in \mathbb{N}$ ), $\mathbb{P}_X$ can be uniquely determined by the following function:
$$F(x) \triangleq F\left(x_1, \cdots, x_m\right) \triangleq \mathbb{P}\left{X_i \leq x_i, 1 \leq i \leq m\right},$$

where $x=\left(x_1, \cdots, x_m\right)$ and $\left(X_1, \cdots, X_m\right)=X$. We call $F(x)$ the distribution function of $X$. If $\mathbb{P}X$ is absolutely continuous w.r.t. the Lebesgue measure in $\mathbb{R}^m$, then by the Radon-Nikodým theorem (i.e., Theorem 2.33), there exists a (nonnegative) function $f \in L^1\left(\mathbb{R}^m\right)$ such that $$\mathbb{P}_X(A)=\int_A f(x) d x, \quad \forall A \in \mathcal{B}\left(\mathbb{R}^m\right) .$$ Particularly, $$F(x)=\int{-\infty}^{x_1} \cdots \int_{-\infty}^{x_m} f\left(\xi_1, \cdots, \xi_m\right) d \xi_1 \cdots d \xi_m .$$
The function $f(x)$ is called the density of $X$. As a special case, if $f(x)$ is of the following form:
$$f(x)=\left[(2 \pi)^m \operatorname{det} Q\right]^{-1 / 2} \exp \left{-\frac{1}{2}(x-\lambda) Q^{-1}(x-\lambda)^{\top}\right}, \quad x \in \mathbb{R}^m,$$
for some $\lambda \in \mathbb{R}^m, Q \in \mathbb{R}^{m \times m}$ with $Q^{\top}=Q>0$, then we say that $X$ has a normal distribution with parameter $(\lambda, Q)$, denoted by $X \sim \mathcal{N}(\lambda, Q)$. We call $X$ a Gaussian random variable (valued in $\mathbb{R}^m$ ) if $X$ has a normal distribution or $X$ is constant.

# 随机过程代考

## 统计代写|随机过程代写stochastic process代考|Signed Measures

1) $\mu(\emptyset)=0$
2) $\mu\left(\bigcup_{j=1}^{\infty} A_j\right)=\sum_{j=1}^{\infty} \mu\left(A_j\right)$ 对于任何序列 $\left(L^{\prime}\left{A_{-} j\right.\right.$ 右 $}$ 互不相交的集合来自 $\mathcal{F}$; 和
3) $\mu$ 最多假定一个值 $+\infty$ 和 $-\infty$.

$$\mu(A)=\int_A f d \nu, \quad \forall A \in \mathcal{F},$$

$$\Omega=A_1 \cup B_1 \quad \text { and } \quad \Omega=A_2 \cup B_2$$

$$\mu\left(E \cap A_1\right)=\mu\left(E \cap A_2\right) \quad \text { and } \quad \mu\left(E \cap B_1\right)=\mu\left(E \cap B_2\right) .$$

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