# 数学代写|概率论代写Probability theory代考|STAT4528

## 数学代写|概率论代写Probability theory代考|Probability Density Function and Distribution Function

In this section, we discuss two simple and useful methods to construct distributions on a locally compact metric space $(S, d)$. The first starts with one integration $I$ on $(S, d)$ in the sense of Definition 4.2.1, where the full set $S$ need not be integrable. Then, for each nonnegative integrable function $g$ with integral 1, we can construct a distribution on $(S, d)$ using $g$ as a density function. A second method is for the special case where $(S, d)=(R, d)$ is the real line, equipped with the Euclidean metric. Let $F$ be an arbitrary distribution function on $R$, in the sense of Definition 4.1.1, such that $F(t) \rightarrow 0$ as $t \rightarrow-\infty$, and $F(t) \rightarrow 1$ as $t \rightarrow \infty$. Then the Riemann-Stieljes integral corresponding to $F$ constitutes a distribution on $(R, d)$.

Definition 5.4.1. Probability density function. Let $I$ be an integration on a locally compact metric space $(S, d)$ in the sense of Definition 4.2.1. Let $(S, \Lambda, I)$ denote the completion of the integration space $(S, C(S), I)$. Let $g \in \Lambda$ be an arbitrary nonnegative integrable function with $I g=1$. Define
$$I_g f \equiv I g f$$
for each $f \in C(S, d)$. According to the following lemmas, the function $I_g$ is a probability distribution on $(S, d)$, in the sense of Definition 5.2.1.

In such a case, $g$ will be called a probability density function, or $p$.d. f. for short, relative to the integration $I$, and the completion $\left(S, \Lambda_g, I_g\right)$ of $\left(S, C(S, d), I_g\right)$ will be called the probability space generated by the p.d.f. $g$.

Suppose, in addition, that $X$ is an arbitrary r.v. on some probability space $(\Omega, L, E)$ with values in $S$ such that $E_X=I_g$, where $E_X$ is the distribution induced on the metric space $(S, d)$ by the r.v. $X$, in the sense of Definition 5.2.3. Then the r.v. $X$ is said to have the p.d.f. $g$ relative to $I$.

## 数学代写|概率论代写Probability theory代考|Skorokhod Representation

In this section, let $(S, d)$ be a locally compact metric space with an arbitrary but fixed reference point $x_0 \in S$. Let
$$\left(\Theta_0, L_0, I\right) \equiv\left([0,1], L_0, \int \cdot d x\right)$$
denote the Lebesgue integration space based on the unit interval $[0,1]$, and let $\mu$ be the corresponding Lebesgue measure. Then we will call $I$ the uniform distribution on $[0,1]$.

Given two distributions $E$ and $E^{\prime}$ on the locally compact metric space $(S, d)$, we saw in Proposition $5.2 .5$ that they are equal to the distributions induced by some $S$-valued r.v.’s $X$ and $X^{\prime}$, respectively. The underlying probability spaces on which $X$ and $X^{\prime}$ are defined can, in general, be different. Therefore functions of both $X$ and $X^{\prime}$, such as $d\left(X, X^{\prime}\right)$, and their associated probabilities are, up to this point, undefined. Additional conditions on joint probabilitiess are needéd to construct one common probability space on which both $X$ and $X^{\prime}$ are defined.

One such condition is independence, to be made precise in a later section, where the observed value of $X$ has no effect whatsoever on the probabilities related to $X^{\prime}$.
In some other situations, it is desirable, instead, to have models where $X=X^{\prime}$ if $E=E^{\prime}$, and more generally where $d\left(X, X^{\prime}\right)$ is small when $E$ is close to $E^{\prime}$. In this section, we construct the Skorokhod representation, which, to each distribution $E$ on $S$, assigns a unique r.v. $X: \Theta_0 \rightarrow S$ that induces $E$. In the context of applications to random fields, theorem 3.1.1 of [Skorokhod 1956] introduced this representation and proves that it is continuous relative to weak convergence of $E$ and a.u. convergence of $X$. We will prove this result for applications in Chapter 6 .
In addition, we will prove that, when restricted to a tight subset of distributions, the Skorokhod representation is uniformly continuous relative to the distribution metric $\rho_{D i s t, \xi}$ on the space of distributions $E$, and the metric $\rho_{\text {Prob }}$ on the space of r.v.’s $X: \Theta_0 \rightarrow S$. The metrics $\rho_{D i s t, \xi}$ and $\rho_{\text {Prob }}$ were introduced in Definition 5.3.4 and in Proposition 5.1.11, respectively.

The Skorokhod representation is a generalization of the quantile mapping, which to each P.D.F. $F$ assigns the r.r.v. $Y \equiv F^{-1}: \Theta_0 \rightarrow R$ on the probability space $\Theta_0$, where $Y$ can be shown to induce the P.D.F. $F$.

# 概率论代考

## 数学代写|概率论代写Probability theory代考|Probability Density Function and Distribution Function

$$I_g f \equiv I g f$$

## 数学代写|概率论代写Probability theory代考|Skorokhod Representation

$$\left(\Theta_0, L_0, I\right) \equiv\left([0,1], L_0, \int \cdot d x\right)$$

Skorokhod 表示是分位数映射的推广，它对每个 PDFF分配 rv $Y \equiv F^{-1}: \Theta_0 \rightarrow R$ 在概率空间 $\Theta_0$ ， 在哪里 $Y$ 可以显示诱导 PDF $F$.

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