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# 经济代写|计量经济学代写Introduction to Econometrics代考|BEA242 Properties of Random Sets Related to Their Capacity Functionals

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## 经济代写|计量经济学代写Introduction to Econometrics代考|Properties of Random Sets Related to Their Capacity Functionals

As we have seen, $\mathrm{T}$ determines uniquely the distribution of $\boldsymbol{X}$, and therefore in principle properties of $\boldsymbol{X}$ can be expressed in terms of $\mathrm{T}$. Here we give such examples (but we also caution the reader that, in other situations, deriving properties of $\boldsymbol{X}$ in terms of $\mathrm{T}$ may be extremely difficult). Specifically, we consider random sets of points, i.e., simple point processes (these are point processes such that, with probability one, no two points of the process are coincident).

Example 1.35 (Binomial process) A random closed set $\boldsymbol{X}$ is a sample of i.i.d. points $\left{\boldsymbol{x}_1, \ldots, \boldsymbol{x}_n\right}$ with the common nonatomic distribution $\mu$, and is called a binomial process, if and only if its capacity functional is

$$\mathrm{T}(K)=1-(1-\mu(K))^n$$
for all compact $K \subset \mathbb{R}^d$.
Example 1.36 (Poisson point process) Define
$$\mathrm{T}(K)=1-e^{-\Lambda(K)}, \quad K \in \mathcal{K},$$
with $\Lambda$ being a locally finite measure on $\mathbb{R}^d$ (that is, each point admits a neighborhood of finite measure) and such that $\Lambda$ attaches zero mass to any single point. The corresponding random closed set $\boldsymbol{X}$ is the Poisson process in $\mathbb{R}^d$ with intensity measure $\Lambda$. Indeed, if $\boldsymbol{X}$ is a point set such that the number of points in any set $K$ is Poisson distributed with mean $\Lambda(K)$, then $X$ hits $K$ if and only if the Poisson random variable with mean $\Lambda(K)$ does not vanish. The probability of this latter event is exactly the right-hand side of (1.9). The random set $\boldsymbol{X}$ is stationary (see Definition 1.38) and then called a homogeneous Poisson process if and only if $\Lambda$ is proportional to the Lebesgue measure on $\mathbb{R}^d$; the coefficient of proportionality is called the intensity of the process.

## 经济代写|计量经济学代写Introduction to Econometrics代考|Weak Convergence

The weak convergence of random closed sets is defined by specializing the general weak convergence definition for random elements in $\mathcal{F}$ or, equivalently, the weak convergence of probability measures on the space $\mathcal{F}$ of closed sets. For a set $K$, denote Int $K$ its interior, that is, the largest open subset of $K$.
Theorem 1.40 A sequence $\left{\boldsymbol{X}n, n \geq 1\right}$ of random closed sets in $\mathbb{R}^d$ converges weakly to $\boldsymbol{X}$ if and only if $\mathrm{T}{X_n}(K) \rightarrow \mathrm{T}_X(K)$ as $n \rightarrow \infty$ for all compact sets $K$ such that $\mathrm{T}_X(K)=\mathrm{T}_X(\operatorname{Int} K)$.

Therefore, the weak convergence of random closed sets is equivalent to the pointwise convergence of their capacity functionals on continuity sets of the limiting capacity functional $\mathrm{T}X$. The condition on $K$ in Theorem $1.40$ is akin to the conventional requirement for the weak convergence of random variables, which asks for convergence of their cumulative distribution functions at all points of continuity of the limit. The condition $\mathrm{T}_X(K)=\mathrm{T}_X($ Int $K)$ means that $X$ “touches” $K$ with probability zero, i.e., $$\mathbf{P}{\boldsymbol{X} \cap K \neq \emptyset, \boldsymbol{X} \cap \operatorname{Int} K=\emptyset}=0 .$$ Recall that $\mathrm{T}_X$ (Int $\left.K\right)$ is defined using the extension of the capacity functional to open sets. If $\boldsymbol{X}_n=\left{\boldsymbol{x}_n\right}$ are random singletons, then $\mathrm{T}{\boldsymbol{X}_n}(K)$ is the probability distribution of $\boldsymbol{x}_n$ and $\mathrm{T}_X(K)=\mathrm{T}_X(\operatorname{Int} K)$ for the weak limit $\boldsymbol{X}={\boldsymbol{x}}$ means that $\boldsymbol{x}$ belongs to the boundary of the set $K$ with probability zero. Therefore, the weak convergence of random singletons corresponds to the classical definition of weak convergence for random elements. The weak convergence is denoted by $\Rightarrow$.

As usual, the weak convergence of $\boldsymbol{X}_n$ to $\boldsymbol{X}$ implies the weak convergence of $f\left(\boldsymbol{X}_n\right)$ to $f(\boldsymbol{X})$ for any continuous map applied to sets. However, many important maps are not continuous, e.g., the Lebesgue measure is not continuous. This can be seen by noticing that a finite set of points dense in the ball converges to the ball, while the Lebesgue measure of any finite set vanishes and the ball has a positive measure. However, the Lebesgue measure becomes continuous if restricted to the family of convex sets.

Since the space of closed sets $\mathcal{F}$ in $\mathbb{R}^d$ is compact, a variant of Helly’s theorem for random closed sets establishes that each family of random closed sets has a subsequence that converges in distribution. Therefore, quite differently to the studies of random functions and stochastic processes, there is no need to check the tightness conditions when proving the weak convergence of random closed sets.

## 经济代写|计量经济学代写Introduction to Econometrics代考|Properties of Random Sets Related to Their Capacity Functionals

$$\mathrm{T}(K)=1-(1-\mu(K))^n$$

$$\mathrm{T}(K)=1-e^{-\Lambda(K)}, \quad K \in \mathcal{K},$$

## 经济代写|计量经济学代写Introduction to Econometrics代考|Weak Convergence

$$\mathbf{P} \boldsymbol{X} \cap K \neq \emptyset, \boldsymbol{X} \cap \operatorname{Int} K=\emptyset=0 .$$

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