定理 3.1.1 (联合分布函数的性质)

定理 3.1.1 (联合分布函数的性质)

设 $\left(\Omega ,\mathcal{F},P\right)$ 为概率空间, 随机向量 $\left(X_1,X_2,\cdots ,X_n\right)$ 的联合分布函数为 $F_{X_1,X_2,\cdots ,X_n}$, 则

1. $0\le F_{X_1,X_2,\cdots ,X_n}\left(x_1,x_2,\cdots ,x_n\right)\le 1,\forall \left(x_1,x_2,\cdots ,x_n\right)\in {\mathbb{R}}^n$.
2. $F_{X_1,X_2,\cdots ,X_n}\left(x_1,x_2,\cdots ,x_n\right)$ 关于每个变元 $x_i$ 单增右连续, $i=1,2,\cdots ,n$.
3. 极限满足
4. $\underset{x_i\to -\infty }{\lim }{F}_{X_1,X_2,\cdots ,X_n}\left(x_1,x_2,\cdots ,x_n\right)=0$, $i=1,2,\cdots ,n$.
5. $\underset{\begin{array}{c}{x}_i\to \infty \\ i=1,2,\cdots ,n\end{array}}{\lim }{F}_{X_1,X_2,\cdots ,X_n}\left(x_1,x_2,\cdots ,x_n\right)=1.$
6. 对任意 $\left(x_1,x_2,\cdots ,x_n\right)\in {\mathbb{R}}^n$ 和 $h_i>0,i=1,2,\cdots ,n$ 有 {\bigtriangleup }_{\left( {x}_{1},{x}_{2},\cdots,{x}_{n}\right) }^{\left( {x}_{1} + {h}_{1},{x}_{2} + {h}_{2},\cdots,{x}_{n} + {h}_{n}\right) }{F}_{{X}_{1},{X}_{2},\cdots,{X}_{n}}\left( {{t}_{1},{t}_{2},\cdots,{t}_{n}}\right) \geq 0. \tag{3.1.2}

证明

定理 3.1.1 的 (i)-(iii) 直观上容易理解, 对于 (iv),我们就 $n=2$ 的情况给出证明. 事实上

$$\begin{array}{rl}0& \le P\left(x_1<X_1\le x_1+h_1,x_2<X_2\le x_2+h_2\right)\\ & =P\left(X_1\le x_1+h_1,X_2\le x_2+h_2\right)-P\left(X_1\le x_1,X_2\le x_2+h_2\right)\\ & -P\left(X_1\le x_1+h_1,X_2\le x_2\right)+P\left(X_1\le x_1,X_2\le x_2\right)\\ & =F_{X_1,X_2}\left(x_1+h_1,x_2+h_2\right)-F_{X_1,X_2}\left(x_1,x_2+h_2\right)\\ & -F_{X_1,X_2}\left(x_1+h_1,x_2\right)+F_{X_1,X_2}\left(x_1,x_2\right)\\ & ={\Delta }_{x_1}^{x_1+h_1}\left(F_{X_1,X_2}\left(t_1,x_2+h_2\right)-F_{X_1,X_2}\left(t_1,x_2\right)\right)\\ & ={\Delta }_{(x_1,x_2)}^{(x_1+h_1,x_2+h_2)}{F}_{X_1,X_2}\left(t_1,t_2\right).\end{array}$$

从以上证明,类似地可以证明 (3.1.2) 左端的 $n$ 阶差分

$$\begin{array}{rl}& {\Delta }_{(x_1,x_2,\cdots ,x_n)}^{(x_1+h_1,x_2+h_2,\cdots ,x_n+h_n)}{F}_{X_1,X_2,\cdots ,X_n}(t_1,t_2,\cdots ,t_n)\\ =& P\left(\omega \in \Omega :\bigcap _{i=1}^n\left\{x_i<X_i(\omega )\le x_i+h_i\right\}\right),\end{array}\text{\hspace{1em}(3.1.3)}$$