实际上联合平稳随机过程和单一的随机过程是十分相似的，联合平稳随机过程用来表征两个随机过程之间的关系。
文章目录
联合平稳随机过程1.概率分布与矩函数2.矩函数3.联合平稳的矩函数联合平稳随机过程
1.概率分布与矩函数
假定有两个随机过程 X(t)X(t)X(t) 和 Y(t)Y(t)Y(t) 概率密度分别为 pn(x1,x2,⋯,xn;t1,t2,⋯,tn)p_n(x_1,x_2,\dotsb,x_n;t_1,t_2,\dotsb,t_n)pn(x1,x2,⋯,xn;t1,t2,⋯,tn)pm(y1,y2,⋯,ym;t1′,t2′,⋯,tm′)p_m(y_1,y_2,\dotsb,y_m;t'_1,t'_2,\dotsb,t'_m)pm(y1,y2,⋯,ym;t1′,t2′,⋯,tm′)联合概率分布函数Fm+n(x1,x2,⋯,xn;y1,y2,⋯,ym;t1,t2,⋯,tn;t1′,t2′,⋯,tm′)=P{X(t1)≤x1,X(t2)≤x2,⋯X(tn)≤xn;Y(t1′)≤y1,Y(t2′)≤y2,⋯Y(tn′)≤yn}\begin{matrix} F_{m+n}(x_1,x_2,\dotsb ,x_n;y_1,y_2,\dotsb,y_m;t_1,t_2,\dotsb,t_n;t'_1,t'_2,\dotsb,t'_m)\\ ~~\\ = P\{ X(t_1) \le x_1,X(t_2) \le x_2, \dotsb X(t_n) \le x_n ;Y(t'_1) \le y_1,Y(t'_2) \le y_2, \dotsb Y(t'_n) \le y_n \} \end{matrix}Fm+n(x1,x2,⋯,xn;y1,y2,⋯,ym;t1,t2,⋯,tn;t1′,t2′,⋯,tm′)=P{X(t1)≤x1,X(t2)≤x2,⋯X(tn)≤xn;Y(t1′)≤y1,Y(t2′)≤y2,⋯Y(tn′)≤yn}联合概率密度函数
pm+n(x1,x2,⋯,xn;y1,y2,⋯,ym;t1,t2,⋯,tn;t1′,t2′,⋯,tm′)=∂nFn(x1,x2,⋯,xn;y1,y2,⋯,ym;t1,t2,⋯,tn;t1′,t2′,⋯,tm′)∂x1∂x2⋯∂xn∂y1∂y2⋯∂ym\begin{matrix} p_{m+n}(x_1,x_2,\dotsb ,x_n;y_1,y_2,\dotsb,y_m;t_1,t_2,\dotsb,t_n;t'_1,t'_2,\dotsb,t'_m)\\ ~~\\ = \dfrac{\partial^n F_n(x_1,x_2,\dotsb ,x_n;y_1,y_2,\dotsb,y_m;t_1,t_2,\dotsb,t_n;t'_1,t'_2,\dotsb,t'_m)}{\partial x_1\partial x_2 \dotsb \partial x_n\partial y_1\partial y_2 \dotsb \partial y_m} \end{matrix}pm+n(x1,x2,⋯,xn;y1,y2,⋯,ym;t1,t2,⋯,tn;t1′,t2′,⋯,tm′)=∂x1∂x2⋯∂xn∂y1∂y2⋯∂ym∂nFn(x1,x2,⋯,xn;y1,y2,⋯,ym;t1,t2,⋯,tn;t1′,t2′,⋯,tm′)过程的统计独立：
pm+n(x1,x2,⋯,xn;y1,y2,⋯,ym;t1,t2,⋯,tn;t1′,t2′,⋯,tm′)=pn(x1,x2,⋯,xn,t1,t2,⋯,tn)pm(y1,y2,⋯,ym,t1′,t2′,⋯,tn′)\begin{matrix} p_{m+n}(x_1,x_2,\dotsb ,x_n;y_1,y_2,\dotsb,y_m;t_1,t_2,\dotsb,t_n;t'_1,t'_2,\dotsb,t'_m) \\ ~~\\ = p_n(x_1,x_2,\dotsb ,x_n,t_1,t_2,\dotsb,t_n) p_m(y_1,y_2,\dotsb ,y_m,t'_1,t'_2,\dotsb,t'_n) \end{matrix}pm+n(x1,x2,⋯,xn;y1,y2,⋯,ym;t1,t2,⋯,tn;t1′,t2′,⋯,tm′)=pn(x1,x2,⋯,xn,t1,t2,⋯,tn)pm(y1,y2,⋯,ym,t1′,t2′,⋯,tn′)
2.矩函数
互相关函数：RX(t1,t2)=E[X(t1)Y(t2)]=∫−∞∞∫−∞∞xyp2(x,y;t1,t2)dxdyR_X(t_1,t_2) = E[X(t_1)Y(t_2)]=\int _{-\infin}^{\infin}\int _{-\infin}^{\infin}xy~p_2(x,y;t_1,t_2)dxdyRX(t1,t2)=E[X(t1)Y(t2)]=∫−∞∞∫−∞∞xyp2(x,y;t1,t2)dxdy互协方差函数：
CXY(t1,t2)=E[X(t1)−mX(t1)][Y(t2)−mY(t2)]C_{XY}(t_1,t_2) = E[X(t_1)-m_X(t_1)][Y(t_2)-m_Y(t_2)]CXY(t1,t2)=E[X(t1)−mX(t1)][Y(t2)−mY(t2)]
3.联合平稳的矩函数
定义：pm+np_{m+n}pm+n 不随时间的改变而变化，称过程 X(t)X(t)X(t) 和 Y(t)Y(t)Y(t) 联合平稳。互相关函数：RXY(t1,t2)=RXY(τ)R_{XY}(t_1,t_2) = R_{XY}(\tau)RXY(t1,t2)=RXY(τ)互相关系数：rXY(τ)=CXY(τ)σXσYr_{XY}(\tau) = \frac{C_{XY}(\tau)}{\sigma_X\sigma_Y}rXY(τ)=σXσYCXY(τ)互协方差函数：CXY(t1,t2)=CXY(τ)C_{XY}(t_1,t_2) = C_{XY}(\tau)CXY(t1,t2)=CXY(τ)正交：对任意 τ\tauτ，rXY(τ)r_{XY}(\tau)rXY(τ) 均为 000，就是说过程 X(t)X(t)X(t) 和过程 Y(t)Y(t)Y(t)完全不相关（线性）。可以用 E[X(t)Y(t+τ)]=0E[X(t)Y(t+\tau)] = 0E[X(t)Y(t+τ)]=0 来表示两过程正交。⭐️ 注意区分独立和不相关，独立的定义要比不相关更加严格。互相关函数性质： RXY(τ)=RYX(−τ)R_{XY}(\tau) = R_{YX}(-\tau)RXY(τ)=RYX(−τ)∣Rxy(τ)∣2≤RX(0)RY(0)|R_{xy}(\tau)|^2\le R_X(0)R_Y(0)∣Rxy(τ)∣2≤RX(0)RY(0)
