例题：随机游走

 
 设 
       
            X 
           
            t 
           
          X_t 
         
      Xt​为一随机游走序列： 
       
            X 
           
            t 
           
           = 
          
            X 
           
             t 
            
             − 
            
             1 
            
           + 
          
            ε 
           
            t 
           
          X_t=X_{t-1}+\varepsilon_t 
         
      Xt​=Xt−1​+εt​, 
       
            X 
           
            0 
           
           = 
          
           0 
          
          X_0=0 
         
      X0​=0。求 
       
           C 
          
           o 
          
           r 
          
           r 
          
           ( 
          
            X 
           
            t 
           
           , 
          
            X 
           
             t 
            
             + 
            
             k 
            
           ) 
          
          Corr(X_t,X_{t+k}) 
         
      Corr(Xt​,Xt+k​)。
  
             C 
            
             o 
            
             r 
            
             r 
            
            ( 
           
             X 
            
             t 
            
            , 
           
             X 
            
              t 
             
              + 
             
              k 
             
            ) 
           
            = 
           
               C 
              
               o 
              
               v 
              
              ( 
             
               X 
              
               t 
              
              , 
             
               X 
              
                t 
               
                + 
               
                k 
               
              ) 
             
                V 
               
                a 
               
                r 
               
               ( 
              
                X 
               
                t 
               
               ) 
              
                V 
               
                a 
               
                r 
               
               ( 
              
                X 
               
                 t 
                
                 + 
                
                 k 
                
               ) 
              
           {\rm Corr}(X_t,X_{t+k})=\frac{{\rm Cov}(X_t,X_{t+k})}{\sqrt{{\rm Var}(X_t){\rm Var}(X_{t+k})}} 
          
       Corr(Xt​,Xt+k​)=Var(Xt​)Var(Xt+k​) 
                      
                     ​Cov(Xt​,Xt+k​)​
 其中 
       
            C 
           
            o 
           
            v 
           
           ( 
          
            X 
           
            t 
           
           , 
          
            X 
           
             t 
            
             + 
            
             k 
            
           ) 
          
           = 
          
           E 
          
           ( 
          
            X 
           
            t 
           
            X 
           
             t 
            
             + 
            
             k 
            
           ) 
          
           − 
          
           E 
          
           ( 
          
            X 
           
            t 
           
           ) 
          
           E 
          
           ( 
          
            X 
           
             t 
            
             + 
            
             k 
            
           ) 
          
          {\rm Cov}(X_t,X_{t+k})=E(X_tX_{t+k})-E(X_t)E(X_{t+k}) 
         
      Cov(Xt​,Xt+k​)=E(Xt​Xt+k​)−E(Xt​)E(Xt+k​)
 易知
  
             X 
            
             t 
            
            = 
           
             ε 
            
             1 
            
            + 
           
            . 
           
            . 
           
            . 
           
            + 
           
             ε 
            
             t 
            
             X 
            
              t 
             
              + 
             
              k 
             
            = 
           
             ε 
            
             1 
            
            + 
           
            . 
           
            . 
           
            . 
           
            + 
           
             ε 
            
              t 
             
              + 
             
              k 
             
           X_t=\varepsilon_1+...+\varepsilon_t \\ X_{t+k}=\varepsilon_1+...+\varepsilon_{t+k} 
          
       Xt​=ε1​+...+εt​Xt+k​=ε1​+...+εt+k​
 那么 
       
           E 
          
           ( 
          
            X 
           
            t 
           
           ) 
          
           = 
          
           E 
          
           ( 
          
            X 
           
             t 
            
             + 
            
             k 
            
           ) 
          
           = 
          
           0 
          
          E(X_t)=E(X_{t+k})=0 
         
      E(Xt​)=E(Xt+k​)=0。而 
       
            X 
           
            t 
           
            X 
           
             t 
            
             + 
            
             k 
            
          X_tX_{t+k} 
         
      Xt​Xt+k​的平方项有 
       
            ε 
           
            1 
           
            2 
           
           + 
          
           . 
          
           . 
          
           . 
          
           + 
          
            ε 
           
            t 
           
            2 
           
          \varepsilon_1^2+...+\varepsilon_t^2 
         
      ε12​+...+εt2​,其余都是 
       
            ε 
           
            t 
           
          \varepsilon_t 
         
      εt​的交叉项（期望为0）。
 那么 
       
            C 
           
            o 
           
            v 
           
           ( 
          
            X 
           
            t 
           
           , 
          
            X 
           
             t 
            
             + 
            
             k 
            
           ) 
          
           = 
          
           t 
          
            σ 
           
            ε 
           
            2 
           
          {\rm Cov}(X_t,X_{t+k})=t{\sigma_{\varepsilon}^2} 
         
      Cov(Xt​,Xt+k​)=tσε2​
 而 
       
            V 
           
            a 
           
            r 
           
           ( 
          
            X 
           
            t 
           
           ) 
          
           = 
          
           t 
          
            σ 
           
            ε 
           
            2 
           
           , 
          
            V 
           
            a 
           
            r 
           
           ( 
          
            X 
           
             t 
            
             + 
            
             k 
            
           ) 
          
           = 
          
           ( 
          
           t 
          
           + 
          
           k 
          
           ) 
          
            σ 
           
            ε 
           
            2 
           
          {\rm Var}(X_t)=t{\sigma_{\varepsilon}^2},{\rm Var}(X_{t+k})=(t+k){\sigma_{\varepsilon}^2} 
         
      Var(Xt​)=tσε2​,Var(Xt+k​)=(t+k)σε2​
 所以
  
             C 
            
             o 
            
             r 
            
             r 
            
            ( 
           
             X 
            
             t 
            
            , 
           
             X 
            
              t 
             
              + 
             
              k 
             
            ) 
           
            = 
           
               C 
              
               o 
              
               v 
              
              ( 
             
               X 
              
               t 
              
              , 
             
               X 
              
                t 
               
                + 
               
                k 
               
              ) 
             
                V 
               
                a 
               
                r 
               
               ( 
              
                X 
               
                t 
               
               ) 
              
                V 
               
                a 
               
                r 
               
               ( 
              
                X 
               
                 t 
                
                 + 
                
                 k 
                
               ) 
              
            = 
           
              t 
             
               σ 
              
               ε 
              
               2 
              
               t 
              
                σ 
               
                ε 
               
                2 
               
               ( 
              
               t 
              
               + 
              
               k 
              
               ) 
              
                σ 
               
                ε 
               
                2 
               
            = 
           
              t 
             
               t 
              
               + 
              
               k 
              
           {\rm Corr}(X_t,X_{t+k}) =\frac{{\rm Cov}(X_t,X_{t+k})}{\sqrt{{\rm Var}(X_t){\rm Var}(X_{t+k})}} \\=\frac{t{\sigma_{\varepsilon}^2}}{\sqrt{t{\sigma_{\varepsilon}^2}(t+k){\sigma_{\varepsilon}^2}}}=\sqrt{\frac{t}{t+k}} 
          
       Corr(Xt​,Xt+k​)=Var(Xt​)Var(Xt+k​) 
                      
                     ​Cov(Xt​,Xt+k​)​=tσε2​(t+k)σε2​ 
                      
                     ​tσε2​​=t+kt​ 
               
              ​
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