Skorokhod's representation theorem

<h2 id="statement">Statement</h2>
<p>Let 
  
    
      
        (
        
          μ
          
            n
          
        
        
          )
          
            n
            ∈
            
              N
            
          
        
      
    
    {\displaystyle (\mu _{n})_{n\in \mathbb {N} }}
  
 be a sequence of probability measures on a <a href="/facts/Metric_space/gnBBRXtc">metric space</a> 
  
    
      
        S
      
    
    {\displaystyle S}
  
 such that 
  
    
      
        
          μ
          
            n
          
        
      
    
    {\displaystyle \mu _{n}}
  
 <a href="/facts/Convergence_in_distribution/pWdatFlY">converges weakly</a> to some probability measure 
  
    
      
        
          μ
          
            ∞
          
        
      
    
    {\displaystyle \mu _{\infty }}
  
 on 
  
    
      
        S
      
    
    {\displaystyle S}
  
 as 
  
    
      
        n
        →
        ∞
      
    
    {\displaystyle n\to \infty }
  
.  Suppose also that the <a href="/facts/Support_(measure_theory)/mBM9wEAF">support</a> of 
  
    
      
        
          μ
          
            ∞
          
        
      
    
    {\displaystyle \mu _{\infty }}
  
 is <a href="/facts/Separable_space/2bmU73bk">separable</a>. Then there exist 
  
    
      
        S
      
    
    {\displaystyle S}
  
-valued random variables 
  
    
      
        
          X
          
            n
          
        
      
    
    {\displaystyle X_{n}}
  
 defined on a common probability space 
  
    
      
        (
        Ω
        ,
        
          
            F
          
        
        ,
        
          P
        
        )
      
    
    {\displaystyle (\Omega ,{\mathcal {F}},\mathbf {P} )}
  
 such that the law of 
  
    
      
        
          X
          
            n
          
        
      
    
    {\displaystyle X_{n}}
  
 is 
  
    
      
        
          μ
          
            n
          
        
      
    
    {\displaystyle \mu _{n}}
  
 for all 
  
    
      
        n
      
    
    {\displaystyle n}
  
 (including 
  
    
      
        n
        =
        ∞
      
    
    {\displaystyle n=\infty }
  
) and such that 
  
    
      
        (
        
          X
          
            n
          
        
        
          )
          
            n
            ∈
            
              N
            
          
        
      
    
    {\displaystyle (X_{n})_{n\in \mathbb {N} }}
  
 converges to 
  
    
      
        
          X
          
            ∞
          
        
      
    
    {\displaystyle X_{\infty }}
  
, 
  
    
      
        
          P
        
      
    
    {\displaystyle \mathbf {P} }
  
-almost surely.
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Convergence_of_random_variables/pWdatFlY">Convergence in distribution</a></li></ul>

<ul><li>Billingsley, Patrick (1999). <a href="https://archive.org/details/convergenceofpro0000bill"><i>Convergence of Probability Measures</i></a>. New York: John Wiley & Sons, Inc. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-471-19745-9. (see p. 7 for weak convergence, p. 24 for convergence in distribution and p. 70 for Skorokhod's theorem)</li></ul>

Skorokhod's representation theorem open-in-new

Skorokhod's representation theorem