V-statistics are a class of statistics named for Richard von Mises who developed their asymptotic distribution theory in a fundamental paper in 1947. V-statistics are closely related to U-statistics (U for "unbiased") introduced by Wassily Hoeffding in 1948.  A V-statistic is a statistical function (of a sample) defined by a particular statistical functional of a probability distribution.
 Statistics that can be represented as functionals 
  
    
      
        T
        (
        
          F
          
            n
          
        
        )
      
    
    {\displaystyle T(F_{n})}
  
 of the empirical distribution function 
  
    
      
        (
        
          F
          
            n
          
        
        )
      
    
    {\displaystyle (F_{n})}
  
 are called statistical functionals. Differentiability of the functional T plays a key role in the von Mises approach; thus von Mises considers differentiable statistical functionals.
 Suppose x1, ..., xn is a sample. In typical applications the statistical function has a representation as the V-statistic
 where h is a symmetric kernel function. Serfling discusses how to find the kernel in practice.  Vmn is called a V-statistic of degree m.
 A symmetric kernel of degree 2 is a function h(x, y), such that h(x, y) = h(y, x) for all x and y in the domain of h. For samples x1, ..., xn, the corresponding V-statistic is defined
 In examples 1–3, the asymptotic distribution of the statistic is different: in (1) it is normal, in (2) it is chi-squared, and in (3) it is a weighted sum of chi-squared variables.
 Von Mises' approach is a unifying theory that covers all of the cases above.  Informally, the type of asymptotic distribution of a statistical function depends on the order of "degeneracy," which is determined by which term is the first non-vanishing term in the Taylor expansion of the functional T. In case it is the linear term, the limit distribution is normal; otherwise higher order types of distributions arise (under suitable conditions such that a central limit theorem holds).
 There are a hierarchy of cases parallel to asymptotic theory of U-statistics. Let A(m) be the property defined by:
 Case m = 1 (Non-degenerate kernel):
 If A(1) is true, the statistic is a sample mean and the Central Limit Theorem implies that T(Fn) is asymptotically normal.
 In the variance example (4), m2 is asymptotically normal with mean 
  
    
      
        
          σ
          
            2
          
        
      
    
    {\displaystyle \sigma ^{2}}
  
 and variance 
  
    
      
        (
        
          μ
          
            4
          
        
        −
        
          σ
          
            4
          
        
        )
        
          /
        
        n
      
    
    {\displaystyle (\mu _{4}-\sigma ^{4})/n}
  
, where 
  
    
      
        
          μ
          
            4
          
        
        =
        E
        (
        X
        −
        E
        (
        X
        )
        
          )
          
            4
          
        
      
    
    {\displaystyle \mu _{4}=E(X-E(X))^{4}}
  
.
 Case m = 2 (Degenerate kernel):
 Suppose A(2) is true, and 
  
    
      
        E
        [
        
          h
          
            2
          
        
        (
        
          X
          
            1
          
        
        ,
        
          X
          
            2
          
        
        )
        ]
        <
        ∞
        ,
        
        E
        
          |
        
        h
        (
        
          X
          
            1
          
        
        ,
        
          X
          
            1
          
        
        )
        
          |
        
        <
        ∞
        ,
      
    
    {\displaystyle E[h^{2}(X_{1},X_{2})]<\infty ,\,E|h(X_{1},X_{1})|<\infty ,}
  
 and 
  
    
      
        E
        [
        h
        (
        x
        ,
        
          X
          
            1
          
        
        )
        ]
        ≡
        0
      
    
    {\displaystyle E[h(x,X_{1})]\equiv 0}
  
. Then nV2,n converges in distribution to a weighted sum of independent chi-squared variables:
 where 
  
    
      
        
          Z
          
            k
          
        
      
    
    {\displaystyle Z_{k}}
  
 are independent standard normal variables and 
  
    
      
        
          λ
          
            k
          
        
      
    
    {\displaystyle \lambda _{k}}
  
 are constants that depend on the distribution F and the functional T. In this case the asymptotic distribution is called a quadratic form of centered Gaussian random variables. The statistic V2,n is called a degenerate kernel V-statistic. The V-statistic associated with the Cramer–von Mises functional (Example 3) is an example of a degenerate kernel V-statistic.