In probability, weak dependence of random variables is a generalization of independence that is weaker than the concept of a martingale[citation needed].  A (time) sequence of random variables is weakly dependent if distinct portions of the sequence have a covariance that asymptotically decreases to 0 as the blocks are further separated in time.  Weak dependence primarily appears as a technical condition in various probabilistic limit theorems.    
 Fix a set S, a sequence of sets of measurable functions 
  
    
      
        {
        
          
            
              F
            
          
          
            d
          
        
        
          }
          
            d
            =
            1
          
          
            ∞
          
        
        ∈
        
          ∏
          
            d
            =
            1
          
          
            ∞
          
        
        
          
            (
            
              
                S
                
                  d
                
              
              →
              
                R
              
            
            )
          
        
      
    
    {\displaystyle \{{\mathcal {F}}_{d}\}_{d=1}^{\infty }\in \prod _{d=1}^{\infty }{\left(S^{d}\to \mathbb {R} \right)}}
  
, a decreasing sequence 
  
    
      
        {
        
          θ
          
            δ
          
        
        
          }
          
            δ
            =
            1
          
          
            ∞
          
        
        →
        0
      
    
    {\displaystyle \{\theta _{\delta }\}_{\delta =1}^{\infty }\to 0}
  
, and a function 
  
    
      
        ψ
        ∈
        
          
            
              F
            
          
          
            2
          
        
        ×
        (
        
          
            Z
          
          
            +
          
        
        
          )
          
            2
          
        
        →
        
          
            R
          
          
            +
          
        
      
    
    {\displaystyle \psi \in {\mathcal {F}}^{2}\times (\mathbb {Z} ^{+})^{2}\to \mathbb {R} ^{+}}
  
.  A sequence 
  
    
      
        {
        
          X
          
            n
          
        
        
          }
          
            n
            =
            1
          
          
            ∞
          
        
      
    
    {\displaystyle \{X_{n}\}_{n=1}^{\infty }}
  
 of random variables is 
  
    
      
        (
        {
        
          
            
              F
            
          
          
            d
          
        
        
          }
          
            d
            =
            1
          
          
            ∞
          
        
        ,
        {
        
          θ
          
            δ
          
        
        
          }
          
            δ
          
        
        ,
        ψ
        )
      
    
    {\displaystyle (\{{\mathcal {F}}_{d}\}_{d=1}^{\infty },\{\theta _{\delta }\}_{\delta },\psi )}
  
-weakly dependent iff, for all 
  
    
      
        
          j
          
            1
          
        
        ≤
        
          j
          
            2
          
        
        ≤
        ⋯
        ≤
        
          j
          
            d
          
        
        <
        
          j
          
            d
          
        
        +
        δ
        ≤
        
          k
          
            1
          
        
        ≤
        
          k
          
            2
          
        
        ≤
        ⋯
        ≤
        
          k
          
            e
          
        
      
    
    {\displaystyle j_{1}\leq j_{2}\leq \dots \leq j_{d}<j_{d}+\delta \leq k_{1}\leq k_{2}\leq \dots \leq k_{e}}
  
, for all 
  
    
      
        ϕ
        ∈
        
          
            
              F
            
          
          
            d
          
        
      
    
    {\displaystyle \phi \in {\mathcal {F}}_{d}}
  
, and 
  
    
      
        θ
        ∈
        
          
            
              F
            
          
          
            e
          
        
      
    
    {\displaystyle \theta \in {\mathcal {F}}_{e}}
  
, we have: 315 
 Note that the covariance does not decay to 0 uniformly in d and e.: 9   
 Weak dependence is a sufficient weak condition that many natural instances of stochastic processes exhibit it.: 9   In particular, weak dependence is a natural condition for the ergodic theory of random functions.
 A sufficient substitute for independence in the Lindeberg–Lévy central limit theorem is weak dependence.: 315   For this reason, specializations often appear in the probability literature on limit theorems.: 153–197   These include Withers' condition for strong mixing, Tran's "absolute regularity in the locally transitive sense," and Birkel's  "asymptotic quadrant independence."
 Weak dependence also functions as a substitute for strong mixing.  Again, generalizations of the latter are specializations of the former; an example is Rosenblatt's mixing condition.
 Other uses include a generalization of the Marcinkiewicz–Zygmund inequality and Rosenthal inequalities.: 314, 319 
 Martingales are weakly dependent[citation needed], so many results about martingales also hold true for weakly dependent sequences.  An example is Bernstein's bound on higher moments, which can be relaxed to only require