Clean

• Setup
  Let $X_1, X_2, \cdots $ be independent variables, and let $w_{ijn}(\cdot, \cdot)$ be Borel functions such that $Var[w_{ijn}(X_i, X_j)]$ is finite. Put $$ W(n) = \sum_{1 \leq i \leq n} \sum_{1 \leq j \leq n} w_{ijn}(X_i, X_j), ~~~ (\text{such as } W(n) = \sum_{1 \leq i \leq n} \sum_{1 \leq j \leq n} a_{ij} X_i X_j) $$ and $$ W_{ij} = w_{ijn}(X_i, X_j) + w_{jin}(X_j, X_i). $$ The index $n$ is suppressed in the notation $W_{ij}.$
  
  
• Definition
  $W(n)$ is called clean if the conditional expectations of $W_{ij}$ vanish : $$ E(W_{ij} | X_i) = 0 ~~~ \text{a.s.} ~~ \forall i,j \leq n. $$ If $W(n)$ is clean, then $W_{ij}$ has zero expectation and the diagonal elements $W_{ii}$ vanish a.s.