Suppose that $(Y_i,X_i)_{i\in\mathbb N}$ is random sequence of equally distributed random pairs, possibly dependent. Let $W_{ij}=X_iX_jY_i^2-X_iX_jY_iY_j$. For any $i\neq j$, does the following hold:
$$E(W_{ij})=E(W_{12})?$$
What happens if the $(Y_i,X_i)_{i\in\mathbb N}$ were strenghtened to be strictly stationary?
Comments
For independent pairs, it is easy: \begin{align} E(W_{ij})&=E(X_iX_jY_i^2-X_iX_jY_iY_j)\\ &\overset{indep.}{=}E(X_iY_i^2)E(X_j)-E(X_iY_i)E(X_jY_j)\\ &\overset{ident.}{=}E(X_1Y_1^2)E(X_2)-E(X_1Y_1)E(X_2Y_2)\\ &=E(X_1X_2Y_1^2-X_1X_2Y_1Y_2)=E(W_{12}) \end{align}
Without the independence, the equality is not obvious for me in both cases: the pairs $(X_i,Y_i)$ being equally distributed or $\{(X_i,Y_i)\}$ being strictly stationary.
This is not true in general. For instance, suppose that $(X_1,Y_1)=(X_2,Y_2)$ a.s., but $(X_1,Y_1),(X_3,Y_3)$ are independent. Then $$ \mathbb{E}[W_{12}] = \mathbb{E}[X_1^2Y_1^2-X_1^2Y_1^2] = 0 $$ while $$ \mathbb{E}[W_{13}] = \mathbb{E}[X_1X_3Y_1^2-X_1X_3Y_1Y_3] = \mathbb{E}[X_3]\mathbb{E}[X_1Y_1^2]-\mathbb{E}[X_1Y_1]\mathbb{E}[Y_3] $$ which is not zero in general (for instance, if $X_1>0$ a.s., $Y_1$ not constant and $\mathbb{E}[Y_1]=0$.)
Maybe the concept your are looking for instead should be interchangeability?