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I'm studying the following term:

$$W_{ij} = \frac{1}{n(n-1)E[f(X_1,X_2)]} \sum_{i\neq j}^n f(X_i,X_j)Y_j,$$

where $(X_i, Y_i)$ is a sequence of equally distributed random pairs and $f$ a measurable function.

Question: Does $E(W_{ij})=\frac{E[f(X_1,X_2)Y_2]}{E[f(X_1,X_2)]}$ hold?

Considerations:

To get this result, one has to show that $E[f(X_i,X_j)Y_j] = E[f(X_1,X_2)Y_2], \forall i \neq j$, which I believe is false in general.

Take for example $Y = 1$ and $f(x, y) = xy$. It's not correct to say that $E(X_iX_j) = E(X_1X_2)$ for $i \neq j$, only using the equality in distribution, right? If we add independecy, $E(X_iX_j) = E(X_i)E(X_j) = E(X_1X_2)$.

Being more specific

In my case, I have $f(x,y)=[g(x)]^2h(x)h(y)-g(x)h(x)g(y)h(y)$, with g,f measurable. For simplicity's sake, denote $g(X_k):=g_k$ and $h(X_k)=h_k$. If I assume $(X_i,Y_i)_{i\in\mathbb{N}}$ is also independent, then each $X_i$ is independent of $X_j$ and $Y_j$, for $i\neq j$.

That said, we have

\begin{align} E(f(X_i,X_j)Y_j)&=E(g_i^2h_ih_jY_j)-E(g_ih_ig_jh_jY_j)\\ &\overset{indep.}{=}E(g_i^2h_i)E(h_jY_j)-E(g_ih_i)E(g_jh_jY_j)\\ &\overset{i.d.}{=}E(g_1^2h_1)E(h_2Y_2)-E(g_1h_1)E(g_2h_2Y_2)\\ &=E(f(X_1,X_2)Y_2). \end{align}

But when $(X_i,Y_i)_{i\in\mathbb{N}}$ is dependent, the above equality does not hold in general.

Are my thoughts right?

Celine Harumi
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  • A naive question to check: By “equally distributed” do you mean “identically distributed” or “distributed according to a uniform distribution”? I’m guessing “identically distributed” but that seems to lead to triviality? – PtH Feb 05 '21 at 14:43
  • I meant $F_{X_i,Y_i}=F_{X_j,Y_j}$ where $F$ is the probability distribution of the pair $(X,Y)$ – Celine Harumi Feb 05 '21 at 14:49

1 Answers1

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Yes, I also think that an assumption of independence is needed.

Suppose $g(x) = x$, $h(x) = x$ and that for some $i \neq j$ we have $X_i = X_j = Y_j$ (which is allowed under dependence). Then at line 1 the second term is $E(Y_j^5)$. But in general, $E(Y_j^5) \neq E(Y_j^2)E(Y_j^3)$, versus the step to line 2 which is valid under assumption of independence.

(Strictly, this example answer doesn’t prove that an assumption of independence is necessary for the original equation proposed for $E(W_{ij})$. But it does indicate that without such an assumption, the equality isn’t obvious.)

PtH
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