# Probability problem (Durrett 4.7.4.) Exchangeable Sequence of Random Variables.

I am having trouble in solving the following problem;

If $$X_1, X_2, . . . \in \mathbb{R}$$ are exchangeable with $$EX_i^2 < \infty$$ then $$E(X_1X_2) ≥ 0.$$

What I know is that the definition of exchangeable sequence;

A sequence $$X_1, X_2, . . .$$is said to be exchangeable if for each $$n$$ and permutation $$\pi$$ of $$\{1, . . . , n\}, (X_1, . . . , X_n)$$ and $$(X_{\pi(1)}, . . . , X_{\pi(n)})$$ have the same distribution.

How could I develop ideas from the above simple definition? Are there any hints available? Thanks in advance.

Define the matrix $$A=(E(X_iE_j))$$; by exchangeability all the diagonal elements are equal (to $$a$$, say) and all the off diagonal elements are equal (to $$b$$, say). This matrix is positive semidefinite. Now evaluate the quadratic form $$q_n=x'Ax$$, where the first $$n$$ elements of $$x$$ are $$1$$ and the rest are $$0$$. You get $$q_n = na+(n^2-n)b$$. But $$q_n\ge0$$ for all $$n$$. Hence $$b\ge0$$.
Equivalently, $$q_n=E(X_1+\cdots +X_n)^2 = nE(X_1^2)+(n^2-n)E(X_1X_2)=na+(n^2-n)b\ge0$$ for all $$n>0$$, so $$b\ge0$$.