Constructing dependent sequences of random variables It is easy, given some random variable $X \colon \Omega \to \mathbb{R}$ on a probability space $(\Omega, \mathbb{P})$, to construct an i.i.d. sequence $X_1, X_2, \ldots$ distributed as the law of $X$. Given some positive operator $V$ on $\ell^2(\mathbb{R})$, we can view it as an infinite covariance matrix. Can we construct a sequence of identically-distributed variables $\{X_i\}$ such that $\text{Cov}(X_i, X_j) = V_{i,j}$? Or even a finite sequence of variables, where $V$ is a square matrix?
Some additional constraints are needed. If $X$ is constant the answer is obviously no, so perhaps we need to assume that the law of $X$ is absolutely continuous with respect to the Lebesgue measure, or at least non-atomic.
 A: First the finite case: Let $V$ be a non-negative-definite symmetric real $n\times n$ matrix. The finite-dimensional version of the spectral theorem implies that $V$ has a non-negative-definite symmetric square root with real entries, which let us call $V^{1/2}$.  Let $Z\in\mathbb R^{n\times 1}$ be a column vector whose entries are i.i.d. random variables the distribution of each of which is $N(0,1)$.  Then
$$
\operatorname{var}(V^{1/2} Z) = V^{1/2} \Big(\operatorname{var}(Z)\Big) \left( V^{1/2} \right)^T = V^{1/2} \left(V^{1/2}\right)^T = V.
$$
(Recall that $\operatorname{var}(X) = \operatorname{E}\left( (X-\mu)(X-\mu)^T \right) \in \mathbb R^{n\times n}$, where $\mu=\operatorname{E}(X)$, whence we get $\operatorname{var}(MX) = M \Big( \operatorname{var}(X)\Big) M^T$.)
Thus you can get the covariances you want. (PS: Of course they won't be identically distributed unless all of the diagonal entries in $V$ are the same.)
I was going to add that some exchangeable sequences cannot be infinitely extended.  For example, suppose you draw a sequence of ten marbles without replacement from an urn that contains $20$ red and $20$ green marbles, and let $X_i$ be the number of red marbles on the $i$th trial (either $0$ or $1$).  The covariance between any two of these $10$ random variables is the same negative number.  You cannot extend any sequence with those variances and those covariances to a sequence longer than some finite number while continuing to have the same variances and the same covariances.
However, you've already got your positive operator on all of $\ell^2$, so I think what I did in the finite case should work.  Use the spectral theorem.
