## 2 Generalizations/Modifications of [PERSON]'s Theorem
### Modifying the Exchangeability Requirement  
There are numerous results that prove some variant of [PERSON]'s theorem (and of its more general version, the Hewitt-Savage theorem) for random structures that satisfy some symmetry requirement similar to exchangeability. For a survey see [PERSON] [11] and [PERSON] [12]. Here we present two characteristic examples. **Partially exchangeable arrays. ** Let \(X=[X_{i,j}]\), \(1\leq i,j<\infty\), be a doubly infinite array (infinite matrix) of random variables, taking values from a Borel measurable space \(S\). Let \(R_{i},C_{j}\) denote the \(i\)th row and \(j\)th column of \(X\), respectively. We say that \(X\) is _row-exchangeable_, if the sequence \(R_{1},R_{2},\ldots\) is exchangeable. Similarly, \(X\) is _column-exchangeable_ if \(C_{1},C_{2},\ldots\) is exchangeable. Finally, \(X\) is _row and column exchangeable (\(RCE\))_, if \(X\) is both row-exchangeable and column-exchangeable. Observe that RCE is a weaker requirement than demanding that all entries of \(X\), listed as a single sequence, form an exchangeable sequence. For RCE arrays, [PERSON] [13] proved a characterization, which contains the de Finetti (in fact, the Hewitt-Savage) theorem as a special case. We use the notation \(X=_{d}Y\) to express that the random variables \(X\), \(Y\) have the same distribution. **Theorem 4** (Row and column exchangeable (RCE) arrays): _If \(X\) is an RCE array, then there exists independent random variables \(\alpha,\xi_{i},\eta_{j},\lambda_{i,j},\)\(1\leq i,j<\infty\), such that all of them are uniformly distributed on \([0,1]\), and there exists a measurable function (see the definition in Appendix A) \(f:[0,1]^{4}\mapsto S\), such that \(X=_{d}X^{*}=[X^{*}_{i,j}]\), where \(X^{*}_{i,j}=f(\alpha,\xi_{i},\eta_{j},\lambda_{i,j})\)._  
When the array \(X\) consists of a single row or a single column, we get a special case, which is equivalent to the Hewitt-Savage theorem (and includes [PERSON]'s theorem):  
**Theorem 5**: _An infinite S-valued sequence \(Z\) is exchangeable if and only if there exists a measurable function \(f:[0,1]^{2}\mapsto S\) and i.i.d. random variables \(\alpha,\xi_{1},\xi_{2},\ldots,\) all uniformly distributed on \([0,1]\), such that \(Z=_{d}\left[f(\alpha,\xi_{i})\right]_{i=1}^{\infty}\)._  
Note that for any fixed \(\alpha=c_{0}\), the sequence \([f(c_{0},\xi_{i})]_{i=1}^{\infty}\) is i.i.d., so with a random \(\alpha\) we indeed obtain an i.i.d. mix. Comparing with the formulations of Theorems 1 and 3,observe that here the potentially complicated mixing measure \(\mu\) is replaced by the simple random variable \(\alpha\), which is _uniform_ on \([0,1]\). Of course, the potential complexity of \(\mu\) does not simply \"evaporate,\" it is just shifted to the function \(f\). **[PERSON]'s theorem for Markov chains. ** [PERSON] and [PERSON] [14] created a version of [PERSON]'s Theorem for Markov chains. The mixture of Markov chains can be interpreted similarly to other sequences, as a Markov chain is just a special sequence of random variables. To elaborate the conditions, consider random variables taking values in a countable state space \(I\). Let us call two fixed sequences \(a=(a_{1},\ldots,a_{n})\) and \(b=(b_{1},\ldots,b_{n})\) in \(I\)_equivalent_ if \(a_{1}=b_{1}\), and the number of \(i\to j\) transitions occurring in \(a\) is the same as the number of \(i\to j\) transitions occurring in \(b\), for every \(i,j\in I\). Let \(X=(X_{1},X_{2},\ldots)\) be a sequence of random variables over \(I\). We say that \(X\) is _recurrent_ if for any starting state \(X_{1}=i\), the sequence returns to \(i\) infinitely many times, with probability \(1\). Then the Markov chain version of [PERSON]'s Theorem, proved by [PERSON] and [PERSON] [14], can be formulated as follows:  
Theorem 6 (Markov chain version of [PERSON]'s theorem): _Let \(X=(X_{1},X_{2},\ldots)\) be a recurrent sequence of random variables over a countable state space \(I\).