a suitable randomized ordering of the random feature allocation $\randfa_{N}$. We achieve this ordering in two steps.

The first step involves ordering the features via a procedure that we refer to as *order-of-appearance labeling*. The basic idea is that we consider data indices $n = 1, 2, 3$, and so on in order. Each time a new data point arrives, we examine the features associated with that data point. Each time we see a new feature, we label it with the lowest available feature label from $k = 1, 2, \ldots$.

In practice, the order-of-appearance scheme requires some auxiliary randomness since each index $n$ may belong to zero, one, or many different features (though the number must be finite). When multiple features first appear for index $n$, we order them uniformly at random. That simple idea is explained in full detail as follows. Recursively suppose that there are $K$ features among the indices $[N-1]$. Trivially there are zero features when no indices have been seen yet. Moreover, we suppose that we have features with labels $1$ through $K$ if $K \ge 1$, and if $K = 0$, we have no features. If features remain without labels, there exists some minimum 