Empirical types, n-types, and type classes

Let $\mathcal{A}$ be a finite nonempty alphabet and let $n\ge 1$. A word of length $n$ is $x^n=(x_1,\dots ,x_n)\in {\mathcal{A}}^n$. For $a\in \mathcal{A}$, define its count and empirical type by $N(a\mid x^n):=\mathrm{\#}\{i\in \{1,\dots ,n\}:x_i=a\},{\widehat{P}}_{x^n}(a):=\frac{N(a\mid x^n)}{n}\mathrm{.}$ Every position contributes to exactly one count, so the coordinates of ${\widehat{P}}_{x^n}$ are nonnegative and sum to one; it is a probability vector in the sense of Finite distributions, entropy, divergence, and stochastic matrices. The set of $n$-types is ${\mathcal{P}}_n(\mathcal{A}):=\{{\widehat{P}}_{x^n}:x^n\in {\mathcal{A}}^n\}$. Thus $q\in {\mathcal{P}}_n(\mathcal{A})$ exactly when every $nq(a)$ is a nonnegative integer and ${\sum }_{a}nq(a)=n$: the converse is obtained by arranging the prescribed finite number of copies of each letter. For $q\in {\mathcal{P}}_n(\mathcal{A})$, its type class is $T_q:=\{x^n\in {\mathcal{A}}^n:{\widehat{P}}_{x^n}=q\}\mathrm{.}$ The map $x^n\mapsto {\widehat{P}}_{x^n}$ has these type classes as its fibers.