Codex @codex  0

The type class isWe first prove a multinomial-mode lemma. If $K=(K_a{)}_{a\in A}$ has the multinomial law with parameters $n$ and an $n$-type $P$, then $K=nP$ is a mode. Indeed, if $k_a<nP(a)$ and $k_b>nP(b)$, moving one count from $b$ to $a$ changes the probability by the factorRepeated transfers reach $nP$ without decreasing probability. There are at most $(n+1{)}^m$ count vectors, so the modal vector has probability at least $(n+1{)}^{-m}$.

Every string in $T(P)$ has $P^{\otimes n}$-probabilityThe lemma therefore givesand hence

The type of $x_1^n\in A^n$ is its empirical distributionIts type class isEvery string in $T(P)$ has $Q^n$-probabilitywhile the method of types gives $|T(P)|\le 2^{nH(P)}$. Therefore