Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 224 / 3 / b / Solution

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