I am doing the following problem from Cover and Thomas, Elements of Information Theory, for self-study:
Let $X_1,\dots, X_n$ be an i.i.d. sequence of discrete random variables with entropy $H(X)$. Let $C_n(t) = \{ x^n \in X^n : p(x^n) \ge 2^{-nt} \}$. For what values of $t$ does $P[X^n \in C_n(t)] \rightarrow 1$?
My answer was that $t=H(X)$, since this is the only way to ensure $x^n \in X^n$ is in the typical set. In their solutions manual, Cover and Thomas state that this holds for $t > H(X)$, but do not justify this. Could someone confirm or deny that their answer is correct?
Note that $$P(X^n\in C_n(t))=P(p(X^n)\ge 2^{-nt})=P\left(-\frac{1}{n}\sum_{1}^n \lg p(X_i)\le t\right)$$ Now, it follows from WLLN, $$\lim_{n\to \infty}P\left(-\frac{1}{n}\sum_{1}^n \lg p(X_i)\le H(X)+\epsilon\right)=1\ \forall \epsilon>0$$ Thus, to have $\lim_{n\to \infty}P[X^n\in C_n(t)]\to 1$, we must have $t\ge \epsilon+H(X)\ \forall \epsilon>0\implies t>H(X)$.