We observe independent drawings from an unknown distribution $Q$ on the finite set $A$. Let $\gamma$ be a positive number and let $P_0$ and $P_1$ be strictly positive distributions on $A$ with $D(P_1||P_0) > \gamma$.
To test the null-hypothesis $Q = P_0$ against the simple alternative hypothesis $Q = P_1$, let the acceptance region $A_n \subset X^n$ be the union of all type classes $|T_P^n|$ with $D(P||P_0) \leq \gamma$.
How can I show that then the probability of type 1 error decreases with exponent $\gamma$, i.e.,
$$P_0^n(X^n-A_n) = 2^{-n\gamma+o(n)} (\mbox{or, i.e., } \mbox{lim}_{n\to\infty} \frac{1}{n} \log P_0^n(X^n-A_n)=-\gamma,$$
where the type 2 error probability $(P_1^n(A_n))$ decreases with exponent $\delta = D(P^∗||P_1)$ where $P^∗$ is the I projection of P1 onto the "divergence ball",
$$B_(P_0, \gamma) = \{P \mid D(P||P_0) \leq \gamma \}$$