# Implications of lower-bounded total variation distance on hypothesis testing

Let $\{X_i\}_n$ be a sequence of $n$ random variables independently and identically drawn from either $P$ or $Q$. Thus the sequence $\{X_i\}_n$ has a product distribution, which is either $P^n$ or $Q^n$. Assume that $P$ and $Q_\theta$ have the same support $\Omega$.

Suppose that the total variation distance between $P^n$ and $Q^n$ is lower-bounded:

$$\delta(P^n,Q_\theta^n)=\frac{1}{2}\int_{\Omega}|f_P^n(x)-f_{Q}^n(x)|dx\geq 1-\epsilon_n$$ (here $f_P^n(\cdot)$ and $f_{Q}^n(\cdot)$ are respective densities of $P^n$ and $Q^n$)

Now suppose that $\epsilon_n$ converges to zero as $n$ gets large, i.e. $\lim_{n\rightarrow \infty}\epsilon_n=0$. We can also assume uniform convergence, if that helps. Basically, by choosing $n$ large enough, we can obtain $\delta(P^n,Q^n)$ arbitrarily close to unity.

Does this imply that there exists a sequence of hypothesis tests $\phi_n(\{X_i\}_n)$ that classifies $\{X_i\}_n$ as drawn either from $P^n$ or $Q^n$ with the sum of error probabilities $\alpha_n+\beta_n\leq\epsilon_n$, where $\alpha_n$ is the probability of mis-classifying $\{X_i\}_n$ that was drawn from $P^n$ as drawn from $Q^n$, and $\beta_n$ is the probability of mis-classifying $\{X_i\}_n$ that was drawn from $Q^n$ as drawn from $P^n$? If so, is there a well-defined procedure on determining how to construct such a sequence of tests? I know that $1-\delta(P^n,Q^n)$ provides a lower bound on the sum $\alpha_n+\beta_n$ for any hypothesis test, but I am not sure what happens if $\delta(P^n,Q^n)$ is lower-bounded close to unity.