There is no such counterexample. Indeed, by Theorem 1, any strong mixing sequence obeys the $0$-$1$ law. Also, any $k$-dependent sequence is obviously strong mixing, for any natural $k$, and hence must obey the $0$-$1$ law.

There is no such counterexample. Indeed, by Theorem 1, any strong mixing sequence obeys the $0$-$1$ law. Also, any $m$-dependent sequence is obviously strong mixing, for any integer $m\ge0$, and hence must obey the $0$-$1$ law.