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Iosif Pinelis
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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.

Iosif Pinelis
  • 127.8k
  • 8
  • 107
  • 229