On the Local Limit Theorem for Lattice Distributions

Yu. V. Prokhorov made a conjecture that the two statements are equivalent: (a) the local limit theorem is applicable to a sequence of integer-valued random variables, (b) these variables obey the central limit theorem and their sums are asymptotically uniformly distributed modulo $h$ for arbitrary $h$.We give an example showing that the above conjecture does not hold true.