A Lower Bound on the Complexity of Testing Grained Distributions

A distribution is called m -grained if each element appears with probability that is an integer multiple of 1 /m . We prove that, for any constant c < 1, testing whether a distribution over [Θ( m )] is m -grained requires Ω( m c ) samples, where testing a property of distributions means distinguishing between distributions that have the property and distributions that are far (in total variation distance) from any distribution that has the property.

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