Summary: We initiate a study of a new model of property testing that is a hybrid of testing properties of distributions and testing properties of strings. Specifically, the new model refers to testing properties of distributions, but these are distributions over huge objects (i.e., very long strings). Accordingly, the model accounts for the total number of local probes into these objects (resp., queries to the strings) as well as for the distance between objects (resp., strings), and the distance between distributions is defined as the earth mover’s distance with respect to the relative Hamming distance between strings.
We study the query complexity of testing in this new model, focusing on three directions. First, we try to relate the query complexity of testing properties in the new model to the sample complexity of testing these properties in the standard distribution-testing model. Second, we consider the complexity of testing properties that arise naturally in the new model (e.g., distributions that capture random variations of fixed strings). Third, we consider the complexity of testing properties that were extensively studied in the standard distribution-testing model: Two such cases are uniform distributions and pairs of identical distributions, where we obtain the following results.

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Testing whether a distribution over \(n\)-bit long strings is uniform on some set of size \(m\) can be done with query complexity \(\widetilde{O}(m/\epsilon^3)\) where \(\epsilon>(\log_2m)/n\) is the proximity parameter. Testing whether two distribution over \(n\)-bit long strings that have support size at most \(m\) are identical can be done with query complexity \(\widetilde{O}(m^{2/3}/\epsilon^3)\).

Both upper bounds are quite tight; that is, for \(\epsilon=\Omega(1)\), the first task requires \(\Omega(m^\epsilon)\) queries for any \(c<1\) and \(n=\omega(\log m)\), whereas the second task requires \(\Omega(m^{2/3})\) queries. Note that the query complexity of the first task is higher than the sample complexity of the corresponding task in the standard distribution-testing model, whereas in the case of the second task the bounds almost match.