Estimating entropy on m bins given fewer than m samples

Consider a sequence p_N of discrete probability measures, supported on m_N points, and assume that we observe N independent and identically distributed (i.i.d.) samples from each p_N. We demonstrate the existence of an estimator of the entropy, H(p_N), which is consistent even if the ratio N/m_N is bounded (and, as a corollary, even if this ratio tends to zero, albeit at a sufficiently slow rate).