  The behavior of the Kozachenko - Leonenko estimates for the (differential)
Shannon entropy is studied when the number of i.i.d. vector-valued observations
tends to infinity. The asymptotic unbiasedness and L^2-consistency of the
estimates are established. The conditions employed involve the analogues of the
Hardy - Littlewood maximal function. It is shown that the results are valid in
particular for the entropy estimation of any nondegenerate Gaussian vector.
