Limit distribution theory for maximum likelihood estimation of a log-concave density

We find limiting distributions of the nonparametric maximum likelihood estimator (MLE) of a log-concave density, that is, a density of the form f0=exp..0 where .0 is a concave function on .. The pointwise limiting distributions depend on the second and third derivatives at 0 of Hk, the .lower invelope. of an integrated Brownian motion process minus a drift term depending on the number of vanishing derivatives of .0=log.f0 at the point of interest. We also establish the limiting distribution of the resulting estimator of the mode M(f0) and establish a new local asymptotic minimax lower bound which shows the optimality of our mode estimator in terms of both rate of convergence and dependence of constants on population values.