Let θ be the mode of a probability density and θn itskernel estimator. In the case θ is nondegenerate, we first specify the weakconvergence rate of the multivariate kernel mode estimator by stating the central limittheorem for θn - θ. Then, we obtain a multivariate law ofthe iterated logarithm for the kernel mode estimator by proving that, with probabilityone, the limit set of the sequence θn - θ suitably normalized is an ellipsoid.We also give a law of the iterated logarithm for the lp norms, p ∈ [1,∞], ofθn - θ. Finally, we consider the case θ is degenerate and give the exactweak and strong convergence rate of θn - θ in the univariate framework.
