Let θ be the mode of a probability density and θn its
kernel estimator. In the case θ is nondegenerate, we first
specify the weak
convergence rate of the multivariate kernel mode estimator by stating
the central limit
theorem for θn - θ. Then, we obtain a multivariate law of
the iterated logarithm for the kernel mode estimator by proving that,
with probability
one, 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,∞], of
θn - θ. Finally, we consider the case θ is
degenerate and give the exact
weak and strong convergence rate of θn - θ in the
univariate framework.