Kernel estimators of density function of directional data

Let X be a unit vector random variable taking values on a k-dimensional sphere [Omega] with probability density function f(x). The problem considered is one of estimating f(x) based on n independent observation X1,...,Xn on X. The proposed estimator is of the form fn(x) = (nhk-1)-1C(h) [Sigma]i=1n K[(1-x'Xi)/h2], x [set membership, variant] [Omega], where K is a kernel function defined on R+. Conditions are imposed on K and f to prove pointwise strong consistency, uniform strong consistency, and strong L1-norm consistency of fn as an estimator of f.