The Strong Uniform Consistency of Nearest Neighbor Density Estimates

Let X₁,⋯, Xₙ be independent, identically distributed random vectors with values in Rᵈ and with a common probability density f. If Vₖ(x) is the volume of the smallest sphere centered at x and containing at least k of the X₁,⋯, Xₙ then fₙ(x) = k/(nVₖ(x)) is a nearest neighbor density estimate of f. We show that if $k = k(n)$ satisfies k(n)/n → 0 and k(n)/log n → ∞ then ₓ|fₙ(x) - f(x)|→ 0 w.p. 1 when f is uniformly continuous on Rᵈ.