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We prove a uniform functional law of the logarithm for the local empiricalprocess. To accomplish this we combine techniques from classical and abstractempirical process theory, Gaussian distributional approximation and probabilityon Banach spaces. The body of techniques we develop should prove useful to thestudy of the strong consistency of d-variate kernel-type nonparametric functionestimators.

A uniform functional law of the logarithm for the local empirical process