Laws of large numbers for classes of functions

J. E. Yukich

Journal of Multivariate Analysis, 1985, vol. 17, issue 3, 245-260

Abstract: Let (X, , P) be a probability space and n, n >= 1, a sequence of classes of measurable complex-valued functions on (X, , P). Under a weak metric entropy condition on n and sup {||g||[infinity]: g [set membership, variant] n}, Glivenko-Cantelli theorems are established for the classes n with respect to the probability measure P; i.e., limn --> [infinity] supg [set membership, variant] n ||[integral operator] g(dPn - dP)|| = 0 a.s. The result is applied to kernel density estimation and a law of the logarithm is derived for the maximal deviation between a kernel density estimator and its expected value, improving upon and generalizing the recent results of W. Stute (Ann. Probab. 10 (1982), 414-422). This result is also used to derive improved rates of uniform convergence for the empirical characteristic function.

Keywords: kernel; density; estimation; empirical; characteristic; function (search for similar items in EconPapers)
Date: 1985
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