Necessary and Sufficient Conditions for the Strong Law of Large Numbers for U-statistics

Under some mild regularity on the normalizing sequence, we obtain necessary and sufficient conditions for the Strong Law of Large Numbers for (symmetrized) U-statistics. We also obtain nasc's for the a.s. convergence of series of an analogous form.Comment: LaTeX, 22 page

Modified empirical CLT's under only pre-Gaussian conditions

We show that a modified Empirical process converges to the limiting Gaussian process whenever the limit is continuous. The modification depends on the properties of the limit via Talagrand's characterization of the continuity of Gaussian processes.Comment: Published at http://dx.doi.org/10.1214/074921706000000833 in the IMS Lecture Notes Monograph Series (http://www.imstat.org/publications/lecnotes.htm) by the Institute of Mathematical Statistics (http://www.imstat.org

Weighted uniform consistency of kernel density estimators

Let f_n denote a kernel density estimator of a continuous density f in d dimensions, bounded and positive. Let \Psi(t) be a positive continuous function such that \|\Psi f^{\beta}\|_{\infty}<\infty for some 0<\beta<1/2. Under natural smoothness conditions, necessary and sufficient conditions for the sequence \sqrt\frac{nh_n^d}{2|\log h_n^d|}\|\Psi(t)(f_n(t)-Ef_n(t))\|_{\infty} to be stochastically bounded and to converge a.s. to a constant are obtained. Also, the case of larger values of \beta is studied where a similar sequence with a different norming converges a.s. either to 0 or to +\infty, depending on convergence or divergence of a certain integral involving the tail probabilities of \Psi(X). The results apply as well to some discontinuous not strictly positive densities.Comment: Published by the Institute of Mathematical Statistics (http://www.imstat.org) in the Annals of Probability (http://www.imstat.org/aop/) at http://dx.doi.org/10.1214/00911790400000006