Some laws of the iterated logarithm for empirical processes rescaled in the "time" parameter are presented. These laws of the iterated logarithm are applied to obtain strong limit theorems for M-estimators. In particular, a law of the iterated logarithm for M-estimators with unusual rates of...

The law of the iterated logarithm for canonical or completely degenerate U-statistics with square integrable kernel h is proved, for h taking values in 1, 7 and, in general, in a type 2 separable Banach space. The LIL is also obtained for U-processes indexed by canonical Vapnik-Cervonenkis...

We give some easy methods to check sufficient conditions for the weak convergence of stochastic processes indexed by smooth functions. The main condition is a moment condition on the increments of the process. We apply this to empirical processes and U-processes in the independent identically...

We study the weak convergence for the row sums of a triangular array of empirical processes under bracketing conditions involving majorizing measures. As an application, we consider the weak convergence of stochastic processes of the form where {Xj}j=1[infinity] is a sequence of i.i.d.r.v.s with...

Exponential inequalities, the law of the iterated logarithm and the bootstrap central limit theorem for U-processes indexed by Vapnik-Cervonenkis classes of functions are derived. These results are then applied to the asymptotics and the bootstrap of U-statistics with estimated parameters, in...

We discuss the large and moderate deviations of a type of empirical processes whose finite-dimensional distributions do not satisfy the Cramer condition. Nonstandard speeds and rate functions appear.

An approximate M-estimator is defined as a value that minimizes certain random function up to a [var epsilon]n, where {[var epsilon]n} is a sequence of real numbers converging to zero. We determine the rate of [var epsilon]n so that the approximate M-estimator is asymptotically normal with rate...

It is shown that if {Xt: t [set membership, variant] T} is a Gaussian process such that (T, [varrho]) is a separable metric space, where [varrho](t, s) = Cov(Xt,Xs), then, with probability 1, no sample path of X can achieve its supremum at two distinct points of T. Conversely if Pr* {supt[set...

We discuss the asymptotic linearization of multivariate M-estimators, when the limit distribution is stable. We consider two different types of kernels: VC and bracketing. When applied to the case of normal limits, our work improves the known results to obtain the limit distribution of...