Evarist Giné

High dimensional probability III

I. Measures on General Spaces and Inequalities.- Stochastic inequalities and perfect independence.- Prokhorov-LeCam-Varadarajans compactness criteria for vector measures on metric spaces.- On measures in locally convex spaces.- II. Gaussian Processes.- Karhunen-Loeve expansions for weighted Wiener processes and Brownian bridges via Bessel functions.- Extension du theoreme de Cameron-Martin aux translations aleatoires. II. Integrabilite des densites.- III. Limit Theorems.- Rates of convergence for Levys modulus of continuity and Hinchins law of the iterated logarithm.- On the limit set in the law of the iterated logarithm for U-statistics of order two.- Perturbation approach applied to the asymptotic study of random operators.- A uniform functional law of the logarithm for a local Gaussian process.- Strong limit theorems for mixing random variables with values in Hilbert space and their applications.- IV. Local Times.- Local time-space calculus and extensions of Itos formula.- Local times on curves and surfaces.- V. Large, Small Deviations.- Large deviations of empirical processes.- Small deviation estimates for some additive processes.- VI. Density Estimation.- Convergence in distribution of self-normalized sup-norms of kernel density estimators.- Estimates of the rate of approximation in the CLT for L1-norm of density estimators.- VII. Statistics via Empirical Process Theory.- Statistical nearly universal Glivenko-Cantelli classes.- Smoothed empirical processes and the bootstrap.- A note on the asymptotic distribution of Berk-Jones type statistics under the null hypothesis.- A note on the smoothed bootstrap.

On stability of probability laws with univariate stable marginals

SummaryExamples of D. Marcus in ℝ2 dispel the belief that a probability measure on ℝd is stable if and only if all its univariate marginals are stable. However, in ℝd (in fact, in fairly general linear spaces), a probability measure whose two-dimensional marginals are all infinitely divisible is stable if and only if all its univariate marginals are stable.

Sums of Independent Random Variables

The theory of decoupling aims at reducing the level of dependence in certain problems by means of inequalities that compare the original sequence to one involving independent random variables. It is therefore important to have information on results dealing with functionals of independent random variables.

Limit Theorems for U-Statistics

The decoupling results of Chapter 3 are very well suited for application in the asymptotic theory of U-statistics and U-processes, as we will see in this chapter and in the next. In this chapter we consider U-statistics. In general, we are interested in the “three gems” of classical probability theory, the law of large numbers, the central limit theorem, and the law of the iterated logarithm. Some inequalities, such as exponential and Hoffmann-Jorgensen-type inequalities, are also considered, mainly as first examples of application of decoupling and randomization, but also for their intrinsic interest.