Probability limit theorems in infinite-dimensional spaces give conditions un der which convergence holds uniformly over an infinite class of sets or functions. Early results in this direction were the Glivenko-Cantelli, Kolmogorov-Smirnov and Donsker theorems for empirical distribution functions. Already in these cases there is convergence in Banach spaces that are not only infinite-dimensional but nonsep arable. But the theory in such spaces developed slowly until the late 1970's. Meanwhile, work on probability in separable Banach spaces, in relation with the geometry of those spaces, began in the 1950's and developed strongly in the 1960's and 70's. We have in mind here also work on sample continuity and boundedness of Gaussian processes and random methods in harmonic analysis. By the mid-70's a substantial theory was in place, including sharp infinite-dimensional limit theorems under either metric entropy or geometric conditions. Then, modern empirical process theory began to develop, where the collection of half-lines in the line has been replaced by much more general collections of sets in and functions on multidimensional spaces. Many of the main ideas from probability in separable Banach spaces turned out to have one or more useful analogues for empirical processes. Tightness became "asymptotic equicontinuity. " Metric entropy remained useful but also was adapted to metric entropy with bracketing, random entropies, and Kolchinskii-Pollard entropy. Even norms themselves were in some situations replaced by measurable majorants, to which the well-developed separable theory then carried over straightforwardly.
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ISBN-13
978-1-4612-0367-4 (9781461203674)
DOI
10.1007/978-1-4612-0367-4
Schweitzer Klassifikation
An exposition of Talagrand's mini-course on matching theorems.- The Ajtai-Komlos-Tusnady matching theorem for general measures.- Some generalizations of the Euclidean two-sample matching problem.- Sharp bounds on the Lp norm of a randomly stopped multilinear form with an application to Wald's equation.- On Hoffmann-Jorgensen's inequality for U-processes.- The Poisson counting argument: A heuristic for understanding what makes a Poissonized sum large.- On the lower tail of Gaussian measures on lp.- Conditional versions of the Strassen-Dudley Theorem.- An approach to inequalities for the distributions of infinite-dimensional martingales.- Random integral representations for classes of limit distributions similar to Levy class L0- III.- Asymptotic dependence of stable self-similar processes of Chentsov-type.- Distributions of stable processes on spaces of measurable functions.- Harmonizability, V-boundedness, and stationary dilation of Banach space-valued processes.- Asymptotic behavior of self-normalized trimmed sums: Nonnormal limits III.- On large deviations of Gaussian measures in Banach spaces.- Mosco convergence and large deviations.- A functional LIL approach to pointwise Bahadur-Kiefer theorems.- The Glivenko-Cantelli theorem in a Banach space setting.- Marcinkiewicz type laws of large numbers and convergence of moments for U-statistics.- Self-normalized bounded laws of the iterated logarithm in Banach spaces.- Rates of clustering for weakly convergent Gaussian vectors and some applications.- On the almost sure summability of B-valued random variables.- On the rate of Clustering in Strassen's LIL for Brownian Motion.- A central limit theorem for the renormalized self-intersection local time of a stationary process.- Moment generating functions for localtimes of symmetric Markov processes and random walks.- Partial-sum processes with random lattice-points and indexed by Vapnik ?ervonenkis classes of sets in arbitrary sample spaces.- Learliability models and Vapnik-Chervonenkis combinatorics.- Nonlinear functional of empirical measures.- KAC empirical processes and the bootstrap.- Functional limit theorems for probability forecasts.- Exponential bounds in Vapnik-?ervonenkis classes of index.- Tail estimates for empirical characteristic functions, with applications to random arrays.- The radial process for confidence sets.- Stochastic search in a Banach space.
