Ebook: Probability in Banach Spaces, 8: Proceedings of the Eighth International Conference

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 Then, modern empirical process theory began to develop, where the collection of half-lines in the line has been replaced by much more general collections of sets in and functions on multidimensional spaces.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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Content:
Front Matter....Pages I-XI
Front Matter....Pages 1-1
An Exposition of Talagrand’s Mini-Course on Matching Theorems....Pages 3-38
The Ajtai-Komlos-Tusnady Matching Theorem for General Measures....Pages 39-54
Some Generalizations of the Euclidean Two-Sample Matching Problem....Pages 55-66
Front Matter....Pages 67-67
Sharp Bounds on the LP Norm of a Randomly Stopped Multilinear form with an Application to Wald’s Equation....Pages 69-79
On Hoffmann-J?rgensen’s Inequality for U-Processes....Pages 80-91
The Poisson Counting Argument: A Heuristic for Understanding What Makes a Poissonized Sum Large....Pages 92-105
Conditional Versions of the Strassen-Dudley Theorem....Pages 106-115
An Approach to Inequalities for the Distributions of Infinite-Dimensional Martingales....Pages 116-127
Front Matter....Pages 128-134
Asymptotic Dependence of Stable Self-Similar Processes of Chentsov Type....Pages 135-135
Distributions of Stable Processes on Spaces of Measurable Functions....Pages 137-151
Harmonizability, V-Boundedness, and Stationary Dilation of Banach-Valued Processes....Pages 152-165
Front Matter....Pages 166-188
Asymptotic Behavior of Self-Normalized Trimmed Sums: Nonnormal Limits III....Pages 189-205
On Large Deviations of Gaussian Measures in Banach Spaces....Pages 207-207
Mosco Convergence and Large Deviations....Pages 209-227
Front Matter....Pages 228-244
A Functional Lil Approach to Pointwise Bahadur-Kiefer Theorems....Pages 245-252
The Glivenko-Cantelli Theorem in a Banach Space Setting....Pages 253-253
Marcinkiewicz Type Laws of Large Numbers and Convergence of Moments for u-Statistics....Pages 255-266
Self-Normalized Bounded Laws of the Iterated Logarithm in Banach Spaces....Pages 267-272
Front Matter....Pages 273-291
Rates of Clustering for Weakly Convergent Gaussian Vectors and Some Applications....Pages 292-303
On the Almost Sure Summability of B-Valued Random Variables....Pages 253-253
On the Rate of Clustering in Strassen’s Lil for Brownian Motion....Pages 304-324
Front Matter....Pages 325-338
A Central Limit Theorem for the Renormalized Self-Intersection Local Time of a Stationary Process....Pages 339-347
Moment Generating Functions for Local Times of Symmetric Markov Processes and Random Walks....Pages 349-349
Front Matter....Pages 351-363
Partial-Sum Processes with Random Locations and Indexed by Vapnik-?ervonenkis Classes of Sets in Arbitrary Sample Spaces....Pages 364-376
Learnability Models and Vapnik-Chervonenkis Combinatorics....Pages 377-377
Nonlinear Functionals of Empirical Measures....Pages 379-389
KAC Empirical Processes and the Bootstrap....Pages 390-402
Functional Limit Theorems for Probability Forecasts....Pages 403-410
Exponential Bounds in Vapnik-?ervonenkis Classes of Index 1....Pages 411-429
Front Matter....Pages 430-450
Tail Estimates for Empirical Characteristic Functions, with Applications to Random Arrays....Pages 451-465
The Radial Process for Confidence Sets....Pages 467-467
Stochastic Search in a Banach Space....Pages 469-478
Back Matter....Pages 479-496
....Pages 497-510