Probability in Banach Spaces, 8: Proceedings of the Eighth International Conference (Progress in Probability, 30)

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 Then, modern empirical process theory began to develop, where the collection of half-lines in the line has been replaced by much more general collections of sets in and functions on multidimensional spaces.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. 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 L P Norm of a Randomly Stopped Multilinear form with an Application to Wald’s Equation....Pages 69-79 On Hoffmann-Jørgensen’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 On the Lower Tail of Gaussian Measures on l p ....Pages 106-115 Conditional Versions of the Strassen-Dudley Theorem....Pages 116-127 An Approach to Inequalities for the Distributions of Infinite-Dimensional Martingales....Pages 128-134 Front Matter....Pages 135-135 Random Integral Representations for Classes of Limit Distributions Similar to Levy Class L 0 . III....Pages 137-151 Asymptotic Dependence of Stable Self-Similar Processes of Chentsov Type....Pages 152-165 Distributions of Stable Processes on Spaces of Measurable Functions....Pages 166-188 Harmonizability, V-Boundedness, and Stationary Dilation of Banach-Valued Processes....Pages 189-205 Front Matter....Pages 207-207 Asymptotic Behavior of Self-Normalized Trimmed Sums: Nonnormal Limits III....Pages 209-227 On Large Deviations of Gaussian Measures in Banach Spaces....Pages 228-244 Mosco Convergence and Large Deviations....Pages 245-252 Front Matter....Pages 253-253 A Functional Lil Approach to Pointwise Bahadur-Kiefer Theorems....Pages 255-266 The Glivenko-Cantelli Theorem in a Banach Space Setting....Pages 267-272 Marcinkiewicz Type Laws of Large Numbers and Convergence of Moments for u -Statistics....Pages 273-291 Self-Normalized Bounded Laws of the Iterated Logarithm in Banach Spaces....Pages 292-303 Front Matter....Pages 253-253 Rates of Clustering for Weakly Convergent Gaussian Vectors and Some Applications....Pages 304-324 On the Almost Sure Summability of B-Valued Random Variables....Pages 325-338 On the Rate of Clustering in Strassen’s Lil for Brownian Motion....Pages 339-347 Front Matter....Pages 349-349 A Central Limit Theorem for the Renormalized Self-Intersection Local Time of a Stationary Process....Pages 351-363 Moment Generating Functions for Local Times of Symmetric Markov Processes and Random Walks....Pages 364-376 Front Matter....Pages 377-377 Partial-Sum Processes with Random Locations and Indexed by Vapnik-Červonenkis Classes of Sets in Arbitrary Sample Spaces....Pages 379-389 Learnability Models and Vapnik-Chervonenkis Combinatorics....Pages 390-402 Nonlinear Functionals of Empirical Measures....Pages 403-410 KAC Empirical Processes and the Bootstrap....Pages 411-429 Functional Limit Theorems for Probability Forecasts....Pages 430-450 Exponential Bounds in Vapnik-Červonenkis Classes of Index 1....Pages 451-465 Front Matter....Pages 467-467 Tail Estimates for Empirical Characteristic Functions, with Applications to Random Arrays....Pages 469-478 The Radial Process for Confidence Sets....Pages 479-496 Stochastic Search in a Banach Space....Pages 497-510 Back Matter....Pages 511-512 Probability limit theorems in infinite-dimensional spaces give conditions unƯ 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 nonsepƯ 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