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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half-lines in the line has been replaced by much more general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."
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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