Dudley, R. M. (1973) Sample Functions of the Gaussian Process
     @article{1973,
jstor_articletype = {research-article},
title = {Sample Functions of the Gaussian Process},
author = {Dudley, R. M.},
journal = {The Annals of Probability},
jstor_issuetitle = {},
volume = {1},
number = {1},
jstor_formatteddate = {Feb., 1973},
pages = {pp. 66-103},
url = {http://www.jstor.org/stable/2959347},
ISSN = {00911798},
abstract = {This is a survey on sample function properties of Gaussian
processes with the main emphasis on boundedness and continuity, including
Holder conditions locally and globally. Many other sample function
properties are briefly treated. The main new results continue the program
of reducing general Gaussian processes to "the" standard isonormal linear
process $L$ on a Hilbert space $H$, then applying metric entropy methods.
In this paper Holder conditions, optimal up to multiplicative constants,
are found for wide classes of Gaussian processes. If $H$ is $L^2$ of
Lebesgue measure on $R^k, L$ is called "white noise." It is proved that we
can write $L = P(D)\lbrack x\rbrack$ in the distribution sense where $x$
has continuous sample functions if $P(D)$ is an elliptic operator of degree
$> k/2$. Also $L$ has continuous sample functions when restricted to
indicator functions of sets whose boundaries are more than $k - 1$ times
differentiable in a suitable sense. Another new result is that for the
Levy(-Baxter) theorem $\int^1_0(dx_t)^2 = 1$ on Brownian motion, almost
sure convergence holds for any sequence of partitions of mesh $o(1/\log n)$. If partitions into measurable sets other than intervals are allowed,
the above is best possible: $\mathscr{O}(1/\log n)$ is insufficient.},
language = {English},
year = {1973},
publisher = {Institute of Mathematical Statistics},
}


Dudley, R. M. (1978) Central Limit Theorems for Empirical Measures
    @article{1978,
jstor_articletype = {research-article},
title = {Central Limit Theorems for Empirical Measures},
author = {Dudley, R. M.},
journal = {The Annals of Probability},
jstor_issuetitle = {},
volume = {6},
number = {6},
jstor_formatteddate = {Dec., 1978},
pages = {pp. 899-929},
url = {http://www.jstor.org/stable/2243028},
ISSN = {00911798},
abstract = {Let $(X, \mathscr{A}, P)$ be a probability space. Let $X_1, X_2,\cdots,$ be independent $X$-valued random variables with distribution
$P$. Let $P_n := n^{-1}(\delta_{X_1} + \cdots + \delta_{X_n})$ be the
empirical measure and let $\nu_n := n^\frac{1}{2}(P_n - P)$. Given a class
$\mathscr{C} \subset \mathscr{a}$, we study the convergence in law of
$\nu_n$, as a stochastic process indexed by $\mathscr{C}$, to a certain
Gaussian process indexed by $\mathscr{C}$. If convergence holds with
respect to the supremum norm $\sup_{C \in \mathscr{C}}|f(C)|$, in a
suitable (usually nonseparable) function space, we call $\mathscr{C}$ a
Donsker class. For measurability, $X$ may be a complete separable metric
space, $\mathscr{a} =$ Borel sets, and $\mathscr{C}$ a suitable collection
of closed sets or open sets. Then for the Donsker property it suffices that
for some $m$, and every set $F \subset X$ with $m$ elements, $\mathscr{C}$
does not cut all subsets of $F$ (Vapnik-Cervonenkis classes). Another
sufficient condition is based on metric entropy with inclusion. If
$\mathscr{C}$ is a sequence $\{C_m\}$ independent for $P$, then
$\mathscr{C}$ is a Donsker class if and only if for some $r, \sigma_m(P(C_m)(1 - P(C_m)))^r < \infty$.},
language = {English},
year = {1978},
publisher = {Institute of Mathematical Statistics},
}

@article{1979,
jstor_articletype = {research-article},
title = {Corrections to "Central Limit Theorems for Empirical Measures"},
author = {Dudley, R. M.},
journal = {The Annals of Probability},
jstor_issuetitle = {},
volume = {7},
number = {5},
jstor_formatteddate = {Oct., 1979},
pages = {pp. 909-911},
url = {http://www.jstor.org/stable/2243316},
ISSN = {00911798},
abstract = {},
language = {English},
year = {1979},
publisher = {Institute of Mathematical Statistics},
}


Dudley, R. M. (1987) Universal Donsker Classes and Metric Entropy
     @article{1987,
jstor_articletype = {research-article},
title = {Universal Donsker Classes and Metric Entropy},
author = {Dudley, R. M.},
journal = {The Annals of Probability},
jstor_issuetitle = {},
volume = {15},
number = {4},
jstor_formatteddate = {Oct., 1987},
pages = {pp. 1306-1326},
url = {http://www.jstor.org/stable/2244004},
ISSN = {00911798},
abstract = {Let (X, A) be a measurable space and F a class of measurable
functions on X. F is called a universal Donsker class if for every
probability measure P on A, the centered and normalized empirical measures
n1/2(Pn - P) converge in law, with respect to uniform convergence over F,
to the limiting "Brownian bridge" process GP. Then up to additive
constants, F must be uniformly bounded. Several nonequivalent conditions
are shown to imply the universal Donsker property. Some are connected with
the Vapnik-Cervonenkis combinatorial condition on classes of sets, others
with metric entropy. The implications between the various conditions are
considered. Bounds are given for the metric entropy of convex hulls in
Hilbert space.},
language = {English},
year = {1987},
publisher = {Institute of Mathematical Statistics},