- Dudley, R. M. (1973) Sample Functions of the Gaussian Process
- 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}, copyright = {Copyright © 1978 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}, copyright = {Copyright © 1979 Institute of Mathematical Statistics}, }

- Dudley, R. M. (1987) Universal Donsker Classes and Metric Entropy

@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}, copyright = {Copyright © 1973 Institute of Mathematical Statistics}, }

@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}, copyright = {Copyright © 1987 Institute of Mathematical Statistics}, }