MaPhySto
Centre for Mathematical Physics and Stochastics
Department of Mathematical Sciences, University of Aarhus

## Empirical Processes

From Monday, August 9, 1999 to Friday, August 20, 1999 MaPhySto organized a summer school on Empirical Processes. There were 4 series of main lectures with the following content:

1. Uniform Central Limit Theorems by R. M. Dudley (MIT)

Summary: One of the main topics of empirical process theory is asymptotic normality of suitably normalized partial sums uniformly over classes of sets and functions. For the uniform convergence to hold there must be a limiting Gaussian process with sample continuity and boundedness. First, these properties of Gaussian processes will be treated in terms of metric entropy and the Talagrand-Fernique majorizing measure theorem. Then, combinatorial properties sufficient for uniform central limit theorems uniformly over all underlying probability measures will be studied. A good property for families of sets is finiteness of the Vapnik-Chervonenkis or VC index, also studied in computer learning theory. The VC property has various extensions to families of functions. Another useful property is bracketing, where families of functions are covered by unions of brackets [f_i,g_i], where [f,g] is the set of measurable functions h with f =< h =< g, and there are suitable bounds of the number of brackets in relation to some distance between f_i and g_i in mean or mean square.
Some of the lectures was based on parts of a book by the author, also called Uniform Central Limit Theorems published by Cambridge University Press.

The notes for these lectures may be fetched in various formats.

2. Empirical Processes at Work in Statistics by A.W. Van der Vaart (Amsterdam) & J.A. Wellner (Seattle)

Summary: The lectures of Van der Vaart and Wellner will focus on the use of empirical process methods in dealing with a variety of questions and problems in statistics. Our examples and applications will be drawn from problems concerning semi-parametric models and non-parametric estimation for inverse problems. We will begin with a review of bounds for suprema of empirical processes, and will then discuss uses of these bounds in establishing:

1. consistency of M- and Z-estimators;
2. rates of convergence;
3. convergence in distribution of maximum likelihood, sieved and penalized maximum likelihood estimators

3. Empirical and Partial-sum Processes Revisited as Random Measure Processes by P. Gänssler (Munich)

Summary: In a general framework of so-called random measure processes (RMP's) we present uniform laws of large numbers (ULLN) and functional central limit theorems (FCLT) for RMP's yielding known and also new results for empirical processes and for so-called smoothed empirical processes based on data in general sample spaces. At the same time one obtains results for partial-sum processes with either fixed or random locations. Proofs are based on tools from modern empirical process theory as presented e.g. in Van der Vaart and Wellner [(1996): Weak Convergence and Empirical Processes; Springer Series in Statistics]. Our presentation will be also guided by showing up some aspects of the development of empirical process theory from its classical origin up to the present which offers now a wide variety of applications in statistics as demonstrated e.g. in Part 3 of Van der Vaart and Wellner [1996].

4. Convergence in Law of Random Elements and Sets by J. Hoffmann-Jørgensen (Aarhus)

Summary: The classical definition of convergence in law of random elements is founded on convergence of the upper expectation of continuous functions. This concept has served very well in the theory of law convergence of empirical processes when the underlying topological space is metrizable or at least has sufficiently many continuous functions. However, in the context of law convergence of random sets associated to empirical processes (e.g. zero-sets or max-sets), the concept trivializes because the natural topology (the upper Fell topology) has no non-constant continuous functions. In the lectures I shall present a new concept of law convergence (convergence in Borel law) which coincides with the classical definition in ``nice'' topological spaces, and I shall demonstrate how this concept provides sensible limit theorems for random sets. In particular, we shall derive new and old results for law convergence of a certain class of estimators (J-estimators) which includes zero estimators and maximum estimators.

### Final Schedule

#### Lectures

The Summer School took place in Aarhus at the Department of Mathematical Sciences, University of Aarhus..

First Week
Time Monday Tuesday Wednesday Thursday Friday
9.00-10.00 Registration JAW JAW PG JAW
10.00-10.10 OEBN: Welcome
10.20-11.20 JHJ RMD JAW PG RMD
11.30-12.30 JHJ RMD Eclipse JHJ RMD
12.30-14.00 Lunch
14.30-15.30 RMD JHJ PG VDP VD
15.45-16.15 FB GP (1 hour) DM TS
16.30-17.00 AB RB

Second week
Time Monday Tuesday Wednesday Thursday Friday
9.00-10.00 AVV JHJ AVV AVV PG
10.20-11.20 AVV PG RMD AVV PG
11.30-12.30 JHJ PG RMD RMD JHJ
12.30-14.00 Lunch
14.30-15.30 WS SVG SEG MBH
15.45-16.15 LM MP MS RMC
16.30-17.00 RH AB

Legend:

AB:= A. Bufetov
AVV:= A. Van der Vaart
DM:= D. Marinucci
FB:= F. Bravo
GP:= G. Peskir
JAW:= J.A. Wellner
JHJ:= J. Hoffmann-Jørgensen
LM:= L. Menneteau
MBH:= M.B. Hansen
RH:= R. Huntsinger
MP:= M. Piccioni
MS:=M. Scavino
OEBN:= O.E. Barndorff-Nielsen
PG:= P. Gänssler
RB:= R. Bilba
RMC:= R. McCrorie
RMD:= R.M. Dudley
SEG:= S.E. Graversen
SVG:= S. van der Geer
TS: T. Schreiber
VD:= V. Dobric
VDP:= V. de la Pena
WS:= W. Stute

Registered participants
(Michigan State University)
• Ole E. Barndorff-Nielsen
(University of Aarhus)
• Luisa Beghin
("La Sapienza", Rome)
• Cecilia Elena Bilba
(University of Iasi, Romania)
("George Bacovia" University, Romania)
• Francesco Bravo
(University of Southampton)
• Alexander Bufetov
(Independent University of Moscow)
• Annalisa Cerquetti
(University of Firenze)
• Chong-Cheul Cheong
(University of Southampton)
• Pier Luigi Conti
(University of Bologna)
• Jose Luis Batun Cutz
(CIMAT, Mexico)
• Omar El-Dakkak
(Milan)
(Lehigh University)
• Richard M. Dudley
(MIT)
• Sara van de Geer
(Leiden University)
• Tue Gorgens
(University of New South Wales)
• Aurea Grané
(University of Barcelona)
• Jorge Graneri
(Centro de Matematica, Uruguay)
• Svend Erik Graversen
(University of Aarhus)
• Nora Gürtler
(University of Karlsruhe)
• Peter Gänssler
(LMU Munich)
• Kasper Daniel Hansen
(University of Copenhagen)
• Martin Bøgsted Hansen
(Aalborg University)
• Niels Væver Hartvig
(University of Aarhus)
• Daniel Hlubinka
(Prague University)
• Reid Huntsinger
(InfoWorks, Chicago)
• Wenjiang Jiang
(University of Aarhus)
• Sven Jesper Knudsen
(University of Southern Denmark)
• Lars Korsholm
(University of Southern Denmark)
• Rodrigo Labouriau
(Danish Institute of Agricultural Science)
• Domenico Marinucci
("La Sapienza", Rome)
• Bo Markussen
(University of Copenhagen)
• Roderick McCrorie
(LSE, London)
• Ludovic Menneteau
(University of Paris VI)
• Marco Perone Pacifico
("La Sapienza", Rome)
• Jan Parner
(University of Copenhagen)
• Paolo Paruolo
(University of Bologna)
• Jesper Lund Pedersen
(University of Aarhus)
• Victor de la Pena
(Columbia University)
• Goran Peskir
(University of Aarhus)
• Mauro Piccioni
(University of L'Aquila)
• Silvia Polettini
("La Sapienza", Rome)
• Alina Posirca
(Michigan State University)
• Daniel Rost
(LMU Munich)
• Birgitte Rønn
(Royal Veterinary and Agricultural University of Denmark)
• Marco Scavino
(Centro de Matematica, Uruguay)
• Tomasz Schreiber
(Nicholas Copernicus University, Poland)
• Ingo Steinke
(University of Rostock)
• Winfried Stute
(University of Giessen)
• Luca Tardella
("La Sapienza", Rome)
(Vrije Universiteit, Amsterdam)
• Jon. A. Wellner
(University of Washington)
• Allan Würtz
(University of Aarhus)
• Paolo Zaffaroni
(Banca d'Italia, Rome)

### Pictures from the Summer School!

Please make further inquiries to MaPhySto (`maphysto@maphysto.dk`) or to the organizer Jørgen Hoffmann-Jørgensen (`hoff@imf.au.dk`).