Thomas Stieltjes Institute for Mathematics

Title:	Course Large Deviations and Course Introductory Course to Time Series Analysis
Organisation	AIO-netwerk Stochastiek zie ook: http://www.cs.vu.nl/~aad/aio.html (A.W. van der Vaart (aad@cs.vu.nl), tel: 020-4447697, J.H.J. Einmahl ( einmahl@win.tue.nl), tel: 040-2472499).
Lecturers:	Frank den Hollander (first course) Thomas Mikosch (second course)
Time and place:	Fall 97 We have planned the dates: September 5 and 19, October 3, 17 and 31*, November 14 and 28, December 12. (*The probability session of 31 October will be moved to 21 November, in Lunteren.) The morning session will be probability and the afternoon session will be statistics. As usual the courses consist of eight lectures of two hours each, given on eight Fridays. The courses will probably be given in Utrecht, in the Academie-gebouw, Domplein, room 10. Morning and afternoon sessions are planned for 10.45-12.30 and 13.30-15.15, respectively. A final announcement will follow.

Abstract: Frank den Hollander (KUN) (denholla@sci.kun.nl)

The theory of large deviations deals with the computation of `small' probabilities, away from the central limit theorem. The reason for studying these probabilities is that they come up in a wide variety of situations, e.g. the study of functionals of random processes, the calculation of tail properties for sums of weakly dependent random variables, error estimates, etc. Large deviation theory has proved to be the crucial tool in handling many questions in statistics, probability and statistical mechanics.

The course will have three parts:

1. Exposition of the foundations of the theory, illustrated for i.i.d. sequences and Markov sequences (3 lectures).
2. Description of the general framework of the theory: large deviation principle, Varadhan's theorem, contraction principle, etc. (2 lectures).
3. Discussion of some applications: hypothesis testing, conditioned limit theorems, random walks in random media, Curie-Weiss model for ferromagnetism, polymers.

References:
J.D. Deuschel and D.W. Stroock, Large Deviations, Academic Press, Boston, 1989.
J.A. Bucklew, Large Deviation Techniques in Decision, Simulation, and Estimation, Wiley, New York, 1990.
A. Dembo and O. Zeitouni, Large Deviation Techniques and Applications, Jones and Bartlett Publishers, Boston, 1993.

Abstract: Thomas Mikosch (RUG) (mikosch@math.rug.nl)

By now, time series analysis is one of the well established theories in probability theory and mathematical statistics with applications in various fields such as hydrology, meteorology, finance, astronomy, economics. The stochastic model underlying it is mostly assumed to be a stationary discrete--time process. The aim of the analysis is to detect the dependence structure in the series, to fit an appriate model and to predict future values in the series. This is done in the time or frequency domain. The first method is based on considerations of the autocorrelations and autocovariances. The second one uses Fourier analytic methods; it is also called spectral analysis.

The analysis of time series is mostly based on Hilbert space techniques. They allow one to represent a stationary process as a stochastic integral with trigonometric functions as integrands. Thus a stationary process, to a first approximation, can be understood as a sinosoid with random coefficients. The order of magnitude of those coefficients is discribed by the spectral distribution function and its density. The course will give a gentle introduction to modelling of time series both in the frequency and time domain, stressing more the understanding of the models than the discussion of technical details. Classical ARMA processes (=autoregressive moving average processes) allow us to give insight into the structure of more complicated models. ARMA models will serve as prime examples for developing a spectral and time domain analysis which is accompanied by an introduction to the statistical tools for fitting data. Prediction of times series is one of the main issues and will be described in detail.

In addition to the basics of classical time series modelling the following recent issues of interest will be discussed:

• Modelling of non-stationary time series. For example, how can we deal with time series if there is a trend or seasonality in it?
• Heavy-tailed time series. This topic has recently attracted much attention, mainly in financial times series modelling where observations fluctuate wildly and can assume very large values. This topic is closely related to extreme value theory.
• Long-range dependent models. This is another topic which has been condidered over the last few years after observing that some real-life data seem to contain some very strong dependencies which cannot be explained by classical methods. Such phenomenon has been considered for example in teletraffic data (world wide web) and hydrology.
• Non-linear time series. Linear models (such as ARMA processes) cannot explain the rapid change of volatility for example in daily financial log--returns. Specific non--linear models were developed for describing such effects. The topic of non--linear models also includes the so--called deterministic or chaotic time series which gained a certain popularity over the last few years.

Recommended references are:
Brockwell, P.J. and Davis, R.A. (1991) {Time Series: Theory and Methods,2nd~edition. Springer, New York.
Brockwell, P.J. and Davis, R.A. (1996) { Introduction to Time Series and Forecasting.} Springer, New York.
The first reference is an introduction to time series analysis for a mathematically oriented person, the second one is for the more applied person who is interested in how things work, but does not ask why.

References:
J.C. van Houwelingen, S. le Cessie, Predictive value of statistical models, Statistics in Medicine 9, 1303-1325, 1990.
S. le Cessie, H.C. van Houwelingen, Testing the fit of a regression model via score tests in random effects models, Biometrics 51, 600-614, 1995
H.C. van Houwelingen, J. Thorogood, Construction, validation and updating of a prognostic model for kidney graft survival, Statistics in Medicine 14, 1999-2008, 1995
P.J.M. Verweij, H.C. van Houwelingen, Cross-validation in survvial analysis, Statistics in Medicine 12, 2305-2314, 1993
J.J. Howing-Duistermaat, B.H.F. Derkx, F.R. Rosendaal, H.C. van Houwelingen, Testing familial aggregation, Biometrics 51, 1292-1301, 1995

Abstract: Gerard Hooghiemstra (TUD) (G.Hooghiemstra@twi.tudelft.nl)

This course is on Brownian motion and the Functional Central Limit Theorem. (FCLT).
We start with an introduction to Brownian motion involving: a full proof of continuity of paths, reflection principle, calculation of distributions of funtionals (by various methods, including exponential martingales). As reference we take the book of Freedman.
In connection with the above topic I intend to treat two FCLT's with applications:

• A Central limit theorem for Martingale-difference arrays and
• A Central limit theorem using strong mixing and the Lindeberg condition. As application of this second theorem I we treat : The asymptotic normality of the L_1 distance between the Grenander estimator and the true density in the monotone case.

References:
Freedman, D. Brownian Motion and Diffusion. Holden Day, 1971, ISBN 0-8162-3024-2
Pollard, D. Convergence of Stochastic Processes. Springer Verlag , 1984, ISBN 0-387-90990-7
Groeneboom, P., Hooghiemstra, G. and Lopuhaa, H. P. Asymptotic normality of the L_1 error of the Grenander estimator. Report 97-01, TWI, TUDELFT.