Probability and Statistics Seminar at Boston University
Seminar on self-similar processes and Long-range dependence

Day:	Spring 2005: Mondays (Fall 2004: Tuesdays)	Click here for directions to the Boston University Department of Mathematics. Campus text map, gif map , and a general area map.
Time:	10am-noon
Place:	Room 135, Department of Mathematics, 111 Cummington St., Boston University

Go to the most recent scheduled talk.

This is a research oriented seminar, coordinated by Professor Murad Taqqu. The main theme, this year will be

In the academic year 2004-2005, we will focus on self-similar processes, long-range dependence and their applications, particularly to networking. The following book is a useful reference:

Paul Embrechts and Makoto Maejima Selfsimilar Processes , Princeton University Press, 2002.

We will have talks on other subjects as well.

In the fall, Professor Haya Kaspi from the Technion (Israel) will give a number of lectures. She will introduce and speak on local time, the Dynkin Isomorphism Theorem which relates symmetric Markov processes to Gaussian processes and on recent results involving infinite divisibility, for example, that the square of the self-similar process fractional Brownain motion is infinitely divisible when its index is in (0,1/2) but not when its index is in the interval (1/2,1).

This seminar has now become a regular feature of Boston University and is also attended by mathematicians, scientists and postdoctoral fellows in the greater Boston area. Announcements will be done by email and through my Web Page:

This fall semester the talks will usually be on Tuesdays (10-12).

Tuesday, September 21, 2004

Self-similarity and computer network traffic: an introduction

Murad S. Taqqu (Boston University)

Ethernet local area network traffic appears to be approximately statistically self-similar. This discovery, made about twelve years ago, has had a profound impact on the field. This is the first of a series of talks on this subject and on self-similarity in general which will be our main topic this year.extremes.

Tuesday, September 28, 2004

Heavy-tailed limits in computer network traffic

Murad S. Taqqu (Boston University)

We will see how heavy-tailed limits appear in computer network traffic setting. We will also describe how to represent stochastic processes that depend on Poisson noise, Gaussian noise and infinite variance stable noise.

Tuesday, October 5, 2004

Convergence to the Telecom process I

Murad S. Taqqu (Boston University)

We will define stochastic integrals with respect to stable and Poisson measures and describe the infinite source Poisson model.

Tuesday, October 12, 2004

Convergence to the Telecom process II

Murad S. Taqqu (Boston University)

We will look at properties of the infinite source Poisson model.

Tuesday, October 19, 2004

Convergence to the Telecom process III

Murad S. Taqqu (Boston University)

We will study the scaling limits of the infinite source Poisson model.

Tuesday, October 26, 2004

Convergence to the Telecom process IV

Murad S. Taqqu (Boston University)

We will study the scaling limits in the slow connection rate situation.

AND

Local time process of Brownian motion

Haya Kaspi (Technion and Boston University)

Professor Haya Kaspi will also start her series of lectures on local time of symmetric Markov processes and their connection to Gauussian processes. In this introductory lecture she will discuss local time process of Brownian motion.

Tuesday, November 2, 2004

Local Times of Symmetric Markov processes

Haya Kaspi (Technion and Boston University)

In this lecture we'll set the basic notation of symmetric (sometimes called time reversible) Markov processes. We shall define additive functionals and local times of symmetric processes and compute moments of their products. Towards the end of stating the Dynkin Isomorphism Theorem (DIT in the sequel) we shall associate with each symmetric process a Gaussian process and, if time permits, give a "guided tour" through the proof of the DIT.

The following paper:

"Sample path behaviour of the local times of strongly symmetric Markov processes via Gaussian processes" by M.J. Marcus and J. Rosen. Special Invited Paper Annals of Probability 1992 Vol.20, No.3 1603-1684.

is recommended reading for this lecture and the one that follows (both lectures will be devoted to the main ideas presented in this paper).

Tuesday, November 9, 2004

Dynkin Isomorphism Theorem

Haya Kaspi (Technion and Boston University)

In this lecture we shall state and give a guided tour through the proof of the Dynkin Isomorphism Theorem for symmetric Markov processes. We shall then use the theorem to study continuity and boundedness properties of the local time process (indexed by the states) of symmetric processes that have a local time at each point.

This lecture is based on; Sample Path properties of the Local Time of Strongly Symmetric Markov Processes Via Gaussian Processes by Michael Marcus and Jay Rosen, Special Invited Paper, The Annals of Probability(1992) Vol.20 No. 4, 1603-1684.

Thursday, November 11, 2004

Isomorphism Theorem for symmetric Markov processes

Haya Kaspi (Technion and Boston University)

In this lecture we'll discuss an Isomorphism Theorem for symmetric Markov processes stopped at the first time that their local time at a given state 0 exceeds t. This theorem extends the second Ray Knight Theorem, that we have discussed for the local time of the Brownian motion, to symmetric Markov processes. If time permits we shall discuss an application of the results to a study of the most visited site of a symmetric stable process.

This lecture is based on: A Ray Knight Theorem for Symmetric Markov Processes by Nathalie Eisenbaum, Haya Kaspi, Michael Marcus, Jay Rosen and Zhan Shi. The Annals of Probability(2000) Vol 28, No.4. 1781-1796.

Tuesday, November 30, 2004

Extension of the Ray-Knight Theorem

Haya Kaspi (Technion and Boston University)

We will start with an overview of Bessel processes. We will then extend the second Ray-Knight Theorem to a general symmetric Markov process and discuss some applications.

Recommended reading: A Ray Knight Theorem for Symmetric Markov Processes by Nathalie Eisenbaum, Haya Kaspi, Michael Marcus, Jay Rosen and Zhan Shi. The Annals of Probability(2000) Vol 28, No.4. 1781-1796.

Tuesday, December 7, 2004

When does a Gaussian process have an infinitely divisible square?

Haya Kaspi (Technion and Boston University)

We shall show that up to a multiplication by a constant function, a Gaussian process has an infinitely divisible square if and only if, its covariance is the potential density of a transient symmetric Markov process.

The lecture is based on the paper:

"A characterization of the infinitely divisible squared Gaussian processes", By Nathalie Eisenbaum and Haya Kaspi. You may download a preprint from Haya Kaspi's homepage at http://ie.technion.ac.il/iehaya/phtml

Tuesday, December 14, 2004

Self-similarity in natural images: the smoothness issue

Francois Roueff (Ecole Normale Superieure des Telecommunications, Paris)

Our primary goal here is to model smoothness in images. Two basic properties for modeling images are often considered: self-similarity and stationarity. A third property, the occlusion phenomenon, should also be taken into account as it is specific to the formation process of images. Although these properties are idealizations of the real world, we will rely on them for building a stochastic model and study its smoothness properties.

The presentation will be two parts:

First we will give a short introduction on the smoothness issue, on the one hand, for stationary self-similar stochastic processes and, on the other hand, for non-parametric statistics. In contrast with long range dependence, the smoothness properties rely on the small scale behavior. In the context of non-parametric statistical processing such as denoising an image, the performance and, sometimes, the method is related to the smoothness. Focusing on natural images, we will provide a short overview of statistical approaches for analysing their smoothness based on statistics such as level sets, jumps or wavelet coefficients. Most of empirical studies exhibit power laws at small scales, which reveals self-similarity in images.

Second we will introduce the standard dead leaves model which is obtained by superimposing random shapes on the plane. This model reproduces the occlusion phenomenon of images, that is the fact that, the objects appearing in an image hide each others depending on where they lie with respect to the camera. However, the standard dead leaves model does not allow a distribution of objects in accordance with empirical studies. More precisely, in order to reproduce the observed objects sizes distribution, a cutoff has to be introduced at small scales.

A limit of the model is thus considered when this cutoff tends to zero. The relevance of the limit field for image modeling is then discussed and its smoothness is studied.

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SPRING SEMESTER 2005

NOTE: During this spring semester, the seminars will take place on MONDAYS (usual time and place: 10-12 in MCS 135)

MONDAY, January 31, 2005

Asymptotic self-similarity and wavelet-based Hurst parameter estimation for stable processes an introduction

Stilian Stoev (Boston University)

We discuss two types of wavelet estimators for the Hurst long-range dependence parameter of Fractional Autoregressive Integrated Moving Average (FARIMA) time series with stable, infinite variance innovations. These estimators are semi-parametric in nature, robust to smooth trends and easy to compute. They are based on the asymptotic self-similarity property of the FARIMA time series, which is closely related to the Linear Fractional Stable Motion (LFSM). We establish the asymptotic normality of these estimators by controlling the rates of convergence of FARIMA to the LFSM. Numerical experiments show that these wavelet estimators perform very well in practice. They can be used, for example, to estimate the Hurst parameter of bursty Internet traffic traces.

MONDAY, February 7, 2005

Asymptotic self-similarity and wavelet-based Hurst parameter estimation for stable processes II

Stilian Stoev (Boston University)

We will start with the wavelet estimation.

MONDAY, February 14, 2005

Self-similarity and Lamperti's theorem

Murad S. Taqqu (Boston University)

We will focus on fractional Brownian motion and on Lamperti's theorem which associates a stationary process to each self-similar process. We will describe a multivariate version and state an open problem.

TUESDAY, February 22, 2005

Self-similarity and Lamperti's theorem II

Murad S. Taqqu (Boston University)

We will consider the integral representation of fractional Brownian motion and discuss Lamperti's theorem which associates a stationary process to each self-similar process. We will describe a multivariate version and state an open problem.

MONDAY, February 28, 2005

No Arbitrage with Transaction Costs and Fractional Brownian Motion

Paolo Guasoni (Boston University)

We establish a simple no-arbitrage criterion for continuous-time market models with transaction costs. We show that proportional transaction costs of any positive size eliminate arbitrage opportunities when asset prices follow either a Geometric Fractional Brownian Motion with Hurst parameter in (1/2, 1), or a general Markov process with regular points.

MONDAY, March 14, 2005

Arbitrage without frictions and no arbitrage with frictions for Fractional Brownian Motion

Paolo Guasoni (Boston University)

We shall discuss the issue of arbitrage for Fractional Brownian Motion models in frictionless markets, examining in details the arbitrage strategies of Rogers and others. Then we shall see how proportional transaction costs eliminate arbitrage possibilities from geometric Fractional Brownian Motion and for a more general class of processes.

MONDAY, March 21, 2005

TIME: 10:30 am - 12pm (MCS 135)

No arbitrage for Fractional Brownian Motion with transaction costs

Paolo Guasoni (Boston University)

We shall discuss how proportional transaction costs of any size eliminate arbitrage possibilities for Geometric Fractional Brownian Motion and for a more general class of processes.

MONDAY, March 28, 2005

TIME: 10:30 am - 12pm (MCS 135)

How rich is the class of multifractional Brownian motions?

Stilian Stoev (Boston University)

The multifractional Brownian motion (MBM) processes are locally self-similar Gaussian processes. They extend the classical fractional Brownian motion processes by allowing their self-similarity parameter 0< H <1 to depend on time. Two types of MBM processes were introduced independently by Peltier and Levy-Vehel (1995) and Benassi, Jaffard and Roux (1997) by using time-domain and frequency-domain integral representations of the fractional Brownian motion, respectively. Contrary to what has been stated in the literature, we show that these two types of processes have different correlation structures when the function H(t) is non-constant. We focus on a class of MBM processes, parametrized by (a1,a2), which contains the previously introduced two types of processes as special cases. We establish the connection between their time- and frequency-domain integral representations and obtain explicit expressions for their covariances. In addition, we show that there are non-constant functions H(t) for which the correlation structure of the MBM processes depends non-trivially on the value of (a1,a2) and hence, even for a given function H(t), there are an infinite number of MBM processes with essentially different distributions.

MONDAY, April 4, 2005

TIME: 10:30 am - 12pm (MCS 135)

How rich is the class of multifractional Brownian motions? II

Stilian Stoev (Boston University)

This will be the continuation of last's week talk.

MONDAY, April 11, 2005

TIME: 10:00 am - 12pm (MCS 135)

Overview of GARCH models

Jianing Di (Boston University)

GARCH type models have been applied in modelling the relation between conditional variance and asset rist premia. In the past 20 years there has been a vast quantity of research uncovering the properties of competing volatility models. I will give a brief review of those models and present some new ideas of random models with the comparison of results.

WEDNESDAY, April 20, 2005

TIME: 10:00 am - 12pm (MCS 135)

Linear Multifractional Stable Motions: Stochastic and Path Properties

Stilian Stoev (Boston University)

We study a family of locally self-similar processes with stable distributions, called Linear Multifractional Stable Motions (LMSM). These processes include, in particular, the classical Linear Fractional Stable Motion (LFSM) process X = {X(t)}, which has stationary increments and is self-similar with self-similarity parameter H, 0< H <1. The LMSM process Y = {Y(t)} is defined by replacing the self-similarity parameter H in the integral representation if the LFSM X by a deterministic function H(t), 0< H(t) < 1. We obtain necessary and sufficient conditions for the continuity in probability of the LMSM Y. We also examine the effect of the regularity of the function H(t) on the local structure of the process Y. We show, under certain Holder regularity conditions on H(t) that Y is locally equivalent to a LFSM process, that is, [ Y(ct + t0) - Y(t0) ]/ ( c^H(t0) ) converges, as c -> 0+, in the sense of the finite-dimensional distributions to a LFSM process {X(t)} with self-similarity parameter H(t0). We also study the path properties of the LMSM process Y. We show that the paths of Y can be bounded, only if 1/alpha < H(t) <1, where alpha is the index of stability if the process Y. We also show that Y has a version of continuous paths if 1/alpha < H(t) <1, provided that H(t) is sufficiently smooth. We establish upper and lower bounds on the point-wise Holder exponents of the paths of Y, when these paths are continuous.

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