Probability and Statistics Seminar at Boston University
TOPICS IN PROBABILITY

Day:	Tuesdays (Sometimes Thursdays)	Click here for directions to the Boston University Department of Mathematics. Campus text map, gif map , and a general area map.
Time:	10am-noon (sometimes earlier)
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 2005-2006, we will focus on various probability topics. These will be extracted from a new book by Allan Gut "Probability: a Graduate Course", Springer Verlag, 2005, ISBN 0-387-22833-0. The goal is to improve one's tools paraphenalia.

There will be talks by invited speakers as well.

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 but sometimes on Thursdays.

Tuesday, September 13, 2005

Power inequalities and applications to the evaluation of the asymptotic codifference of log-fractional stable noise.

Murad S. Taqqu (Boston University)

We will review some basic power inequalties and apply them to obtain the asymptotic behavior of the codifference for log-fractional stable noise. Log-fractional stable noise is a time series with infinite variance and hence the covariance cannot be used to describe its dependence structure. We will use the codifference instead, which is an extension of the covariance for infinite variance stable processes. We will show that the codifference decreases like a power function, so slowly that log-fractional stable noise displays long-range dependence. This is joint research with Joshua Levy.

THURSDAY, September 22, 2005

Large deviation for Markov processes and related analysis

Jin Feng (University of Massachusetts at Amherst)

Large deviation is a type of probabilistic limit theorem which describes atypical behaviors in the corresponding scaling limit. In the case of sequence of Markov processes, I will describe a method for characterizing large deviation in the path space through verifying convergence of some transformed generators. This method effectively connects the probabilistic large deviation issue with analytical tools and techniques in optimal control theroy, and the viscosity solution theory for Hamilton-Jacobi equations. The two subjects then can interact beneficially by offering intuitions from each side.

I plan to illustrate techniques through examples, which range from small random perturbations of ODEs (the Freidlin-Wentzell theory) to multi-scale averaging problems, in finite and infinite dimensions. Time permits, I will close up the talk by a biostatistics example which has significance in clinical trial applications.

Tuesday, September 27, 2005

Stationary processes I

Mamikon Ginovyan (Boston University)

We will define the basic characteristics of stochastic processes, and will state Kolmogorov's theorem on existence of a stochastic process with given family of finite-dimensional distribution functions. Then, we will consider some special classes of stochastic processes: second order processes, strictly and wide sense stationary processes, Gaussian processes. Some examples of stationary processes will also be presented. This is the first of a series of lectures.

THURSDAY, October 6, 2005

Stationary processes II

Mamikon Ginovyan (Boston University)

We will prove the existence of a Gaussian process with given covariance function. This is the second of a series of lectures.

Tuesday, October 11, 2005

Stationary processes III

Mamikon Ginovyan (Boston University)

We will consider the spectral representations of covariance functions of discrete- and continuous - parameter stationary processes (Herglotz's and Bochner-Khintchine theorems, respectively). The Hilbert spaces associated with given second order stationary processes will be introduced. This is the third of a series of lectures.

THURSDAY , October 20 , 2005 , 10:30 am

Stationary processes IV

Mamikon Ginovyan (Boston University)

We will consider Hilbert spaces associated with second order processes (the Hilbert space of random variables generated by the underlying process, the reproducing Hilbert space generated by the covariance function, and the Hilbert space generated by the spectral distribution function), and will establish isometric isomorphism between these spaces.

Then, we will formulate the linear least squares prediction problem for second order processes and will introduce the classes of deterministic and non-deterministic processes. We will discuss Cramer-Wold theorem on decomposition of a second order process into deterministic and non-deterministic components.

THURSDAY , October 27 , 2005 , 10:30 am

Stationary processes V

Mamikon Ginovyan (Boston University)

In this lecture we will define and study the basic properties of reproducing kernel Hilbert spaces (RKHS) . Then we will consider RKHS, generated by the covariance functions of second order processes.

Tuesday , November 1, 10:00 am

Stationary processes VI

Mamikon Ginovyan (Boston University)

Consider RKHS, generated by the covariance functions of second order processes. We will establish isometric isomorphism between the reproducing kernel Hilbert space (RKHS) and the Hilbert space of random variables generated by the underlying process.

Tuesday , November 8, 10:00 am

Stationary processes VII

Mamikon Ginovyan (Boston University)

We will define and study the basic properties of second order stochastic integrals with respect to an orthogonal increment process. Then we will establish an isometric isomorphism between the Hilbert space of random variables generated by the underlying process and the space generated by spectral distribution function . We will also prove the spectral representation theorem.

Tuesday , November 15, 10:00 am

Stationary processes VIII

Mamikon Ginovyan (Boston University)

We will consider some consequences of spectral representation theorem: uniqueness of the orthogonal increment process, the real-valued case, the continuous-parameter case, inversion formulae. Then we will consider linear transformations of stationary processes and will prove Kolmogorov's theorem on existence of a spectral density function.

Tuesday, November 29, 2005

Motivating Large Deviations

Scott Robertson (Boston University)

This talk will provide motivation and background materials needed for the study of large deviations of families of measures and their associated random variables. After introducing the topic via an example, a brief overview of many of the prerequisites crucial to understanding large deviations will be given. Following this overview, the general definitioin of a large deviations principle will be given and Cramer's theorem for real valued random variables will be proved. Time permitting, Schilder's Theorem and the Freidlin-Wentzell theory will be introduced via an example of using large deviations to solve a problem from Mathematical Finance.

Tuesday , December 6, 2005

Stationary processes IX

Mamikon Ginovyan (Boston University)

This will be the last lecture this semester.

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SPRING SEMESTER 2006

Tuesday, January 17, 2006

Wiener chaos and Wick powers: an introduction to multiple integrals with respect to Brownian motion

Murad Taqqu (Boston University)

Norbert Wiener, in the 1940s, wanted to develop a non-linear time series theory. The hope was that the success of the linear time series theory could be replicated in this more realistic setting. Wiener's approach was to start with independent Gaussian variables (white noise) and define non-linear transformations of these variables. The only restriction on these transformations was that the resulting random variables or processes possess a finite variance. Wiener was able to expand his potentially complicated functionals in an orthonormal basis involving Hermite polynomials or, more generally, multiple integrals with respect to Brownian motion. This expansion is now known as the Wiener chaos. The moments of these functionals can be calculated explicitely using so-called "Wick powers" and "Feynman diagrams".

This talk will introduce these notions, provide an overview, and will serve as background to subsequent talks by Anna Amirdjanova (University of Michigan, Ann Arbor) and Giovanni Peccati (Universite de Paris VI).

Tuesday, January 24, 2006

Multiple stochastic fractional integrals and their application to nonlinear filtering theory with fractional Brownian motion noise

Anna Amirdjanova (University of Michigan, Ann Arbor)

The talk will focus on the new properties of multiple Ito and Stratonovich type integrals with respect to persistent fractional Brownian motion (fBm) and their use in representation and approximation of the optimal filter in the context of nonlinear filtering with fBm observation noise. A couple of new results pertaining to the properties of multiple integrals with respect to a general Gaussian process will be also discussed.

Tuesday, January 31, 2006, TIME: 9:15 am - 11:15 am

On Delay Estimation and Testing for Stochastic Differential Equations

Yury Kutoyants (Universite du Maine, France)

We present a review of some results concerning delay estimation and tetsting by continuous time observations of solutions of stochastic differential equations in two asymptotics. The first one corresponds to small noise limit and the second to the large samples limit.. In both cases we describe the asymptotic properties of the maximum likelihood and bayesian estimators with a special attention to the asymptotic efficiency.We show that the first asymptotics corresponds to regular estimation problems and the second is similar to some non regular problems of classical statistics. We propose several asymptotically uniformly most powerful tests in the problem of hypotheses testing.

Tuesday, February 7, 2006, TIME: 9:15 am - 11:15 am

Stable convergence of multiple stochastic integrals, I: diagonal measures, cumulants and martingales

Giovanni Peccati (Universite de Paris VI, France)

In a series of recent works by D. Nualart, M. Taqqu, C.A. Tudor and myself (2004-2006), several necessary and sufficient conditions have been established, ensuring the stable convergence of multiple stochastic Wiener-Itô integrals towards mixtures of Gaussian distributions. After some general motivations and examples, I will discuss such convergence results from two different standpoints: (i) martingale theory (stable convergence of martingales);(ii) the general combinatorial approach to stochastic integration, developed by Rota and Wallstrom in a famous paper of 1997 (Annals of Probability 25(3), 1997, p. 1257-1283), with special attention to the notions of "diagonal sets" and "diagonal measures" associated to a random integrator.

Thursday, February 9, 2006, TIME: 9:15 am - 11:15 am

Stable convergence of multiple stochastic integrals, II: decoupling, filtrations on Gaussian spaces and applications to statistics

Giovanni Peccati (Universite de Paris VI, France)

In this talk, we explore the connections between limit theorems for multiple Wiener-Itô integrals, and a decoupling technique, known as "principle of conditioning", formalized by Jakubowski and al. during the eighties. We will also show that quite general theorems can be obtained by "constructing a filtration" on a given Gaussian space, and that such a construction is tightly related to a class of representation theorems, known as (abstract) Clark-Ocone formulae. Some applications to limit theorems arising in statistics (asymptotic theory of tests) will be also discussed. The core of the talk is based on joint papers with M. Taqqu, and with P. Deheuvels and M. Yor.

Tuesday, February 14, 2006, TIME: 9:15 am - 11:15 am

Various modes of convergence for random variables I

Jeff Hamrick (Boston University)

This will be a basic talk. We will review the various ways in which a sequence of random variables can be said to converge. After reminding ourselves of some basic definitions and notions, we will explore the interconnections between these various modes of convergence, and give proofs or counterexamples where appropriate. Special attention will be paid to the notions of uniform integrability and of weak convergence (Helly's selection theorem, tightness, the vague topology, et cetera).

THURSDAY, February 23, 2006, TIME: 9:15 am - 11:15 am

Various modes of convergence for random variables II

Jeff Hamrick (Boston University)

This will be a basic talk. We will continue the review of the various ways in which a sequence of random variables can be said to converge. After reminding ourselves of some basic definitions and notions, we will explore the interconnections between these various modes of convergence, and give proofs or counterexamples where appropriate. Special attention will be paid to the notions of uniform integrability and of weak convergence (Helly's selection theorem, tightness, the vague topology, et cetera).

THURSDAY, Thursday, March 2, 2006, TIME: 9:15 am - 11:15 am

Various modes of convergence for random variables III

Jeff Hamrick (Boston University)

We will continue the review of the various ways in which a sequence of random variables can be said to converge. After reminding ourselves of some basic definitions and notions, we will explore the interconnections between these various modes of convergence, and give proofs or counterexamples where appropriate. Special attention will be paid to the notions of uniform integrability and of weak convergence (Helly's selection theorem, tightness, the vague topology, et cetera).

Tuesday, March 14, 2006, TIME: 9:15 am - 11:15 am

Spatial Contagion in financial markets

Murad S. Taqqu (Boston University)

One commonly believes that international markets are more dependent during a crisis than they are are in more tranquil times. This extra dependence is often referred to as contagion between markets. We present a definition of contagion between financial markets based on local correlation and propose a test to detect the presence of contagion. The test does not require the specification of crisis and normal periods. As such, it avoids difficulties associated with testing for correlation breakdown, such as hand picking subsets of the data, and it provides a better understanding of the degree of dependence between financial markets. Using this test, we find evidence of contagion between developed and U.S. equity markets and evidence of flight to quality from the U.S. equity market to the U.S. government bond market. The practical application of the test involves nonparametric estimation techniques. This is joint work with Brendan Bradley.

Tuesday, March 28, 2006, TIME: 9:15 am - 11:15 am

Constructing Measures on Sigma Algebras Using Outer Measure and the Caratheodory Extension Theorem.

Scott Robertson (Boston University)

This will be a basic talk. We will give the necessary conditions on both the non-negative set function mu, as well as the class of sets, A, on which mu is defined that allow us to extend mu to a measure on the sigma algebra of sets generated by A. This process is known as the Caratheodory Extension and uses heavily the concepts of outer measure and measurability. After stating and proving the main extension theorem examples constructing specific measures will be given.

Tuesday, April 4, 2006, TIME: 9:15 am - 11:15 am

Conditional laws of semimartingales

Esko Valkeila (Helsinki University of Technology)

The theory of initial enlargement of filtrations can be applied to model asymmetric information in pricing models. We explain how some of the results in the initial enlargement can be obtained from conditional laws of semimartingales. The approach based on conditional laws also connects the theory of initial enlargements to the notion of the weak information by Fabrice Baudoin. We also show how progressive inlargement can be sometimes obtained from initial enlargement by filtration shrinkage.

The talk is is based on joint work with Dario Gasbarra (Helsinki) and Lioudmila Vostrikova (Angers).

Thursday, April 6, 2006, TIME: 9:15 am - 11:15 am

Charaterization of Brownian motion and fractional Brownian motion

Esko Valkeila (Helsinki University of Technology)

A continuous process X is a Brownian motion if and only if X is a martingale and the process X^2_t - t is a martingale. We extend this charaterization to fractional Brownian motion. One part of the extension is based on weighted quadratic variation. We will also give a direct proof that fractional Brownian motionis not a semimartingale based on related notion.

The talk is based on joint work with Yulia Mishura (Kiev).

Tuesday, April 11, 2006, TIME: 9:15 am - 11:15 am

Weak convergence of n-particle systems using bilinear forms I

Joerg-Uwe Loebus (University of New Hampshire)

Symmetric Markov processes can be characterized by means of Dirichlet forms. Mosco convergence of the forms is equivalent to strong convergence of the resolvents of the processes. Additional conditions even guarantee weak convergence of the Markov processes in the Skorohod space. The first part of the talk will be devoted to introducing this theory. The second part presents an extension designed to establish weak convergence of n-particle systems to stationary paths. Applications will be discussed.

Tuesday, April 18, 2006, TIME: 9:15 am - 11:15 am

Weak convergence of n-particle systems using bilinear forms II

Joerg-Uwe Loebus (University of New Hampshire)

We will discuss Dirichlet forms, associated Markov semigroups and the Mosco convergence of Dirichlet forms.

Tuesday, April 25, 2006, TIME: 9:15 am - 11:15 am

Weak convergence of n-particle systems using bilinear forms III

Joerg-Uwe Loebus (University of New Hampshire)

We will consider applications to the weak convergence of n-particle systems.

Go to the top of the list for time and place.