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Mathematics & Statistics

# Probability and Related Fields (Leader: Prof. Arunava Mukherjea)

## Wednesday, April 20, 2005

Title
Speaker
Time
Place

Toeplitz matrices, Part III
Edgardo Cureg
4:00pm-5:00pm
PHY 013

## Wednesday, April 13, 2005

Title
Speaker
Time
Place

An Application of Curtiss' Continuity Theorem to Random Integer Partitions
Lyuben Mutafchiev
4:30pm-5:30pm
PHY 013

## Wednesday, April 6, 2005

Title
Speaker
Time
Place

Cybenko's results on approximation by superpositions of a sigmoidal function, Part II
Dmitri Prokhorov
4:00pm-5:00pm
PHY 013

## Wednesday, March 30, 2005

Title
Speaker
Time
Place

Cybenko's results on approximation by superpositions of a sigmoidal function, Part I
Dmitri Prokhorov
4:00pm-5:00pm
PHY 013

## Wednesday, March 23, 2005

Title
Speaker
Time
Place

TBA
Norbert Youmbi
4:00pm-5:00pm
PHY 013

Abstract

The statement $$XY$$ and $$Y$$ have the same distribution, where $$X$$ and $$Y$$ are two independent $$S$$-valued random variables, is well-understood when $$S$$ is a (multiplicative) group. An equivalent problem is one of studying the Choquet convolution equation $$P*Q=Q$$ for probability measures $$P$$ and $$Q$$. We'll consider this question when $$S$$ is a hypergroup. (Concepts such as hypergroups and convolutions in hypergroups will be introduced first.)

## Wednesday, March 9, 2005

Title
Speaker
Time
Place

Levy continuity theorem on moment generating functions
Arunava Mukherjea
4:30pm-5:30pm
PHY 013

Abstract

Many graduate probability texts contain this theorem. The most general version (see J. H. Curtiss, A note on the theory of moment generating functions, Ann. Math. Stat. 13, 1942) available in printed form is: If a sequence of mgfs converges in an interval CONTAINING 0, then it must converge uniformly in every closed subinterval of that interval, and the limit function must, itself, be a mgf. Furthermore, the corresponding sequence of distribution functions must converge weakly to the distribution function that corresponds to the limiting mgf.

## Wednesday, March 2, 2005

Title
Speaker

Time
Place

Weak Limits
M. Rao
Department of Mathematics
University of Florida
4:00pm-5:00pm
PHY 013

Abstract

Weak convergence of measures is a concept of great importance in Probability Theory. The Central Limit Theorem is just one example. In this preliminary note, we will discuss weak convergence of measures with "boundary" conditions.

## Wednesday, February 16, 2005

Title
Speaker
Time
Place

Toeplitz matrices, Part II
Edgardo Cureg
4:00pm-5:00pm
PHY 013

Abstract

We will establish asymptotic equivalence between Toeplitz and circulant matrices.

## Wednesday, February 9, 2005

Title
Speaker
Time
Place

Toeplitz matrices, Part I
Edgardo Cureg
4:00pm-5:00pm
PHY 013

Abstract

Examples of such matrices are covariance matrices of weakly stationary stochastic time series. The aim of the talk is to relate these matrices to their simpler, more structured cousin — the circulant matrices.

## Wednesday, February 2, 2005

Title
Speaker
Time
Place

Identification of the parameters by knowing the minimum, Part III
John C. Davis, III
4:00pm-5:00pm
PHY 013

## Wednesday, January 26, 2005

Title
Speaker
Time
Place

Multivariate Analysis, Part II
John C. Davis, III
4:00pm-5:00pm
PHY 013

Abstract

Let $$X$$ be an $$n$$-variate non-singular normal vector whose parameters are not known. However, the pdf of $$Y$$, the minimum of the entries of $$X$$, is known. Is it then possible to identify the parameters knowing only this pdf? This general problem, though relevant in numerous practical contexts, has remained unsolved for many years. A special case, when all the correlations are negative, will be discussed.

For practical examples, think of (Supply, Demand) as an unknown bivariate normal, where you actually observe the minimum, the actual amount passing from the sellers to the buyers. You can also think of a machine with multiple parts where the survival times of the parts is a unknown multivariate normal; in case this machine fails as soon as one of its parts fails, then again you know only the minimum of the survival times.

## Wednesday, January 19, 2005

Title
Speaker
Time
Place

Multivariate Analysis, Part I
Arunava Mukherjea
4:00pm-5:00pm
PHY 013