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Title: Toeplitz matrices, Part III Speaker: Edgardo Cureg Time: 4:00pm–5:00pm Place: PHY 013

Title: An Application of Curtiss' Continuity Theorem to Random Integer Partitions Speaker: Lyuben Mutafchiev Time: 4:30pm–5:30pm Place: PHY 013

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

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

Title: TBA Speaker: Norbert Youmbi Time: 4:00pm–5:00pm Place: PHY 013

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.)

Title: Levy continuity theorem on moment generating functions Speaker: Arunava Mukherjea Time: 4:30pm–5:30pm Place: PHY 013

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.

Title: Weak Limits Speaker: M. Rao, Department of Mathematics University of Florida Time: 4:00pm–5:00pm Place: PHY 013

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.

Title: Toeplitz matrices, Part II Speaker: Edgardo Cureg Time: 4:00pm–5:00pm Place: PHY 013

We will establish asymptotic equivalence between Toeplitz and circulant matrices.

Title: Toeplitz matrices, Part I Speaker: Edgardo Cureg Time: 4:00pm–5:00pm Place: PHY 013

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.

Title: Identification of the parameters by knowing the minimum, Part III Speaker: John C. Davis, III Time: 4:00pm–5:00pm Place: PHY 013

Title: Multivariate Analysis, Part II Speaker: John C. Davis, III Time: 4:00pm–5:00pm Place: PHY 013

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.

Title: Multivariate Analysis, Part I Speaker: John C. Davis, III Time: 4:00pm–5:00pm Place: PHY 013