Symmetric Functionals on Random Matrices and Random Matchings Problems

This book is drawn from the recent literature on the asymptotic behavior of random permanents and random matchings. In particular, the authors present an elegant connection between the problem of an asymptotic behavior for a certain family of functionals on random matrices and the asymptotic results in the classical theory of the so-called U-statistics -- objects of fundamental importance in the non-parametric statistical inference.

This book is self-contained and accessible to any mathematics, statistics or engineering graduate student who has taken basic introductory courses in probability theory and mathematical statistics.

Dr.Grzegorz A. Rempala is a Professor of Statistics in the Department of Mathematics at the University of Louisville in Louisville, KY. Dr. Jacek Wesolowski is a Professor of Mathematics and Associate Dean for Research at the Faculty of Mathematics and Information Science, Warsaw University of Technology in Warsaw, Poland.

The volume is a result of the authors' collaborative effort initiated at the IMA during the Institute's 2003/04 annual program on "Probability and Statistics in Complex Systems: Genomics, Networks, and Finance Engineering".

Résumé
This IMA Volume in Mathematics and its Applications SYMMETRIC FUNCTIONALS ON RANDOM MATRICES AND RANDOM MATCHINGS PROBLEMS During the academic year 20032004, the Institute for Mathematics and its Applications (IMA) held a thematic program on Probability and Stat- tics in Complex Systems. The program focused on large complex systems in which stochasticity plays a signi?cant role. Such systems are very diverse, and the IMA program emphasized systems as varied as the human genome, the internet, and world ?nancial markets. Although quite di?erent, these s- tems have features in common, such as multitudes of interconnecting parts and the availability of large amounts of high-dimensional noisy data. The programemphasized the development and application of common mathem- ical and computational techniques to model and analyze such systems. About 1,000 mathematicians, statisticians, scientists, and engineers participated in the IMA thematic program, including about 50 who were in residence at the IMA during much or all of the year. The present volume was born during the 20032004 thematic program at the IMA. The two authors were visitors to the IMA during the program year, withthe?rstauthorresidentforthe entiretenmonths. Thisvolumeisaresult of the authors' interactions at the IMA, and, especially their discussions with the many other program participants in the program and their involvement in the numerous tutorials, workshops,and seminars held during the year.

Contenu
Basic Concepts.- Properties of P-statistics.- Asymptotics for Random Permanents.- Weak Convergence of Permanent Processes.- Weak Convergence of P-statistics.- Permanent Designs and Related Topics.- Products of Partial Sums and Wishart Determinants.