## Probability and Statistics

Department of
Mathematics and Statistics

Research interests of faculty members in probability and statistics include probability on finite groups and other discrete structures, ergodic theory, and Bayesian inference.

### Martin Hildebrand

"My research interests involve probability on finite groups and other discrete structures. One of my papers answered a question which had been open for about 20 years regarding a random process on the integers modulo an odd number p. Start at 0, and at each step of the random process, multiply the previous position of the random process by 2 and then add 1, 0, or -1 chosen at random independent of previous choices. How long does it take for the random process to be close to uniformly distributed on the integers modulo p? This paper showed that it takes at least c \log_2 p steps for some constant c>1.

"I also am interested in ``random random walks" on finite groups. Here the specifications for one step of a random walk are chosen at random (e.g., by choosing at random from all sets of k elements of the group), and then a random walk is performed. For a typical choice of the specification, how long does it take for the random walk to get close to uniformly distributed.

"I also have some interests in combinatorics.

"My Ph.D. students have worked in areas involving probability on finite groups. Joseph McCollum, whose Ph.D. thesis involved random walks on the dihedral group as well as some results involving random random walks on finite abelian groups, has had several papers published and works at Siena College in the Department of Quantitative Business Analysis. Richard Neville, whose Ph.D. thesis extended the lower bounds on the Chung--Diaconis--Graham random process, now teaches at Dean College in Massachusetts. Scott Bianco, whose Ph.D. thesis involved a generalization of the Chung--Diaconis--Graham random process, currently is in a visiting position at Siena College in its Department of Quantitative Business Analysis."

### Karin Reinhold

"My research is in Ergodic Theory with connections to Probability and Harmonic Analysis. I am interested in convergence of a variety of ergodic-type of averages, including random ergodic theorems and averages given by convolutions. My plans for the future include problems in time-frequency analysis for convolution averages and explorations of other topics in applications of probability."

### Carlos Rodriguez

My general area of research is Mathematical Data Analysis. I have produced original results in Density Estimation, Monte Carlo, Quantum Probability, Entropic Priors and Information Geometry. I have also studied the geometry of probability models for sequences of bits (bitnets), where I have found that the average Ricci scalar, for all the models that I have been able to compute, to be always half an integer. I have discovered a new kind of boundary associated to any smooth (regualar) parametric probabilistic model encoding an objective notion of ignorance that produces the general classes of priors that actually work in practice. I have found finite length confidence intervals for the SD of a 1D gaussian observation x, e.g. (0,17|x|) has probability at least 95% of covering the true SD.