## Projects for Statistics 251/551

Here are some possible topics, with some references to get you started. Please consult with DP about your project before you get too involved in any particular topic.

An attached name means that someone has already begun work on a topic. In principle, several students might work on the same topic, but are expected to work independently of each other.

Card shuffling [Dan G.]
How many riffle shuffles are needed to "thoroughly shuffle" a deck of cards? Reference: Aldous and Diaconis, Shuffling cards and stopping times, American Math. Monthly 93 (1986), 333-348.
Markov chain Monte Carlo
Find a paper from the web site http://www.stats.bris.ac.uk/MCMC/
Rayleigh's monotonicity law
Give a purely probabilistic explanation of the result described in Chapter 4 of Doyle and Snell, Random walks and electric networks. Try to use the methods related to cycles as descibed in the Lectures. Do not appeal to electricity.
Bayesian restoration of images
Explain what Geman and Geman (Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images, IEEE Transactions on Pattern Analysis and Machine Intelligence PAMI-6: 721-741) were doing. Possible tasks: implement their method; or prove connection between Markov random fields and Gibbs distributions.
Rates of convergence to stationary distributions
Write a short account of how existence of higher moments of recurrence times in an aperiodic, irreducible Markov chain gives arate of convergence in total variation distance to the stationary distribution. Reference: Pitman, Uniform rates of convergence for markov chain transition probabilities, Z. Wahrscheinlichkeitstheorie verw. Gebiete 29 (1974) 193-227.
Girsanov's theorem [Dan K.]
Produce a semi-technical explanation for how the Balck-Scholes formula can be derived by means of an equivalent martingale measure.
Simulated annealing
Get it to work on a specific problem. reference: Kirkpatrick, Gelatt, and Vecchi, Optimization by simulated annealing, Science 220 (1983) 671-680.