## 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.