Abstract:
 (1) The decentralized quickest change detection problem is studied in sensor networks, where a set of sensors receive observations from a hidden Markov model ${\bf X}$ and send sensor messages to a central processor, called the fusion center, which makes a final decision when observations are stopped. It is assumed that the parameter $\theta$ in the hidden Markov model for ${\bf X}$ changes from $\theta_0$ to $\theta_1$ at some unknown time. The problem is to determine the policies at the sensor and fusion center levels to jointly optimize the detection delay subject to the average run length (ARL) to false alarm constraint. In this article, a CUSUMtype fusion rule with stationary binary sensor messages is studied and a simple method for choosing the optimal local sensor thresholds is introduced. Further research is also given. This is a joint work with Dr. Yajun Mei
(2) The idea of cutoff phenomenon was introduced by D. Aldous and P. Diaconis to capture the fact that some ergodic Markov chains converge abruptly to their invariant distributions. Such a phenomenon is closely related to the mixing time behavior which can be sensitive to measurement mechanics on the convergence of Markov chains to their stationarity. The first example where a cutoff in total variation was proved is the random transposition Markov chain on the symmetric group studied by Diaconis and Shahshahani in 1981. One of the most precise and interesting cutoff result concerns repeated riffle shuffles which was proved by D. Aldous in 1983 and improved by D. Bayer and P. Diaconis in 1992. In this talk, we consider families of reversible Markov processes with the assumption on the existence of infinitesimal generators and reversible probability measures. This includes classical examples such as families of ergodic finite Markov chains and Brownian motions on families of compact Riemannian manifolds. We will display criteria on the existence of an $L^2$cutoff and formulas on the $L^2$mixing time in terms of spectral information of the infinitesimal generators. Wellknown examples, such as BirthandDeath chains and Ehrenfest processes, will be discussed.
