Gibbs measures asymptotics

Let (Ω, B, ν) be a measure space and H:Ω→ R+ be B measurable. Let ∫ Ωe-Hdν < ∞. For 0 < T < 1 let μH,T (·) be the probability measure defined by μH,T (A) = (∫A e-H/T dν)/(∫Ω e-H/T dν ), A ∈ B. In this paper, we study the behavior of μH,T (·) as T ↓ 0 and extend the results of Hwang (1980, 1981). When Ω is R and H achieves its minimum at a single value x0 (single well case) and H(·) is Holder continuous at x0 of order α, it is shown that if XT is a random variable with probability distribution μH,T (·) then as T ↓ 0, i) XT→ x0 in probability; ii) (Xt - x0)T-1/α converges in distribution to an absolutely continuous symmetric distribution with density proportional to e-cα|x|α for some 0 < cα < ∞. This is extended to the case when H achieves its minimum at a finite number of points (multiple well case). An extension of these results to the case H : Rn→ R+ is also outlined

On the Geometrical Convergence of Gibbs Sampler inRd

AbstractThe geometrical convergence of the Gibbs sampler for simulating a probability distribution inRdis proved. The distribution has a density which is a bounded perturbation of a log-concave function and satisfies some growth conditions. The analysis is based on a representation of the Gibbs sampler and some powerful results from the theory of Harris recurrent Markov chains

Anticipating Exponential Processes and Stochastic Differential Equations

Exponential processes in the Ito theory of stochastic integration can be viewed in three aspects: multiplicative renormalization, martingales, and stochastic differential equations. In this paper we initiate the study of anticipating exponential processes from these aspects viewpoints. The analogue of martingale property for anticipating stochastic integrals is the near-martingale property. We use examples to illustrate essential ideas and techniques in dealing with anticipating exponential processes and stochastic differential equations. The situation is very different from the Ito theory