Gibbs measures asymptotics

Athreya, K. B. ; Hwang, Chii-Ruey (2010) Gibbs measures asymptotics Sankhya A - Mathematical Statistics and Probability, 72 (1). pp. 191-207. ISSN 0976-836X

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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.

Item Type:Article
Source:Copyright of this article belongs to Indian Statistical Institute.
Keywords:Entropy Maximization; Gibbs Measure; Hamiltonian; Holder Continuous; Laplace's Method; Simulated Annealing; Temperature; Weak Convergence
ID Code:1132
Deposited On:05 Oct 2010 12:53
Last Modified:16 May 2016 12:17

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