Athreya, K. B. ; Hwang, ChiiRuey (2010) Gibbs measures asymptotics Sankhya A  Mathematical Statistics and Probability, 72 (1). pp. 191207. ISSN 0976836X

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Official URL: http://www.springerlink.com/content/q035641771t044...
Related URL: http://dx.doi.org/10.1007/s1317101000065
Abstract
Let (Ω, B, ν) be a measure space and H:Ω→ R^{+} be B measurable. Let ∫_{ Ω}e^{H}dν < ∞. For 0 < T < 1 let μ_{H,T} (·) be the probability measure defined by μ_{H,T} (A) = (∫_{A} e^{H/T} dν)/(∫_{Ω} e^{H/T} dν ), A ∈ B. In this paper, we study the behavior of μ_{H,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 X_{T} is a random variable with probability distribution μ_{H,T} (·) then as T ↓ 0, i) X_{T}→ x0 in probability; ii) (X_{t}  x0)T^{1/α} converges in distribution to an absolutely continuous symmetric distribution with density proportional to e^{cαxα} for some 0 < c_{α} < ∞. 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 : R^{n}→ 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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