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Limit theorems and parameter estimation for theq-state Curie-Weiss-Potts model
Abstract
The q-state Curie-Weiss-Potts model, where q $\ge$ 3 is an integer, is a useful statistical mechanical model. It is an exponential family parametrized by the inverse absolute temperature $\beta$ and the external magnetic field h. As this dissertation shows, the model has a fascinating probabilistic structure. For q = 2, the model is equivalent to the classical Curie-Weiss model. The first part of the dissertation studies limit theorems for the empirical vector, $L\sb{n}(\omega)$, of the model. These limits include the law of large numbers, a central limit theorem when $\beta$ $<$ $\beta\sb{c}$ and h = 0, and a conditional central limit theorem when $\beta \ge \beta\sb{c}$ and h = 0, where $\beta\sb{c}$ is the critical inverse temperature. Also a central limit theorem with random centering is proved. The phase transition at $\beta\sb{c}$ is first-order, in contrast to a second-order phase transition in the classical Curie-Weiss model. All these limit theorems imply similar limits for the sample mean $n\sp{-1} S\sb{n}(\omega)$. Some limit theorems for the classical Curie-Weiss model are also presented. The second part of the dissertation studies the large sample behavior of the maximum likelihood estimator, $\ h\sb{n}$, of the external magnetic field h. I will study $\ h\sb{n}$ when $\beta$ is given and the true value of h is known to be 0. Under suitable conditioning, it is found that $\ h\sb{n}$ exists and is unique. It is also found that under suitable conditioning, $\sqrt{n}{\ h\sb{n}}$ has a normal limit when $\beta<\beta\sb{c}$ and a discontinuous limit when $\beta\ge\beta\sb{c}$. Despite this discontinuous limit, $\ h\sb{n}$ is always consistent for h whenever $\beta$ is given. I will also summarize some results on the maximum likelihood estimator, $\\beta\sb{n}$, of the inverse absolute temperature $\beta$. These results have been proved in Ellis-Wang (1990b).
Subject Area
Statistics|Mathematics
Recommended Citation
Wang, Kongming, "Limit theorems and parameter estimation for theq-state Curie-Weiss-Potts model" (1991). Doctoral Dissertations Available from Proquest. AAI9132930.
https://scholarworks.umass.edu/dissertations/AAI9132930