In this thesis, first we develop a result of Stein's method for non-normal approximation via exchangeable pair approach. By this result, we derive Berry-Esseen bounds for total magnetization in Curie-Weiss model and Blume-Emery-Griffiths model with size-dependent inverse temperature. We find out general forms of limiting distributions as well as sharper Berry-Esseen bounds. We also provide Berry-Esseen bound for Curie-Weiss Model of self-organized criticality under boundness condition. Moreover, we establish general results of Berry-Esseen bounds for positively and negatively associated random variables. Our general results are applied to some special cases. When covariance of random variables decays exponentially as the distance of indexes of random variables increasing, ou...[ Read more ]

In this thesis, first we develop a result of Stein's method for non-normal approximation via exchangeable pair approach. By this result, we derive Berry-Esseen bounds for total magnetization in Curie-Weiss model and Blume-Emery-Griffiths model with size-dependent inverse temperature. We find out general forms of limiting distributions as well as sharper Berry-Esseen bounds. We also provide Berry-Esseen bound for Curie-Weiss Model of self-organized criticality under boundness condition. Moreover, we establish general results of Berry-Esseen bounds for positively and negatively associated random variables. Our general results are applied to some special cases. When covariance of random variables decays exponentially as the distance of indexes of random variables increasing, our Berry-Esseen bounds have same rate with previous ones. When random variables have power law decay covariance, our Berry-Esseen bounds are sharper than all existing ones in the literature. The proof of Berry-Esseen bounds for associated random variables are based on Stein's method via concentration inequality approach. In particular, when the covariance decays slowly, we use the idea of "block sum" in the proof.