用Stein方法研究拟单二聚体平均场模型的极限定理
Limit Theorems in the Imitative Monomer-Dimer Mean-Field Model via Stein's Method
摘要：
我们考虑[1]中引入的完全图上的模仿单体-二聚体模型。可以理解，该模型是由单体密度描述的，并且沿着一定的临界线具有相变。通过将模型还原为具有硬核相互作用的加权Curie-Weiss模型，利用Stein的可交换对方法，在全参数空间上建立了单体密度涨落性质的完整描述.我们证明了这个量在远离临界线处具有中心极限定理，在临界点处具有归一化指数$3/4$的非正态极限定理。此外，我们的方法还允许获得沿临界线的条件中心极限定理。在所有这些结果中，给出了Kolomogorov-Smirnov距离的Berry-Esseen不等式.
英文摘要：
We consider the imitative monomer-dimer model on the complete graph introduced in [1]. It was understood that this model is described by the monomer density and has a phase transition along certain critical line. By reverting the model to a weighted Curie-Weiss model with hard core interaction, we establish the complete description of the fluctuation properties of the monomer density on the full parameter space via Stein's method of exchangeable pairs. We show that this quantity exhibits the central limit theorem away from the critical line and enjoys a non-normal limit theorem at criticality with normalized exponent $3/4$. Furthermore, our approach also allows to obtain the conditional central limit theorems along the critical line. In all these results, the Berry-Esseen inequalities for the Kolomogorov-Smirnov distance are given.
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