# Studies on the weak convergence of partial sums in Gibbs-Markov dynamical systems

Open Access
Author:
Zhang, Xuan
Mathematics
Degree:
Doctor of Philosophy
Document Type:
Dissertation
Date of Defense:
March 05, 2015
Committee Members:
• Manfred Heinz Denker, Dissertation Advisor
• Manfred Heinz Denker, Committee Chair
• Yakov B Pesin, Committee Member
• Federico Rodriguez Hertz, Committee Member
• Guodong Pang, Committee Member
Keywords:
• weak convergence
• Gibbs-Markov
• dynamical systems
• Poisson limit theorem
• central limit theorem
• first return times
Abstract:
This dissertation investigates distributional limit theorems of partial sums of the form $$f_{n,1}+f_{n,2}\circ T_n+\cdots+f_{n,n}\circ T_n^{n-1}$$ for Gibbs-Markov dynamical systems $(\Omega, \mathscr B, T,\mu,\alpha)$ and arrays of functions $f_{n,m}:\Omega\to \mathbb R$ of certain classes. We show a Central Limit Theorem (CLT) for this array, a CLT of Lindeberg type under a uniform-bound condition and we also investigate the Poisson limit case. We relate the Poisson limit theorem to escape rates of sweep-out sets and the CLT is applied in various situations, in particular to some statistical functions.