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

Open Access
Author:
Zhang, Xuan
Graduate Program:
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.