Central Limit Theorems for Randomly Modulated Sequences of Random Vectors with Resampling and Applications to Statistics
Open Access
Author:
Bagyan, Armine
Graduate Program:
Statistics
Degree:
Doctor of Philosophy
Document Type:
Dissertation
Date of Defense:
May 28, 2015
Committee Members:
Arkady Tempelman, Dissertation Advisor/Co-Advisor Bing Li, Dissertation Advisor/Co-Advisor Francesca Chiaromonte, Committee Member Alexei Novikov, Special Member
Keywords:
limit theorems random modulation ergodic stationary sequences sequences of vectors resampling
Abstract:
In many situations when sequences of random vectors are under consideration, it is of interest to study the asymptotic distribution of their (normalized) sums and to determine the conditions for the limit theorems, such as the Central Limit Theorem (CLT), to hold. In the simplest case when the variables are independent and identically distributed and have finite variance, the CLT is satisfied. Some CLT generalizations with weakened independence assumptions exist as well. For example, the CLT holds for stationary random sequences with strong mixing. However, in many situations when there is dependence, the CLT does not hold.
%In particular when we consider stationary sequences of random variables.
This happens for stationary random sequences even with the weak mixing condition.
In our research we propose a method of random modulation of ergodic stationary random sequences that allows us to prove limit theorems for such sequences without any mixing conditions. These theorems present an opportunity to construct asymptotic confidence intervals for parameters, test parametric and non-parametric hypotheses with the significance level close to the required one and to calculate the approximate power of the test.
More general analogs of the CLT are proved and the speed of convergence is estimated for sequences of random vectors in spaces of non-decreasing dimensions.
