Sample Mean
Uwe Hassler · Wiley series in probability and statistics · 2018
This chapter addresses the issues related to the sample mean. It presents a general central limit theorem (CLT) ensuring limiting normality. The chapter analyses the variance of the arithmetic mean. It discusses some aspects of inference about the mean. The chapter focuses on the behavior of the sample autocorrelations under nonstationarity. CLTs are the reason behind limiting normality of sample means. Classical CLTs have to be adjusted to meet the special needs of time series analysis where serial correlation is rather the rule than the exception. The chapter provides the so-called functional CLTs for type I and type II processes. There is an important issue associated with the estimation of expected values under long memory, namely, the rate of convergence. In fact, the slow rate of convergence under fractional integration with long memory is not a property of the arithmetic mean exclusively.