Convergence in Distribution
Sidney I. Resnick · Modern Birkhäuser classics · 2013
This chapter discusses the basic notions of convergence in distribution. Given a sequence of random variables, when do their distributions converge in a useful way to a limit? In statisticians’ language, given a random sample $$X_1, \ldots, X_n,$$ the sample mean $$\bar{X}_n$$ is CAN; that is, consistent and asymptotically normal. This means that $$\bar{X}$$ has an approximately normal distribution as the sample size grows. What exactly does this mean?