Convergence in Distribution and Central Limit Theorem in \(\mathbb {R}^d\)

In this chapter, you first learn about the notion of a distribution function of random vectors and about properties of such distribution functions. The first main result is the Portmanteau theorem, which provides several equivalent conditions for the weak convergence of Borel measures on \(R^d\) . The convergence in distribution of a sequence of random vectors \(X_n\) to a random vector X is then defined via the convergence \(E(f(X_n))\to E(f(X))\) of expectations for each bounded real-valued continuous function f defined on \(R^d\) , that is, via the weak convergence of the associated distributions. Further contents of this chapter are the mapping theorem, Slutsky’s lemma, the concepts of tightness and relative compactness, and Prokhorov’s theorem, according to which tightness and relative compactness are equivalent notions. The chapter proceeds with stochastic Landau symbols, the continuity theorem of Lévy-Cramér, and the Cramér-Wold device. It concludes with two central limit theorems, and with applications of these theorems, including asymptotic considerations regarding the chi-square goodness-of-fit test, a test for equiprobability of the winning numbers in lotteries, and the delta method.