Stat 5101 Lecture Notes - School of Statistics

166 Stat 5101 (Geyer) Course NotesThere is a much more general notion of convergence in distribution (alsocalled convergence in law or weak convergence) that is equivalent to the conceptdefined in Definition 6.1.1.Theorem 6.1 (Helly-Bray). A sequence of random variables X 1 , X 2 , ...converges in distribution to a random variable X if and only ifE{g(X n )}→E{g(X)}for every bounded continuous function g : R → R.For comparison, Definition 6.1.1 says, when rewritten in analogous notationE{I (−∞,x] (X n )}→E{I (−∞,x] (X)}, whenever P (X = x) =0. (6.1)Theorem 6.1 doesn’t explicitly mention continuity points, but the continuityissue is there implicitly. Note thatmay fail to converge toE{I A (X n )} = P (X n ∈ A)E{I A (X)} = P (X ∈ A)because indicator functions, though bounded, are not continuous. And (6.1)says that expectations of some indicator functions converge and others don’t(at least not necessarily).Also note that E(X n ) may fail to converge to E(X) because the identityfunction, though continuous, is unbounded. Nevertheless, the Theorem 6.1 doesimply convergence of expectations of many interesting functions.How does one establish that a sequence of random variables converges indistribution? By writing down the distribution functions and showing thatthey converge? No. In the common applications of convergence in distributionin statistics, convergence in distribution is a consequence of the central limittheorem or the law of large numbers.6.1.2 The Central Limit TheoremTheorem 6.2 (The Central Limit Theorem (CLT)). If X 1 , X 2 , ... is asequence of independent, identically distributed random variables having meanµ and variance σ 2 andX n = 1 n∑X i (6.2)nis the sample mean for sample size n, then√ (n Xn − µ ) D−→ Y, as n →∞, (6.3)where Y ∼N(0,σ 2 ).i=1