2 Convergence in Distribution The most fundamental concept is that of convergence in distribution Let ibr be a sequence of scalar random variables with cumulative distribution function Fr). If Fr(a)-F(z) as T-o for every continuity point z, where F is the(cumulative) distribution of a random variable Z, then br converge in distribution to the random variable Z, written as br -Z When br -Z, we also say that b converges in law to Z, written as br -Z, or that br is asymptotically distributed as F, denoted as br F Then F is called the limiting distribution of br Example Let (Zt be ii d. random variables with mean u and finite variance 02>0 Define bT 2-E(21)T-1∑1(z1-p)√T(21-p) (Var(zr))1/2 hen by the Lindeberg-Levy central limit theorem, bT A N(O, 1). See the plot of Hamilton p 185 The above definition are unchanged if the scalar br is replaced with an(kx 1) vector br. A simple way to verify convergence in distribution of a vector is the Proposition( Cramer-Wold device) Let br be a sequence of random k x l vector and suppose that for every real k×1 vector X( such that a'入=1?, the scalar Abr a'z where z is a k×1 vector with joint(emulative) distribution function F. Then the limitting distri bution function of bt exists and equals to F. O2(1)2 Convergence in Distribution The most fundamental concept is that of convergence in distribution. Definition: Let {bT } be a sequence of scalar random variables with cumulative distribution function {FT }. If FT (z) → F(z) as T → ∞ for every continuity point z, where F is the (cumulative) distribution of a random variable Z, then bT converge in distribution to the random variable Z, written as bT d −→ Z. When bT d −→ Z, we also say that bT converges in law to Z, written as bT L−→ Z, or that bT is asymptotically distributed as F, denoted as bT A∼ F. Then F is called the limiting distribution of bT . Example: Let {Zt} be i.i.d. random variables with mean µ and finite variance σ 2 > 0. Define bT ≡ Z¯ T − E(Z¯ T ) (V ar(Z¯ T ))1/2 = T −1/2 PT t=1(Zt − µ) σ = √ T(Z¯ t − µ) σ . Then by the Lindeberg-Levy central limit theorem, bT A∼ N(0, 1). See the plot of Hamilton p.185. The above definition are unchanged if the scalar bT is replaced with an (k×1) vector bT. A simple way to verify convergence in distribution of a vector is the following. Proposition (Crame´r-Wold device): Let {bT} be a sequence of random k × 1 vector and suppose that for every real k × 1 vector λ (such that λ 0λ = 1 ?), the scalar λ 0bT A∼ λ 0 z where z is a k × 1 vector with joint (cmulative) distribution function F. Then the limitting distri￾bution function of bT exists and equals to F. Lemma: If bT L−→ Z, then bT = Op(1). 7