Central Limit Theorem using Characteristic functions
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1 Cetral Limit Theorem usig Characteristic fuctios RogXi Guo MAT 477 Jauary 20, 2014 RogXi Guo (2014 Cetral Limit Theorem usig Characteristic fuctios Jauary 20, / 15
2 Itroductio study a radom variable Let Ω with measure m, m (Ω = 1 ad F (Ω measurable fuctios. To radom variables X F (Ω we associate distributio fuctios F (x := P (X < x := m (ξ Ω : X (ξ < x ad f (x = F (x is the probability desity fuctio (shortly pdf. We assume exist fiite: Expected value of X (mea value µ = E (X = R xdf (x = Ω X (ξdm Variace ad the stadard deviatio σ 2 = V (X = R (x µ2 df (x. RogXi Guo (2014 Cetral Limit Theorem usig Characteristic fuctios Jauary 20, / 15
3 Covetio: P (... := m (... ad for {A j R} 1 j ad {X j F (Ω} j set X 1 A 1,, X A := {ξ Ω : X j (ξ A j, j} For our X j distributio fuctio, expected value ad variace are the same ad {X j } j are idepedet, idetically distributed (shortly iid, i.e. P (X 1 A 1,, X A = Π j=1 P (X j A j. For iid {X j } j set S := E (S = E V (S = V 1 X i ad also ( 1 X i 1 = E(X i = µ = µ ( 1 X i = V( 1 X i V (X = = V (X 2 2 = σ2. RogXi Guo (2014 Cetral Limit Theorem usig Characteristic fuctios Jauary 20, / 15
4 Law of Large Numb: lim P ( S µ > ɛ = 0, ɛ > 0 Proof: σ 2 = V (S = R (x µ2 df (x x µ >ɛ (x µ2 df (x ɛ 2 P ( S µ > ɛ P ( S µ > ɛ δ for > σ2 ɛ 2 δ. Def: X d X, i.e. coverge i the sese of distributios meas bouded ad cotiuous fuctio f : Cetral Limit Theorem (shortly CLT: R fdf R fdf. (S µ σ d N (0, 1, where S = 1 X i 1 ad N (0, 1 is the rv with pdf e 2 x 2 2π of Gauss distributio RogXi Guo (2014 Cetral Limit Theorem usig Characteristic fuctios Jauary 20, / 15
5 Next, ote that N(0, 1 has expected value R xdf (x = 0 ad variace R x 2 df (x = 1. Also, {X j } j beig iid s of course (page 3 E (S = µ, V (S = σ2. To prove the theorem we ll use the characteristic fuctios ϕ(t = E(e itx = R eitx df (x, shortly cfs Note: rvs always admit cfs; ϕ (0 = iµ ad µ = 0 ϕ (0 = σ 2. Also ϕ (0 = E (1 = 1, ϕ(t = R eitx df (x R e itx df (x = 1. Fact1: F (b F (a = 1 2π lim x e ita e itb x x it ϕ (t dt. Easy if F (x : f (x = 1 2π R e itx ϕ (t dt. RogXi Guo (2014 Cetral Limit Theorem usig Characteristic fuctios Jauary 20, / 15
6 Properties of characteristic fuctio ϕ X1 +X 2 (t = E(e it(x 1+X 2 = E(e itx 1 E(e itx 2 = ϕ X1 (t ϕ X2 (t ϕ ax +b (t = E(e it(ax +b = e itb E ( e i(atx = e itb ϕ X (at ad also the uiform cotiuity of cf with µ = 0 ad σ = 1 : ϕ(t + h ϕ(t = E(e i(t+hx e itx E( e ihx 1 0 Characteristic fuctio for Gauss distributio is e t2 2, page 14. RogXi Guo (2014 Cetral Limit Theorem usig Characteristic fuctios Jauary 20, / 15
7 Covergece of F implies covergece of cfs Propositio : X d X ϕ (x ϕ (x x R. Proof. : e itx d is bouded ad cotiuous ad X X imply R eitx df R eitx df. To show (see page 12 we eed to prove first a so called tightess of our rvs. Tightess of a family of Radom Variables. Def: a family of rvs X is tight whe ɛ > 0 M such that P ( X > M < ɛ for all. RogXi Guo (2014 Cetral Limit Theorem usig Characteristic fuctios Jauary 20, / 15
8 Claim: covergece of cfs implies tightess of rvs. Proof of 1st step : we show that for ay distributio X := X, ɛ > 0 M, P ( X > M < ɛ 2. Ideed, every cf has a value of 1 at 0 (page 5 ad is cotiuous ɛ > 0 δ > 0 such that t < δ, 1 ϕ (t < ɛ 4 δ ɛ δ 1 ϕ (t dt < 2δ 4 = ɛ δ 2 δ 1 δ δ 1 ϕ (t dt < ɛ 2. O the other had, for some large M δ 1 δ δ 1 ϕ (t dt is a upper boud o P( X M : RogXi Guo (2014 Cetral Limit Theorem usig Characteristic fuctios Jauary 20, / 15
9 δ 1 δ δ (1 ϕ (t dt = δ 1 δ ( ( δ 1 E e itx dt = δ 1 (2δ R ( = 2 1 R ( we have 2 1 R 2 R ( x 2 δ 1 si(δx δx ( si(δx x si( δx x df (x si(δx δx df (x. Replacig 1 by R 1dF (x si(δx δx df (x = 2 ( R df (x 2 ( x 2 δ 1dF (x = P ( X 2 δ 1 si(δx δx 1 si(δx δx df (x. df (x δ 1 δ δ 1 ϕ (t dt P ( X 2 δ. Together with above ɛ 2 > δ 1 δ δ 1 ϕ (t dt P ( X 2 δ. RogXi Guo (2014 Cetral Limit Theorem usig Characteristic fuctios Jauary 20, / 15
10 Step 2 : covergece of cfs implies tightess i its rvs ϕ (x ϕ (x meas ɛ > 0, x R atural umber N s. th. > N holds ϕ (x ϕ (x < ɛ 4 ɛ, δ N such that N we have δ 1 δ δ ϕ (t ϕ (t dt < ɛ 2 (fact from aalysis. Also, (page 9 we may choose δ to satisfy δ 1 δ δ 1 ϕ (t dt < ɛ 2 N we have P ( X 2 δ δ 1 δ δ 1 ϕ (t dt RogXi Guo (2014 Cetral Limit Theorem usig Characteristic fuctios Jauary 20, / 15
11 ( δ 1 δ δ 1 ϕ (t dt + δ δ ϕ (t ϕ (t dt < ɛ. Also, for smaller tha N δ such that P ( X 2 δ δ 1 δ δ 1 ϕ (t dt < ɛ choose δ mi := mi {δ 1, δ 2,, δ, δ}. we have the that P ( X 2 δ mi < ɛ for ay rvs with coverget cfs are tight, the claim is proved. RogXi Guo (2014 Cetral Limit Theorem usig Characteristic fuctios Jauary 20, / 15
12 Proof of ϕ (x ϕ (x x implies X d X usig Fact 2. Tightess of rvs implies compactess i the sese of covergece of distributios ( Prokhorov s Theorem. Proof of from page 7 : Pick ay coverget, say to F 1, subsequece {F 1 } of distributios. Say {ϕ 1 } are their cfs. ϕ (x ϕ (x x implies covergece of all {ϕ 1 } to the same ϕ ad proved o page 7 implies that ϕ is the cf for ay F 1 exists uique F 1 =: F ad, usig Fact 2., X d X, i.e. X d X ϕ (x ϕ (x x is proved. RogXi Guo (2014 Cetral Limit Theorem usig Characteristic fuctios Jauary 20, / 15
13 Coclusio of the proof of Cetral Limit Theorem For a series of iid X i, let Y = 1 X i µ σ ϕ Y (t = ϕ 1 X i µ σ (t = ϕ 1 X i µ ( t σ =: ϕ ( t σ. Let s := t σ, as s 0. Recall: ϕ (0 = iµ = 0, ϕ (0 = σ 2, see page 5. From Taylor expasio: ϕ (0 + s ϕ (0 + s2 2 ϕ (0 ϕ (s = o(s 2 lim {ϕ Y (t = ( 1 σ2 +o(1 2 ( t σ 2 } = lim ( 1 σ2 2 ( t σ 2 = e t2 2 the limit of the cfs is the cf of a Gauss distributio (S µ σ = Y d N(0, 1, as required. RogXi Guo (2014 Cetral Limit Theorem usig Characteristic fuctios Jauary 20, / 15
14 Appedix. cf of Normal distributio, calculatio: f (x = 1 2π e 1 2 x2. ϕ (t = 1 2π e 1 2 x2 e itx dx = 1 2π e 1 2 x2 +itx 1 2 (it (it2 dx = 1 2π e 1 2 (x it2 e t2 2 dx = e t π e 1 2 (x it2 dx. y = x it dy dx = 1 ϕ (t = e t π e 1 2 y 2 dy = e t2 2. RogXi Guo (2014 Cetral Limit Theorem usig Characteristic fuctios Jauary 20, / 15
15 Abbreviatios rv : radom variable rvs : radom variables pdf : probability desity fuctio iid : idepedet, idetical distributed rvs cf : characteristic fuctio cfs : characteristic fuctios RogXi Guo (2014 Cetral Limit Theorem usig Characteristic fuctios Jauary 20, / 15
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