Law of large numbers

Transcription

1 Law of larg umbrs Saya Mukhrj W rvisit th law of larg umbrs ad study i som dtail two typs of law of larg umbrs ( 0 = lim S ) p ε ε > 0, Wak law of larrg umbrs [ ] S = ω : lim = p, Strog law of larg umbrs Wak law of larg umbrs W study th wak law of larg umbrs by xamiig lss ad lss rstrictiv coditios udr which it holds. W start with a fw prlimary cocpts that ar usful.. Trucatio: W rplac th radom squc {X } with a trucatd vrsio {X I [ X ]}.. Tail quivalc: A commo proprty of th trucatd squc that w will try to xploit is tail quivalc. Two squc {X } ad {X } ar tail quivalt if (X X ) <. W will prov somthig ic about th trucatd squc {X I [ X ]} amd th prov it is tail quivalt to {X }. ropositio 0.0. Suppos {X } ad {X } ar tail quivalt. Th ) (X X ) covrgs 2) X ad X covrgs or divrgs or X covrgs iff X covrgs 3) If thr xists a squc {a } such that a ad thr xists a radom variabl X such that if roof. For () w us Borl-Catlli a X j X a X j ([X X ] i.o. ) = 0 (lim if [X X ]) = X. so if w st ω {lim if [X X ]} this implis X (ω) = X (ω) or N(ω). For (2) X (ω) = X (ω). =N =N For (3) a (X j X j) a.s Th followig thorm provids cssary ad sufficit coditios for wak law of larg umbrs. Ths ar th wakst coditios rquird.

2 2 Thorm 0.0. (Gral law of larg umbrs) Suppos {X, } ar idpdt radom variabls ad S = X j. If i) ( X j > ) 0 2 E(X2 j I [ X j ]) = 0 th for a = E(Xj 2 I [ Xj ]), S a roof. W prov sufficicy. Dfi Obsrv so X j = X j I [ Xj ], S = X j. [X j X j ] = ( X j > ) so ( S S ε) [S S ] ( [X j X j ]) (X j X j ) S S Sic Var(X) = E(X 2 ) (EX) 2 E(X 2 ) so ( S ES ) > ε Var(S ) 2 ε 2 2 ε 2 St a = ES = E(X2 j I [ X j ]) so 2 ε 2 E(X 2 j) E(Xj 2 I [ Xj ]) S a S S + S a S a Th followig xampl illustrats that o ca hav a law of larg umbrs v if th first momt is ot boudd. Exampl 0.0. F is a symmtric distributio fuctio such that F (x) = 2x log(x), x, ad F (x) = 2x log( x), x.

3 3 First obsrv that EX + = EX = so th first momt dos ot xist sic St τ(x) = X( X > x) = EX + = 2 log(x) dx = 2 dy y =. log x Also st a = 0 sic F is symmtric so S th wak law of larg umbrs holds, th strog law dos ot. I th followig w wak coditios udr which th law of larg umbrs hold ad show that ach of ths coditios satisfy th abov thorm. Exampl (Boudd scod momt) If {X, } ar iid radom variabls with E(X ) = µ ad E(X) 2 < th X µ. i) ( X > ) E(X2 ) E(X 2 I [ X ]) E(X2 ) 0 Exampl (Khitchi s WLLN) If {X, } ar iid radom variabls with E(X ) = µ ad E( X ) < th X µ. i) ( X > ) = E(I [ X >]) E( X I [ X >]) 0 E(X2 I [ X ]) ε2 ( ( ) ( E X 2 I [ X ε ] + E X 2 I [ X ε X ])) + E( X I [ε X ]) ε 2 + E( X I [ε X ]) ε 2 So E(X I [ X ] S E(X I [ X ] 0 EX E( X I [ X >]) Exampl (Fllr s WLLN) If {X, } ar iid radom variabls with lim x( X > x) = 0, x th S E(X I [ X ]) Strog law of larg umbrs W wat to udrstad th coditios udr which S E(S ) b Start with a fw rsults that w will d i provig SSLNs.

4 4 Thorm (Lévy) If {X, } is a idpdt squc of radom variabls th X covrgs i probability iff X covrgs almost surly ad for S th followig ar quivalt ) {S } is Cauchy i probability 2) {S } covrgs i probability 3) {S } covrgs i almost surly 3) {S } is almost surly Cauchy. Th followig covrgc critrio will b usd. Thorm (Kolmogorov) Suppos {X, } is a idpdt squc of radom variabls. If V ar(x j ) < th covrgs almost surly. (X j E(X j )) roof. Without loss of grality st E(X j ) = 0, so E(X2 j ) <. This implis that {S } is L 2 Cauchy so S S m 2 L = Var(S 2 S m ) = EXj 2 j=m+ {S } is L 2 Cauchy so {S } is Cauchy i probability ad so covrgs almost surly. Lmma 0.0. (Krockr s lmma) Giv squcs {x k } ad {a } such that x k R ad 0 < a x k k= a k covrgs th x k = 0. lim a k=. If Krockr s lmma with th Kolmogorov covrgc critria immdiatly provids a SLLN. Corollary 0.0. {X, } is a idpdt squc of radom variabls such that E(X) 2 <. Giva mooto squc b. If Var( X k ) < b k th k S E(S ) b roof. By th Kolmogorov cpvrgc critrio X EX b covrgs by Krockr s lmma (Xk EX k ) b W ow provid SLLN rsults for iid squcs. W first d th followig lmma. Lmma {X, } is a iid squc of radom variabls. Th followig ar quivalt

5 5 ) E X < 2) lim X = 0 almost surly 3) ε > 0 ( X > ε) <. = Thorm (Kolmogorov s SLLN) If {X, } is a iid squc of radom variabls ad S = X. Thr xists c R such that S c iff E( X ) < ad c = E(X ) roof. W show S c E( X ) <, X = S S = S S c c = 0. Sic X 0 this implis E( X ) <. Almost sur covrgc ca b prov wh th Kolmogorov covrgc critrio dos ot hold, Var(X j) <. This is giv by th thr sris thorm of Kolmogorov. Thorm (Kolmogorov) Lt {X, } b a squc of idpdt radom variabls. I ordr for S = X to covrg almost surly it is cssary ad sufficit for thr to xist a c > 0 such that ) ( X > c) < 2) Var(X I [ X c]) < 3) E(X I [ X c]) covrgs. roof. W prov sufficicy. Dfi X = X I [] X c. To prov () (X X ) = ( X > c) < so {X } ad {X,} ar tail quivalt ad X covrgs iff X covgs To prov (2) obsrv Var(X ) so (X j E(X j ) covrgs To prov (3) w s that E(X ) covrgs.