II of large Number Lattin in probability almost convergence - PowerPoint PPT Presentation

Chapters : Proofs of the laws II of large Number

Lattin in probability almost convergence ⇒ convergence sure - - " weak " strong " " cow corn Two tests for stray convergence - O ⇒ almost sure conveyance . ) - X Ise ① Fe > 0 All Xu i. o - ② VE > O E P ( IX n - Xl > E) as ⇒ almost sure convergence

Newsprint ① Prove S LL N

Then I WLLN ) a sequence of independent random variables If X , ,Xn , is . - IEC Xi ) and That m , VER there and exist so - m - Ruden variables W ( Xi ) EV all , then the fer i in probability - - - t Xa ) converge to : - I ( X , t m Sn - - ml > E) =D . IP ( Isn " I V-E > 0 , . ) on Chebyshev PI ( Proof relies

- THEN .lt .at#Xnl/ - lE( tallit - - it Xd ) Observe IECSN ) : - = M . t-etxnll-ntwlxit.tk ) - NC 's Non ) - need 'T ta ( NIX , )t . - - TNCXNDEI n indef Chebyshev : says a- *l s - Ea E - ml > e) IP ( Isn . sides both " I Take Ed can . .

Then ( SUN ) a sequence of independent random variables If X , ,Xn , is . - IECXI ) and m , a EIR That there and exist - m so - Ruden variables " ) Ea , then the all fer ⇐ ( Ni - m ) i almost surely - - - tha ) converge to - I ( X , t : Sh m - m ) - 1 PC " ash - - - - O . Pt assume m - WLOG we

all fact ZEIR that for have Its ' we a z2sz4tl Hence . LEI Xi ' ) IE ( Xi 't 1) E att = N ( Xi ) E . we'll study - txn , and : let X , t Strategy Ta - - - - ⇐ ( Tn ' ) isn't " too We'll " ) show ⇐ ( Tn . able to use we 'll be which big " from , result The to Markov prove .

- - t Xn ) ( X , t - ut Xn ) ( X , t - - t Xu ) ) IE ( Tn 4) - - - t Xn ) ( X , t = IE ( Nit t ¥ - E ( Ih Xi , .bg?,i2Wt..?..KfixiXk " - Vakil + § 4 Xi ' Xi 24 Xi Xi Xk Xe ) t e ; , .IE#7leYxi%Efxd ECxi.IE Hilt 12 ?g - IE " ) t 6 IE ( Xi - , let xi7Ex 24 EE Exile + 4.7

- IE " ) t 6¥ ECxi.IE ( Xi ) " ) ⇐ ( Tn IECXI - , 6 ÷ ← ? Cath Catt ) a t , - Math ' nut Kalla ' 6 (2) ( att ) = t = na ' - atlatl ← K - - ( at , ) ' ← n' a na k 3 n t s ⇐ ( Tn 4) En - K

: IEC Tn 4) E n - k So -5ptsfrAn Markov says : lP(l9nl)=lP(HHt-tXal)=lPlIlTnl)hh IP ( Itn I > ne ) a Itn Is E ) = IP ( IP ( I Sn Is E) ' - - E nn¥g e ⇐ n!I = Kant " ) " E " s ( Tn n = , . H ( Isak E) E E E : q . In . c is So .

of Cartelli version Borel By The - almost fer sure conveyance test the , p##t get we . DUE almost 0 Sn to converge surely . . Bh