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i:*ha:i : in probability conveyance almost Tests for sure - PowerPoint PPT Presentation

Sep 08, 2022 •369 likes •497 views

Chapters Strong II : weak versus lastlimci notions of Two conveyance his XnH=XlwB ) =/ means lP({ w " Almost " : sure ' ' In probability - Hss ) IP ( Hn " He > 0 " - means - , random variables

  1. Chapters Strong II : weak versus

  2. lastlimci notions of Two conveyance his XnH=XlwB ) =/ means lP({ w " Almost ① " : sure ' ' In probability ① - Hss ) IP ( Hn " He > 0 " - ① means - , random variables which an example of converge We saw almost surely in probability , but not 0 to . talk about ways to nuanced ' Need Motivation more : " approaching certain behavior variables " random

  3. Then I WLLN ) a sequence of independent random variables If X , ,Xn , is . - IEC Xi ) and That m , ve IR 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 hun V-E > 0 ,

  4. i:*ha÷:÷÷÷÷÷i : in probability conveyance almost ② Tests for sure convergence

  5. Thm_(weakversusStney6nuegm ' almost surely convey to X Suppose Xi , XL , . - - - probability converge to X in Xi , XL ; . Then - - " I lP( Hu - Xl > e ) - O ¥ let want . We - be given . es o = X ) =L ' ' am Xn Pl know we . - Xlw ) I > E ) I Xnlw ) . We want = { w Er Bn : let continuity of probability . We'd like te " nm IPL Bn ) - O un - .

  6. aren't nested . : the sets { Bn } Problem - Hulke } - { wer I Xmlw ) : Fm > n An Solution : define so - { An } I ? An Bns Ain Further : We then get . . ' IMB . ) ' am MAN )=' we get IPC ? An ) a- Hence if - O , - = O IP ( fan ) that WTS So we . " um X. ( w ) * Nw ) we 1 An This means let . { Xn ( w ) ) regularly them " " point jump away ( Since the Kw ) )

  7. " I Xnlw ) t Kw ) ) ) =D But All w Er ! almost surely Convey to X the : Xn since - X ) - I IPC ' im Xa - - . Xnlw ) 't Kw ) ) : l 'T = { wer Heme ? An PC ? An ) = 0 and TBH so . convergence is Notational almost sure consequence : " weak in probability " is and convergence " " strong

  8. so desirable is almost Since convergence sure , to test for it . we'd like ways so new keltesttorslneyconvergmletxxi.kz . If for random variables be - - - , i. o ) IP ( Hn - Xl > E = 0 have all a > 0 we , almost surely . converge to X Xi , Xt , Then . - -

  9. - X ) - l BC If we ' :X . want - . Xnlw ) = Xlw ) } " I { wer : Now - Hulke } we get lxnlw ) = { wer : V-E > OF NEIN fn3N a. a . } = { wer - Xlv ) KE the > o IX. ( w ) : i. o . } ' - Xlw )l3E = { wer IX. Lw ) : Fe > o i. o . } But earn - Kw )l3E lxnlw ) : FE > o { wer 'M so arch Jtag " " N%= ¥ - Hwy > g. i. o . } lxnlw ) { wer : . 20

  10. , for all get ' By assumption > 0 we q - o . ) =D IP ( - Xl > q ' I Xn i sub additivity monotonicity : countable and by Hence - Xlw > I > E I Xnlw ) IP ( { wer i. o : FE > 0 so - Xml > q i. o . } ) lxnlw ) ' IP ( Y { wer ' : - , E § - Hull sq ' i -0.3 ) : lxnlw ) Ipf Ewer - Eo .

  11. test for stroy convergence ) Cor ( Bord - Cartelli Ft random variables That be X , Xi , Xz , so i - - have § - Xl > e) < is , all lptxn E > 0 for we almost surely { Xa ) X the to Then conveys . have all - Cartelli , for E > 0 PI By Borel we last result , i. o . ) - Xl > E . By The - O Rl l Xu - we get almost conveyance sure ④ .

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