Chapter distributions Conveyance of I : goal for weeks : - PowerPoint PPT Presentation

Chapter distributions Conveyance of I :

goal for weeks : understand last the two Our theorem ) ( Central limit Thon m kn ) and Ippon with iid Xi , K2 , mean are . . - v ka ) , then variance Lf Hth?-mI ) " " weakly converge to Marian . NwstvfffrtodayWDef.me distribution . for weak conveyance

random of discussed variables : already We're convergence - X ) - l IP ( im Xu - says - " " strong ( Teresa ! ) - Xl > e ) " I Pl Hn - O l " " " - says weak distribution ? in what is anyone But

of distribution ) Det ( weak conveyance on ( IR , 93 ) . Then distributions be let pi , µ , . - ⇒ µ µ , denoted { un ) to weakly convey pun , have banded , continuous f : IR - SIR we if for all = Eplf ) " I Eun tf ) - T T i as § , Http Cdt ) same £ fit ) Mulde )

- X . EI R D= Laid and let - - ( , I ] ' k ' k ) ' lo . , Az A let - , Aa :( 4,11 As :( I. 4 ) - Ey , 's ] Ay As :[ oily ) - - lo , 's ] - l 's , IT , Aa Ag - - - . - , 0 Xn away weakly to - HA . Define we've . seen Xu - . then pin ⇒ µ = Llxn ) Claim : if let we µ . , the tea function . the distribution of where is µ So ) lie .

bounded , continuous fraction be f let . a Eplf ) Then = IE , fffx ) ) f- to ) - IE ,( flat ) - = . XY ) function and is the ( here zero X , hand other the On , ftp.lfl-IEnff/XnD--fIo)1fAnc)tfll)HAn ) Hoti ) -11 Alan ) and so knew As a -3N , - , we = Emf ) . = flo ) by Ep . (f) s . Bd

of distributions ? determine weak Hau conveyance can we of equivalent formulations . lots wth a theorem stale we 'll at A bounty is The then AER , Recall : if - fete )AA¥oB and ( - E , # E) AA -70 - { XEIR ( x : Teso x JCA ) , -

weak distribution comymu ) Tqm ( Equivalent characterizations for are equivalent following : The Mn ⇒ M ( " with platt ) - petal - O ' " A " I all pin IN for - - - O MIKI ) x with - aid ) ( tax ) for all - pull by l " - - pen Y , Ya , Ya , variables (4) ( Skorohod 's theorem ) there random - - - are and Llyn ) measure I with LIY ) - fun - M - ( oil ] - under lebesgue ( ou strongly Yn → Y and functions f banded , Boel - measurable am Enuff ) - Emf ) (5) " for all µ ( LxHRsdif)=0 " " Df with

Preotstntegy : ( 2 ) ( t ) w (5) ( 3) \ , as

prove For now , we convergence ) weak implies Cor ( weak convergence Ippon s . that variables random X , Xi , Xz , are . . . L ( Xn ) ⇒ LIX ) X. Then convey weakly His to . z with NWo : for showing by this PI will any prove - pelts , Ed ) " nm Mn ( t - ah ) have - we - LH ) - L ( Xn ) and where µ - un - .

a ( IH ) - O with be let - given . z Then let > O E . u { Xszte } - X Ise } E l IX . { Xu Ez ) probability gives Taking : t Pl Xszte ) - Xl > e) Xu Ez ) E IP ( l Xu IP ( sides both of Take 4ms up IP ( XE ate ) - Xl > e) t " m :P p ( xn ez ) stuns - r p ( kn

- Xl > e) =P " nm p( Ha have weakly , Xu to X Since convey we " m :P lplxnez ) shuttle ) t lpfxezte ) to 0 let : Now E go = IPIXEZ ) Ip ( Xu ez ) " m :P H H - a. ED Mt " m :P yall - ah )

hand other Da the u { Xnsz ) - Xl > E } E { Hn - e ) { Xs z So : - e) E - XI 're ) t RC Xasz ) IP ( l Xu IP ( X Ez kouinf : Take " mint Blintz ) ' limit P( Ka - Hse ) t - e ) IP ( xez s X weakly " mint RANEE ) - e) ⇐ IP ( xez convey H : Xn since

get As to 0 we goes e , " mint p( Xn et ) A ( Xcz ) E - lplxez ) MHz ) we get MH ) - assumed - O , we've - since have So we - IP ( X Cz ) ' " mint lpfxnee ) E IPCXEE ) " must plxnez ) - - " mint RANEE ) that implies :P pans -2 ) km - Heme = µ ( fats ) = IPLXEZ ) nm Mall - at ) - " I RLXNEZ ) " DAM