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Chapter Theorem limit Central I : tasting characterizations of weak convergence of equivalent Proof of distribution distribution follows theme conveyance of Observation : weak expectation interact with limits : of how IE ( " 7 Xn


  1. Chapter Theorem limit Central I :

  2. tasting characterizations of weak convergence of equivalent Proof of distribution distribution follows theme conveyance of Observation : weak expectation interact with limits : of how IE ( " 7 Xn ) MCT : if { Xn ) TX " nm IE ( Xn ) = Then weak " " - ⇐ impact ) nm Emf ) : it rn " then III. - ⇒ p when for nice f

  3. our goal of to end semester : Recall Theorem ) Thou ( Central limit Tt m and finite with iid be mean X . , Xu , . . . Then finite - X , t - t Xn Sn variance let v . - - . Lfsnjnm ) ⇒ Marion , ← stand :L , distribution

  4. weak wants to some Good prove CLT us news : a- lot knew of distributions , and we convergence of kind conveyance that ways to of measure . to pave at tool need new Bad a we : news and define to tool easy the is Good new news : " to weak conveyance nicely " . relates Defy ( characteristic faction ) characteristic function its . then random variable be let X a = Efoosltx ) ) t i E- fault XD E- Leith ) 40 × 4 ) is = .

  5. of X distribution by the determined so , It ) oh Nate : , d. striated identically , Thu Y if and X are 01 × 14--4 , It ) . characteristic fndien ? about the " characteristic " What's so ( Continuity Theorem ) Thun . Then variables random Tt be , Xz , X , Xi . . . - oh iff " nm Kult ) , It ) L ( Xn ) ⇒ LIX ) .

  6. Theorem ) Pf ( half of continuity EM in ' bounded L ( Xn ) ⇒ L ( x ) . Since cosine is Assume a conveyance gives n of weak function , def ' Coatney Eun ( cos It xD " I " nm IE ( cos ltxn ) ) = = IE µ ( cos It xD IE ( cos It X ) ) = - E ( salt XD IE ( smltxn ) ) follows with Kam - Same idea .

  7. have Huwe we i IEC Scutt Xa ) ) IE ( cos It Xa ) ) t " Y " nm Ox . It ) - - i IE ( Sia Lt X ) ) IE ( eos HX ) ) t " = oh , It ) µ . hand , strategy for continuity theorem with our show ky4s÷H)=4ma in to is at

  8. 9*14 for Hyun , " duraotwsti Anakin compute lets First , : definition By : it × ) - Erm . " ( e it " ) IE ( e OHH - - - " % § , it " Hd x ) e- ± = f : ein tr e - " % dx . compute this . need We to

  9. compute It 's let : dx ] . It ! - " ¥ Loh " ' e " " , Hif E. e = ! ! # Lei dx ) " " " e = f ? ixeitx # e - × % dx . du = - teitx ieitx II. ¥ × e- × %d* parts ! lets integrate by " " re :* e-

  10. ⇐ " × # e - f : ix e - " k ' 10 × 14 d x - . - " I ? - f ? = ÷ " * e- " t t.ci , cite dx . - t oh , It ) O = . ' H ) So : oh talk , It ) ) - th . By integrating , , - t It = .

  11. ( after exponentially ) So get we qft)=e-t%whmXrN# function hand , strategy for with this our show uy9s÷÷lH= in to is at

  12. do this ? We'll need will How we 0 × 14 ) ( Taylor for Them expansion Tet E- ( Xk ) is finite with variable random be x . a - Ipo "Y 01 × 4 ) IEC Xi ) Then junk - t t - so where ttlk ( junk ) → o as .

  13. serie , expansion for exponential ( sketch ) Pf El e it × ) ! # ( Ecitf÷ ) - oh H ) - - , unit Eli ) Z " address details to ( easy do ) to numbers ① deal complex with not might we ② not all terms so neg , ar non - So : truncate . e 't E countable twenty get D .

  14. , sketch form ) ( CLT Pf Theall : with and variance v iid m mean X , Xy . are . . . have - - t k - X , t Su fr we Want : - to :{ %y÷H=et% L ( sniff ) ⇒ Equivalent Mmm , . - Xi and NAH Ii Deline Etxi ) -0 Then - - . . should . - - TXT In - It variable the random Then - same ravttforgn - t% . If hum & It ) have - e so , - ,

  15. - I and So : m :O assume r WLOG - . - th olsyr . It ) Observe Want : → e . .lt/--fE(eit " ) . . . - eitxnlrn ) Olsyr ⇐ ( eitxtrneitx.la i #t= " ) . E feit 'T ⇐ ( eitxtra ) # feit Mra ) . . i :# µ ei÷ - ④ × .lt/rn ) ) " ) ) " " -

  16. So : " nm ( Ox .lt/rnl ) " in 0% It ) " = t litY Elk . ]tli% EH :) trunk ) hnm ( ⇐ I n Tay nm ( - EI " , t junk ) " I + O . I = - IT ) writing and ' Hospital ( after a tu kg leg - aatml I Naw - t% you hid equals this e . BEBO

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