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independent , 2cm - I so and are So Wn ⇐ ( Xu - it win @ Cn - t ) ) ⇐ ( Xn ) = - t ) ) ⇐ ( Wn @ en . . ) IE ( Xn t = ⇐ ( Wn ) ⇐ ( kn - t ) IE ( Xu - i ) t = - q ) IELWN )lp " IE ( Xn . . ) t - 30 - IEC # So : if a - IECX , ) then - I - - - = - p - , a > IECX , )3IECXz ) 's if petz , then - - - ( X. Is # ( Xz ) E if z , thin - as - - ' p >
if bum IE( Xn ) sa pet this gives Nate . win double til you saw , with , we contrast In " nm Xn ) - att ⇐ ( - . - tellin Xa ) ? by # Xn ) when - is MCT of EXIT , then know I ;mXn We nm Xa ) ⇐ ( " linm # ( Xn ) says - . -
( Bounded convergence Theorem ) Thou almost surely . If Ippon exists lynx . that , then IX. Is k have all for we n KEIR so " nm Xu ) E- ( nm Elk ) " = . say , for ;4 The triage inequality - X PI Dude in Xu - . - Xml ) E IE ( IX - Xn ) / I ⇐ ( X - IE ( Xn ) I . I ⇐ ( x ) = so for all NEIN have all : for e > 0 we want We - E- ( Xn ) l K E IIECX ) we get . n z N - Xal ) - a -50 s O El IX as do this shewing by . we
- Xnlwll > e } - { wer I Xlw ) Define let E > 0 An : - . . have all we .r have for we We - Xnlw ) Is et 2K Han ( w ) I Xlw ) expectations Take IEC et 2K Iancu ) ) - Xnlw ) l ) E ⇐ ( lxlw ) et 2K IPC An ) = sides both take kmsup Naw on " TYP MAN ) set 2K Pl ' 'm :P An ) - Xml ) Et 2K ⇐ ( IX " map E
- X. Cull > e ) - " must { wer : lxlw ) An limsup But - line Xnlw ) t Xlw ) with of , is the set w DNE ) { wer : hnmxnlw ) just which is . is measure zero . set hypothesis this by But Hence " TYP MANI ' 'm :P An ) ' et 2K Pl - Xml ) Et 2K ⇐ ( IX " mhm E - E E . " I'm IEHHXND fer " msn.PE/lX-Xal ) < E all s , s . S . = Off
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