03 Strong laws of large numbers

03 Strong laws of large numbers - 7’“ m SH‘Oni law...

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Unformatted text preview: 7’“ m: SH‘Oni law 013 large numbers LC; “hawk - 0”th Lemma: Mimi»...u ”mm”. H ..,. 3,a.1 M, ku'n kecMMBul Jm 9mm cs\\vwa:§ 5mm: convergence. 0 ‘— 1 SEECQYK‘e—L ‘l~13—200t 6} -IS — zooc, SQQOSQ‘ (an) ‘ (A49: VOLVO) £1an 0 anA | 4 WW“ A09! “M (in :- 0 mean? Onha fi'n“te\3 W2 1'5. n—aoo \VYZSE 0" 5 1" Inflm’tely mam} 1's; . \im. a“? a“ = l n' ‘4‘ a O S‘ n «a. F V\ | £113, MA MA swp q“ :: 1 (Mr \emfi on. 1 Q§S§£§J up NAT “2‘ Ld A” A}. ‘3: sch oné gonsdev‘ in” 3A7; 5%? 1AM a 1 .12 M A“ km W E = 10A“ (W “1' 0 all»! iv»? 1“” (w\ 2 5 i (g 00 Q Q“, a“ a} 3 l {\A (00) n11 L E) 0/w n \ “M 6m? 1 A ( ) P 1 if 09% A“ QM «.0. Namfi VI n" 00 '1‘ ~ { O \um (“Q 1 A“ (m) = 42 1, i? 03 e A“ For null MA ‘gm‘hfid New} w ham 0 “MA 3%, LM MC 6? .16“? \‘M 5“? h“ E a: 00°. 03 G. (M Q” {ngmkehd wows ”\ft {am ‘0} T\ ~a on “‘ng‘mkdfi 69‘6“ “M ”C [\n E \ Y (/0 OJ Q A“ $1 a“ box“ gm'lcdd wow} VI? n A M «a GED Shaw . n A“ c, \w «9 A a “N me A“ 9 U m . “:3 Y\ —~ on n —-\ u: “N . \mm'w? hm :1 3‘ 3“ Am new 00 a “"3“? RH ‘2 m U Am "(‘0 n | m?“ 7L“ 943,200! 3,“.2 449- 2.505 .___.__...’-—— “P432 —2000 -MMWM Lax Anni, e, H. I? 2: (MM < w) w M ‘iiW A“) MAN) 2 o nu EYES? LQA‘ N = Z lAvx' nu Mn NON) (3 Wu. nwmbav up n (Cm wwch m «z: A». BM EM == ($7. 5:; 1pm Mow/“7km; Com eraeVlce, Jflneovem @P?Ued +6 PoxJn’al SLAMS: H N W3 p 3’ )I 31"” a A Z ”Wm ‘P<N=oo)to,\ BK“ =. 2 “M s (A -_-, «M 00': «Ear w‘m’ch we 0:“ {M gas “A: fl 7 3w fo'nv'th ”fi'fi '15- U 7'“ (FIB—2019i 3.a.3 $134000; Uswé ”Wm Bord— CMM: Lemma, we can conclude Had: WCOM ‘30,) : Q_ How Abel '“V(S he“: qr? Cowwnfleuu’. ogwiefwfrcgs in firms :1? (“Cmflei 09+: O a,“ -‘3 Q ‘4‘, RT CL“ E. > O 7 +\IL°.VQ EKG“? N ,5de *km‘k‘ H > N) imeke: “A“ \an ~a\ 4 a. a OK“ -> 0,, (g, Cw a.“ £>0, \Om—a\ 7 E“. on“ gt'm'hkj mam} ’n’mer 0 (km a (A (Q. £o'f ox“ 5 5 o , {\Qn ~ ad > g (n§n\'«\c‘3 ““3 km“) Jae, 33% W RT “~11th purposes, “HMS (5 Jmo Marti {-0 sth, \mM- we c913 Shaw (+ Rm “SEEMS W393 Vodwu a? 8'. V os\"\‘\'\l¢. Lei: (8m) Le quemee, Wed: 00an as ‘h 0, o a“ .4. 0L {(2 , gar a“ m, [ \qn ~m‘ > Em {nrx’nlh‘a ogkm} 4236: my} WW‘ {0&37» am“ 6:90, choose m m ‘kkM Em < 8. Thu {lah-al >8 infmf 4:, .ceen Aces no" M996“. 0 0m —--I 0« (I? { \Qn ..(1\ > a“ ingm'klfi QQ‘CR} Joey n01? happen. Tws (s Cssen’n’allx‘} 47H sBeete (My, agar same. pom? N) \Qh~<1\ fi 8mg" o\\fl.>l~ Nah de hut. flu, sulosuilfi on an oyd 8,. (3 Wu Jame, 7‘41 q—I3-200(p 3.5.1 —-- CH’reda 'Qm" a‘mos’r awe. convergence. 6? a SaiuenCC o? ravfiom an'aHes nw~¢> W (9M?) Ld X) X.) X2, ‘oc IcaLVoluch raunclom vourmgkg, Unéer (”hank concxx'fi‘ons can we concluae ‘HAGJ‘ )(h ,., X almos+ swdj? ' Swpposc (an) {s pous've. and wnvevjes 4o O as WNW, w: ’cka’r W \x“— ><\ > an Lo.) = 0. 1'9 4" b Q ! sf \Xn—Xi > an adj Q‘mfidj manjfi'mer}_ Mn WCSL) :4 \) omA 52w a“ m in 52°, ‘Xh((l~.)) -~ )(CW) ‘ > an gar adj ‘R'YWA'GIJ may [1) \ cmA so )(nhp) -~> X0») as haw”) esme'MKj 53 Hay. Saucy; ‘aw- Thus, m “m ><h 2x) z WP<51A=L “—500 0 Sumo“. Half, pr»! 01“ 8>O, W< lxn~><\ >8 L0.) =1 0. W5, ‘Cot‘ (iv-“ti! €>OI WVHA probabv'\"+j 1) an SEQSAQHCL X“ evenlruanj rye}: wv’arh'vw s: a? X. Bud“ This WM 'hakc \onfler ¥or some 00's Hmom. U «700m. Mx’céM Pr be. No: Cm‘nWl‘jfiYmt (n pmbob’ilké, NW” HAW! a‘wmk mm”- For €0C‘(\ ‘m 6. 1N; [6" S14“ :3 I; \Xn “ X ‘\ > 'Lm {-0-} . "War. PC Qm') :: O b? assuw‘fiw‘on’ .30 {PC ‘3 52m) :1 O 0: MM“. M5! gulf {kg WPIQMen-Kr «up 'Hm's union 5w: pmbm ENE; 1) once I \( ‘ 00 U “94m?“ J =1 0* 51: ch Mn‘ram W: h ,x/ m. 1‘ 3/ r’ I? i ¢ é" ‘1 fl . , 2'3: 3? “MS: 82’; ‘2 3m 9,91 WDER‘J) (,0 I5 [70+ W), 52m} : {LO 2 SI», UK“ M, {\Xn'X‘ > 7“; $0.} Ant: 1):: happen} 7- Stu) I (Car (1“ m} \XnQAflr'XOJO) \ 7 Jvfi QCOY‘ ’Rnflm‘n’} "“1va 7L 7: {00 -. “Wt )(yJw) =~ Xlw)}, haw mm, mm meabctak‘a 1, W ><n : ><. Adm 1|.” 0|—~\%~700D 3.0.1 4.7.? $231 MEW;L:: \>. “‘5 Show \nw a? \av? nwmkevr :4?ng Ld x.,x., u, «A Wm v; xf' < 00- 113’:sz- Lelr 5n : %.+ k x“. 1M9.“ .573 ”.3 Ex, oAmx’r suwhé. w: 153 “£er X: 3 X; 'EXH M Maud awlquL, Wfl’kwtk (an GfJe/Mmt'r‘jd) Hm? EX. : 0‘ N1 MAM h: 5W {ANA 3:3 ---a 0 AMA sunka. Viol 670. MA new Th. MGM {an we flu. pm? I: find PU'S'nnP £3 = PC \5M > me) ’ gawk vewvfiiwg {I E. ‘5”? \a Undo shev's inc duh" , 1’ ’0. (new 3 j l 3 Using )9: 2, WASH" : VoMsn) = WVMKA, 30¢er “£225!“ Amman! 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O‘MM’S‘ SUN-L33 Mn )(n —--a X. in ‘bmbm‘d‘A‘a' pmsg w ('1’) Le" “u :\ I Y ‘ Var emck 41>! Hart is a numb“ “k Sud» “06‘, P( \Xnk ._ 'X \ > 11:: ) 4 1:) V (\oecowuc va '9 X M “m‘omb{\\ 3)‘ . WA “k > “by! a Y)“ :4 \P C \ XVHL ~ X \ > ‘k ) < k: '75: : 1' ( an I ‘ k$( 5| ’ ‘01 ‘WU. Qjmre‘ .. Can-‘8“; Lem-ma.) P ( \Xnk-X\ > 1': Lo.) = O. ~Hum X“; A) X dvmfi and» (11' L A co- 1H \xwxx > e) ,_ E Liam,“ Th». rmvaowx vormbk: Mn om Aommdeé b} L a»; E L l on. Harem“, a: Vlad!) M CNN/“3J7- 4\ O «(wwfi Juflbav ‘5} DC“ “M ‘PHxn—xlw’.) .,. E0 :0, n *oo. (1.; ”Ewing!“ . D (DCQYN‘TOW Sukguguhce. . Lek 01m. “IV-1,773, _.., be 0., ”Emerge. quom n; N -—9 N {5 5%"ch \‘mrwdvta' m MBABYV‘L thk)’ 1:52,; W, {s CoJAQA m JUL scgwntt A— QVM mclfl‘}, fl. - W4. flormofij wtfi-c. Qhk , M 5mm c‘eCwa'Wm/n «'3 Made. \Chr 4 SubSQCCLCVKQI «3’ (7L Seiuencfi «2* mnAmw vanfobles. '1‘“ 04—25—2002 3,4.2 Q—w’zow Lama 5.0.3 1" H7 ReaA Case. . LQ’\' 3‘, (J1, -~ I ‘3 Q R. I? mug mhwymme 5Q C3“) M: on gum swksewnu Hm? wuvergu +0 fl! ‘qus 3“ .4 3 a: n '5‘ 0‘" WE (D Le’r E. >0. Consx'c‘u‘ flu. set or? a“ v» Cw wkm,‘ \3“ ‘1‘ 2. a, \QSSULme Hm's se‘r f5 \‘nQinHe. Ca“ A S. “rum kj“ ) ne S mvss’n'wku a su‘oseguvxu’. Ge (3“), 30 H has a Cuan-u sgbsebumnce WHOM mavevaer h: ‘j‘ Bu’t HA3: camaaAkk M (lad- ero'f \‘dn «\3‘ >__ 8 Car 04‘ ‘n e 5, Ms, S is Q'M’rc, omA Q“ N ‘9'?" cmuah, “1M u’va'cJ ‘3““3. ( Er Thu! L5“ ——) (d- . (2) LA 0 = \imm? 3'“ h-sm Om can s‘mwxl M ex(s’reme at o SUAXC‘ELQKW 3W whfck converge: 413 a. BM Unis micscgmnm mam have 0L Emu" «subsegum'i “W" 00"""33‘ +0 ‘3 Ed ““3 Marmot «9 3'1 mark oonmax h a a: we“, anal :0 61:3. $1qurky “M Nf 3“ :«j‘ (NRA “HUM \{m .\ en's“: am; (“g/mgr 3' U ham haw Rewou'k ’nm's 5am pad I: W 3% manna: u'n arblfmrj me‘m’c :Pacex; Hm End Pm? Wo'kl HM» 59‘“ Way. is H— a.\so Jnm. Q0 ahmsk sure, mnvcvgemfi at a seam-me of YnnAom vanhbler? 1.2,me (Vuht’recl) . L“ 3) Ei‘vyzwn' be clemen‘v OP 0. thg ”ML“ mm (3“) «Mme! +0 3, ‘4’, <1th (ml; ~89, every subscgmmcc CW9 has m gar'HIu/r am’butucmt (gnu) whisk mnwryu h g. 7*“ q~27—2o02 3.4.5 Ohm—7.005 Theorem L (a. Z Lc¥ )(\,X1, m, X be vea\—vadu¢c‘ YanAom \lovfob‘cr on (S1,H;W). .TMLM Xv. 9X M ”5’5“”3 Q? 0"; <1an u‘, S1“ everj SubSetylLewce a; (Xm), ‘HNLN. {s m Rr‘fiur Subxegnnu. m“w"3"“3 «k, X Amy!» Hula. Qevnovk (“mud sum c‘nnwvaeme Ace: no’r wovk Nb. cpnvujemc u? "A numbflfl) 91‘ even \x'h. mnvujeme .‘n a mahft 39mm. The“ f: ‘HMM m mafia ‘PN’ Q\Mn:’c sum mnvegence. (Mn an “Kama thck (unvxvuax. in Pnbahllfi bvk— m‘” ms.) 3er 63 5099056. Xn ~9 X (h pmbabill’xj. Le’v 0%” be, a subsccguence. «Curmw (“um th a X 'W\ Qfiba‘oflfij 01‘ ‘1 Am, so +Wm EXI'S‘tJ GASUJOSQ ence Km“ mnveVSMj h: X alMurk sturdy ax Q a», on. @ 3mm 4m My may” «9 (m m a 9mm Jutuzme mveramgr +n>< 0.5. LA a >0. (13min 1;“ = I? C XX, w x \ > 8). "Wm (M (s « rml segsumcc. Consvéer q subsengnce '(PH‘). ’ mm en's’fi a :u'omglence (.thl) a? (gut) “mama +0 X alum NM], £50 X“! #9 x an Pmbokilimw ml mu PM «90 o: flaw. ms, Q" evevj S\Jbse?)u€nc& (PM o? (m, mm “.34; m (Inmr sm‘oxtguenct (13%;) cameramcg *6 0 cu K 400‘ Thus, 1)“ —~>O M “”60- D 7"“ w—zce- 400a 3AA Sumwse Xn —+ X M thbflx‘kj (m3 5‘1 ‘R *R (a oawh'wwow ancl. houm‘ul. T‘m \EHM 4 E904.) a: M m. 339:? Le’r Ylk ‘01 (L subseygmg. m“ X“ comaje: h x m wwwm-J. p do Mm. wash 0‘ subsugueflcfi XNQ mnvugima ’m )( Await WM}- “53 Mammy ngnu‘y “merger ha $003 alwmr Jurdfl. Rr «m m in go, Wm“ Wm) =\, we have Kn“ (w) «MM at Slam “a“ 5‘0““ (M) 2 ¥(XIM)\, m h»: “a Kr a.“ m m 32¢». (53 HM EamktA omega“ Wanna, Egon“) _, E 83 w M 1 «zoo. m“ k. awn} suLIeBuewu Ewa) 39, Euxnx, a (£4er gohxeguwu. (if: 5' (in. 1) owvuau‘ h E ‘1 00. “AK E: $06,) -) EQUQ 0.: “a“. u ) DegflVHoVk Lei: X.)$Z;l,..,X hflu VOL/Kr {k a. +upa\aat‘em\ (or bud—He.) or ELLLL'JGM) space 8 Km CDYNEYSU {'3 X {D A\'s’mbu¥\bn 59) 90+ a“ \oqunhA mMmow ¥;S—*R, .fimm. ._m_..; E Hm d EE¥<><L We hone Show“ “find Comet ewe. i0 mmmur in \(e: comer ence. in Ax'dukmh'on- 3 V 3 " 3 Reiaknhsh’? be’rween mode: 69 chl amt Le’v Xwfi, ,,, , X Ea nod~Vo~luunl {WI/\J/fim Variables. We have Mrs I'Mchah'ovu e X“-9)< um'govmla :9 Km “>X pomhm'm ==’> )(“ax 4,5, 5? XnaX "K P‘J’ =>)(*._, XL“. clv'S'h‘l M'f's'an ° X“ "’ X h“ LP) +70 =7 >01 4 ‘K M PNBQBMJ‘J 2? 561 '7 Y. M ((13+T/[9M'f‘l’0‘4/ S‘mrm (A SEEN/“CC 6? End-Com on (to, t], @Bu] ) 1m) mkkk : a Converge: in Pmbqu'kfi 19qu noF othwonL sunk} 0 Comwtau‘ \‘h pmbabdflj \mfi— nw’s in L‘ a: CphVQ/Fafiy \‘n Abikh’bm‘hbn 5A“ Vlo‘? {n pmbabflt4} 7"” OI, 234002 3A. 5 Conveflcnggfijgmfincficfi_la jubse “TEES . Le¥ £1 [0.00) '€ “2. amA L6 R C‘m’m 2'20: gm) 2L. (9 «A 0&3 r0, gar ox“ 5&3“,th 7‘5. «r ‘9) WA £9“) 2 L.‘ k—JOD 139:? G) Sufibgose. 71ch $0») 2 L (”A [3* 'Xl‘ an ac, Ld' a> O, T‘NLM exfs‘: M Such vaA, {low a“ x>M, “31%) _L[ < 8. W em’s’r; \4 such fixed, QM all k > K. Xi) H For Suck k, Hun, \¥(’)CL) ~L‘ ( a, an SO in“ gka) 2 L. G) 8mm Jam, «9“ a“ “Fence: och ~>°° we have “M Hm =L- ha» For each h =k,2, ,-,.) sd' ah r. in? {-(x) . xzk Bk ‘~‘ 5“? 5:0!) , xzk 1" {S (4‘th de' 0.“ fl bk pm all k, (Gk) is ihcveoshnfl, (M) :5 decreasing. FM k. Thu exisb och > k suck Had S'Ulk) Is close 1% ah. in Hue reuse. Hmfi ak g $0“) $ ah + 1'. Ww‘vmg Hum as Hot“ "E g Qk 0M; ah 4‘ R“) ("[4 mm} Ha. Sven: Law) Wt QM] Hm+ [Hm ah : Luv 3mm, kw» b = L. ‘ 3 km “ Le} 8. >0. mm was \< Wk wt, Pm mu (wk, -5 ( ak-L muck, bh~L 4 6. {5% Hum . L63 < 0k = in? RM 5 5“? 530x) 2 13k < L+g, x24: Ink Thus, Qav m“ OLE KM, L-g < ’EC‘K) < L+€, m3 so “M We) 2 L. D Mam 7L“ (1—27-2002 3.4,(9 LCmHs a? :ezuences arc wage . Lek (in) be a 41:8,“:an of mo] numbur. Sufisom “I“ x“ c a) “M 9C“ 1 b, OMA 504% a WA b (278 £34011. 0"” naoo Thu) at = b. Rang (.(N\"\h°\’\* Mina Hm. mdkotk 6: CAM‘IITRchHon) Lek 8. >0. “nut eus’r N‘ 0AA N; such HMA- {31" all n >N,’ [Mad 4 5i (AAA gov mu “7'01, \xu’- b! < Ell Le" n 7 max (NUNI'), TWA \a~b\ = lama!“ r Kn —'o\ i \Q’k‘m‘ 47 ‘b'xn' +h'GflJ‘2 I'hez'fifil'q'j 4. EL +— £2 0 6’ TM) la.~L;| < E, Cm! a“ 8 > O, 83 HM order Pmper+j a? red number); exacfij out a? ’f‘m Cc:“ow»'\\j Ma‘AJ‘ 1 \q~b\ < O) \a—~b| : o larb‘ > OI l W gal: cammk kappen bf} flax AeR‘niHan «42 aubso‘uh vulva. Fer 0L“ C> 0, Hum 955*! an im‘c‘jw n suck +kn+ J; 4 C. , o. mnsegwnu er? “UL ‘60th Upper Emmi pm‘wrka, BW“ QOVF o.“ n, vk >0) 5;) \Q'H < I“; 9 amA So \q—‘M is no’t >0, Thu \Q—M:O, My <L~L :0 MA 01:19. Nah: m pm? (’Lou’r ‘a-bl 5‘0 is do“). b3 unhflmmve: ("A «Jr mnhméccfion: f—V {(3 6. >0 Ham Mme emh 0m {thaw n > O suck Hw¥ '3: < C. ) 50 m i? 9m 0;“ ithuy n>0, L < C can"; *5 he“, ‘Hrwn C- % 0' V‘. . 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