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Mathematical Statistics

by: Mrs. Triston Collier

Mathematical Statistics STAT 709

Mrs. Triston Collier
GPA 3.57


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This 3 page Class Notes was uploaded by Mrs. Triston Collier on Thursday September 17, 2015. The Class Notes belongs to STAT 709 at University of Wisconsin - Madison taught by Staff in Fall. Since its upload, it has received 41 views. For similar materials see /class/205087/stat-709-university-of-wisconsin-madison in Statistics at University of Wisconsin - Madison.


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Date Created: 09/17/15
Lecture 13 Weak convergence A sequence P7 of probability i neasures on R SBk is tight if for every 6 gt 0 there is a compact set C C Rk such that inf PAC gt 1 e If Xn is a sequence of random k vectors then the tightness of is the same the boundedness of in probability Op1 Proposition 117 Let P7 be a sequence of probability i neasures on Rf 8quot i Tightness of P7 is a necessary and suf cient condition that for every subsequence Pm there exists a further subsequence PM C Pm and a probability i neasure P on Bquot such that PW gtw P j gt ii If P7 is tight and if each subsequence that converges weakly at all converges to the same probability measure P then P7 gtw P The proof can be found in Billingsley 198639 pp 392 395 The following result gives some useful suf cient and necessary conditions for convergence in distribution Theorem 19 Let X X1X2 be random k vectors i X7 gtd X is equivalent to any one of the following conditions a EhXn gt EhX for every bounded continuous function 11 b limsupn g PX for any closed set C C Rk c liminfn PX 0 Z PX O for any open set 0 C Rk ii Levy Cramer continuity theorei n Let bx bxI in be the chf s of XX1X2 respectively X7 gtd X if and only if limH00 bxAt bxt for all t 6 Rk iii Cramer VVold device X7 gtd X if and only if can gtd CTX for every c C Rk Proof First we show X7 gtd X implies By Theorem 18iv Skorohod s theorei n there exists a sequence of random vectors YR and a random vector Y such that Pyn PX for all n Py PX and 1 a Y For bounded continuous 11 MY gtM MY and by the doi ninated convergence theorei n EMS3 gt EhY Then follows from EhXn EMSTa for all n and ElwX EMSH Next we show a implies b Let C be a closed set and fog infx y 6 Then f0 is continuous For 139 12 define Ikoom 1 jt10gtjs Then 170 jfcx is continuous and bounded by 2 11341 139 1 2 and hamquot gt 100 j gt Hence limsupn g limH00 EULAXRH EljX for each 139 by By the doi ninated convergence theorei n EMU X gt EUO X PX This proves b For any open set 0 O is closed Hence b is equivalent to c Now we show b and imply X7 gtd X For x x1xk C Rquot let 30r 30r1 X 30rk and 302 oox1 X X 30rk From b and c PX rgtox liminfn PXn rgto g liminfn FXn g limsupn limsupn PXn fgt0 FX If x is a continuity point of FX then FX This proves X7 gtd X and completes the proof of i llgtlt ii From of part i X7 gtd X implies an gt bxt since a th cosUTx 1 sinUTx and costfx and sintfx are bounded continuous functions for any fixed t 1 Z pammua ap uopnqmsgp 113 30 3113 113 s IJ IAA 31 9 lt 1 W 114 312113 Inmqo am 4 lt a Sup gsgws no GJUanGS x31dmm Km m 9 lt 1101 a 1 mugs lt u 21 9 1 Km 10 11019113 1 u LL 30 3113 m1 uaql 39ng 71 amqm 0 lt lil0 1771 1quot HOV3 1 2 XcJ SGySQRS 1X 30 3113 113 312113 sn1m1m ug 31mm p mm mono 31 gt 1Xg 3121 asoddng 2 1 H X 1X LL pm 39j39p39a uommm p 81mm SG11R XRA mopmax ummxadapug aq X m lX 331 39gz391 91dumxg 3911 mud mm mono 31mm 113 uaql 331 9 4 KJGAG 103 X199 03 X q 30 aatm mum o3 3tm1m1nba 81 Kg 03 Xe3 30 mtm wwm aauaH quot3121 9 4 Km pm 21 9 71 Km 10 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9 1 XMAG m 1 1 3 lt 1 Xc 3121 pm 1 31 3121 mou asoddng Zlt magmas tt HHSGJ mmmdttt tw st sttfl fp39d RIILXOU pmpttms atp st f 1th f lt f 372tp aspwxa Amtp um atto quotquot g 1 77 Z39 afqzzl 1 ttltgtt3txqtt3stp 3 atp f0 f fp39d an saqaq atp mptsum afdttmxa tw 8V 39tttamatp aatm taAttm pawtttttmp atp matf smtmof HHSGJ atp n 39639 0 lt fi pup 2 up pf 52 S uff S 0 mugs wMMWammm14mm um mm WWWWW mm 2776771 30on 390 nplw WI 1 Wm tmtfl 77 39339t39m fp39d 72 st 2 pup a 39639 my uff ooH ttttf nap asoddng 77 GJHSRGIH 72 39339t39m flu U0 s fp39d f0 tptmnbas 72 aq uf 331 39ttmmatp S QQGQJS 911 uogggsodoxd 39 f fp39d 72 Sptf X pup f fp39d 72 Sptf X tmtpn X 1 X tatpatpn gtpatp 03 past aq um HHSGJ Stttmtmof atfl 0171 931 saspwxg ttt tht aw safdttmxa tatpo 1321 afdttmxg tt unsm 3mg atp 3J pRX3UOJ HHSGJ Sgtp sang 391 00 ttopnqmstp Ktpng atp Sptf X 1th X 1 773 nap apnpttm am gf afqzzl 1mg ttonnqmsgp Ktpng atp f0 ftp atp 8 my mugs 77 39EL 91 MJ lt lt0 m x7gtttt 8 quot X 1X 711 m3 atp pup g aspwxg 0 1 x7gtttt st lX f0 ftp atp tmtfl 39ZIJ2SOJ 1 my fp39d an satpq ttmtttttm 72 3U ARQ safqzzmm IIIOpURX 3tmpttadaptt aq X quotquot 1 X 331 331 afdmexg 393 0 VN 1 Mm 1 mm apnpttm am 13 aspwxg ttopnqmstp WINCH atp f0 A mdotd 72 pup mm391 tttamatfl mm 1 493 0N P7 71f77 LIfa nap ttotsstpstp snotmm atp matf smtmof 3t 32L 9 4 paxg Ktw 101 nutty aw 1Xt7gtA 3 pup ng 71 pup SX03JGA j tttoptmt aw X quotquot 1 X 372tp mou asoddng 39zv 0N 1 M u 1 03 spmf 71 LX 5 f0 IXOQRIILXOJSIXRD 72 0 71 f1 30 0N Pt M LL 3th go 0N f0 ftp atp z 04 lt 13 th3 nap sagdtttt IJ IAA 2L 9 1 KIN mf 333 7 7 0 g I y 77 u 30 Sfdtttt 0c gt 1Xt7gtA Zr pup 0 71 Ktmftttttg 3971 dlt 7711 nap smatp 0an Sgtp ggAg39 ttwoatfl mm 1 3971 1 7711 tt639 Immatfl Kg 3971 37gt GJHSRGIH mgtqpqoxd 88mm 3tttod atp quot31 71 37gt


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