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## Mathematical Statistics

by: Mrs. Triston Collier

41

0

3

# Mathematical Statistics STAT 709

Mrs. Triston Collier
UW
GPA 3.57

Staff

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COURSE
PROF.
Staff
TYPE
Class Notes
PAGES
3
WORDS
KARMA
25 ?

## Popular in Statistics

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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