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# PROBABILITY THEORY 2 STA 6467

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This 5 page Class Notes was uploaded by Golden Bernhard on Friday September 18, 2015. The Class Notes belongs to STA 6467 at University of Florida taught by Staff in Fall. Since its upload, it has received 22 views. For similar materials see /class/206576/sta-6467-university-of-florida in Statistics at University of Florida.

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Date Created: 09/18/15

63 STRONG LAWS OF LARGE NUMBERS 227 Because 6 gt 0 was arbitrary this proves that 1 n71 a1 ab gtboO asn gtoo an 1 1 J and thus by 66 Webb 7000 asn gtltgto aquot k1 El Remark 631 Note the following equivalent statement of Kronecker s lemma for real sequences xm n 2 l and amn 2 l with 0 lt an T 00 if 2 xk converges k1 then 1 VI 0 angakxke asneoo This follows by applying Kronecker s lemma to an and le with x anxn The following result is a simple 1 of V 39 s criterion and Kronecker s lemma Theorem 633 Let Xquot n 2 1 be independent random variables wth EXn 0 and EX lt 00 n 2 l Iffor some sequence 0 lt an T 00 then 1 VI EX gt0 asasn gtltgto an 11 Proof Note that EXa 0 n 2 l and 0 X 0 EX2 ZVar quot ltltgto n1 aquot nl aquot Thus by Kolmogorov s convergence criterion 00 X n 2 converges 35 an nl 228 CHAPTER 6 RANDOM SERIES WEAK AND STRONG LAWS and so by Kronecker s lemma Theorem 634 Marcinkiewicz Zygmund Strong Law Suppose that X1Xz are iial random variables analO lt p lt 2 Let Sn 21Xi Then Squot nc nlP a 0 as 67 for some nite constant c ifanal only ifEX1P lt 00 In this case necessarily c EX1 l S p lt 2 whereas c is arbitrary anal hence may be taken as 0 ifO lt p lt 1 Remark 632 Note that 0 lt p lt 2 implies that gt We will see that n 2 is the critical order of magnitude Proof only if If 67 holds then Xquot Sn Sn71iSn nc Sn71 nc quotup quotup quotup quotup Sn nc n l UP Sn1 n lc c 0 gt nlp n n 11p n 11p as h V gt0 as Al gt0 as gt0 Thus PLXn1P gt 1 io 0 and since the an are independent the divergence half ofthe BorelCantelli lemma implies that P 1 n Xn nlP gt1 ZIPanIP gt n lt oo n1 which implies in turn that ELX1 P lt oo i If E X1 IP lt 00 then by the Marcinkiewicz Zygmund convergence theorem either i o lt p lt 1 and 231 quot 12quot ii 1 lt p lt 2 and 221 X2539 converges as or COIlVCI gCS a S 01 iii 7 1 and 221 Xquot EXI X In case i by Kronecker s lemma COIlVCl gCS as Sn 1 VI nl Pnl PZIXi gtOas 63 STRONG LAWS OF LARGE NUNIBERS 229 and from this it is clear that 67 holds for any value of c and in particular for c 0 In case ii by Kronecker s lemma squot nEX1 7 1 71 quotUp nlp ZltXi ElX1 gt 0 as i1 and it is clear that 67 holds ifand only ifc EX1 since for c i EX1 we have SW SquotnEX1 nHPc EX1 W nlp nlP W 100 gt0 as In case iii by Kronecker s lemma s 1 quot 1 quot quot ZElXIIHXllsk Xk ElX1IX1 kl gt 0 21S n quot k1 quot k1 Now by the dominated convergence theorem EX1X1Sk gt EX1 as k gt 00 Thus by the the theorem on Cesaro averages 1 VI ZEX1XIISk gt EX1 as n gt oo k1 Hence Snn gt EX1 as or equivalently Sn nEX1 n gt Gas and it is clear as in case ii that 67 holds if and only ifc EX1 El Corollary 635 Classical Strong Law of Large Numbers LetX1Xz be iid ran dom variables and letSn 21Xi n 2 1 Then i Snn a c as for some nite number 0 ifanal only ile 6 L1 and in this case necessarily c EX1 ii IfEX1 exists possibly 00 or ltgto then Snn a EX1 as Proof Part i follows directly from the Marcinkiewicz Zygmund strong law for the case p l IfX1 6 L1 then ii is just a restatement of i so it remains only to prove ii in case EX1 00 orEX1 ltgto 256 CHAPTER 7 WEAK CONVERGENCE Corollary 724 Ifthe sequence ofprobability measures pm n 2 1 converges weakly then it is tight Corollary 725 Ifp n 2 1 is tight and ifeach weakly convergent subsequence converges to the sameprobability measure p then pr w p Proof Letg 6 CAR and let an fgdin n 2 l and a fgdi Since pmn 2 l is tight given any subsequence pnkk 2 1 there exists a further subsequence pnk j 2 1 that converges weakly and by assumption nk w n Thus ankj fgdpnkj efgalpa asj gtltgto This shows that every subsequence of am n 2 1 has a further subsequence converging to a which implies that an gt a ie fgdun fgdL Since this holds for all g 6 CAR it follows that pquot w y El De nition 722 A sequence ofrandom variables Xm n 2 l is said to be uniformly tight or bounded in probability if their associated distributions are tight ie for any 6 gt 0 there eXists a constantM gt 0 such that 130an SM2 1 6 Theorem 726 IfXn n 2 l is bounded in probability and Yquot w 0 then XnYn w 0 Remark 724 It is implicit here that Xquot and Y n are de ned on the same probability space since otherwise the product X quotYquot would not be de ned Of course XnYn w 0 is equivalent to XnYn 3 0 Proof Let e gt 0 Since Xm n 2 l is bounded in probability given any 6 gt 0 there eXists Mgt0suchthat PXn gtM lt 6 foralln 21 Since Yquot w 0 is equivalent to Yquot i 0 we see that PanYn gt 6 PanYnl gt 61an gt M PanYnl gt 61an S M E s PLXn gt M PYn gt A7 Since 6 gt 0 was arbitrary it follows that PLXnYngte gt0 asn gtltgto El Billingsley 3rd ed Exercise 218 a Suppose that X and Y have rst moments and prove EY 7EX PX lt t g Y 7 PY lt t g Xdtl b Let X Y be a nondegenerate random intervali Show that its expected length is the integral with respect to t of the probability that it covers ti Note Let Q 9quot P be the probability space under consideration In doing this prob lem you will end up looking at integrals with respect to P x Leb meas of functions like 3w t where say B wt Xw lt t g Yw Of course you need 3 to be measurable 9quot x 1 or equivalently B E 9quot x 1 Since B wt Xw lt t O wt t g Yw WM 3 t S Xwc wi i t S Yw7 it suffices to show that w t t Xw E 9quot x Q1 for a general random variable X Consider the mappings fQxRgtR gQxRgtR and mat Xw WM 75 Note that for any H 6 1 r1011 X 1H x R e 9 x Q1 and 91H n x H e 9 x 1 Thus 1 and g are measurable 9quot x QU l Now by a theorem proven in class last semester see below wt t Xw wt gwt fwt E 9 X 1 I certainly do not claim that the argument above is the best one but it is at least illustrative of the ideas that one might use I do not really expect you to verify these kind of measurability details in all exercises but it is easy to make the mistake of assuming that a function or set is measurable when in fact it is not Thus you should always be aware of measurability questions and be able to construct the necessary arguments if you have to Note Recall that for a general measurable space 9 9quot if f g Q a R are measurable l then w 3 NO lt 9M m w fw lt T lt 9M TEQ my m lt a lt w rltgwgt e 9 reQ reQ Q denotes the rationals This proof has the advantage that it also works for extended real valued functions considering f 9 is problematic at points where both are 00 or foo

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