Weak Laws of Large Numbers for Sequences or Arrays of Correlated Random Variables

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1 Iteratioal Mathematical Forum, Vol., 5, o. 4, HIKARI Ltd, Weak Laws of Large Numers for Sequeces or Arrays of Correlated Radom Variales Yutig Lu School of Mathematical Scieces, Beijig Normal Uiversity Beijig 875, People s Repulic of Chia Copyright c 5 Yutig Lu. This is a ope access article distriuted uder the Creative Commos Attriutio Licese, which permits urestricted use, distriutio, ad reproductio i ay medium, provided the origial work is properly cited. Astract I this paper we estalish several weak laws of large umers for sequeces or arrays of correlated radom variales ased o estimates o variaces of weighted sums. Mathematics Suject Classificatio: Primary 6F5, 6F5, 6G9 Keywords: variace, correlated radom variale, weak law of large umers Itroductio Without special statemets all radom variales uder cosideratio are defied o a proaility space Ω, F, P). A sequece of radom variales {ξ k ; k N} with fiite expectatios is said to oey weak law of large umers WLLN) i the classical resp. moder) sese if ξ k k= Eξ P k resp. k= ) P ξ k a for some sequeces a ) R ad ) R + with ). Because of history reasos the laws of large umers for sequeces of idepedet radom variales have ee studied sufficietly, see [, 4, 3, 9] ad refereces therei. However, sequeces of correlated radom variales are very complex; ad there k=

2 66 Yutig Lu are few papers to study the laws of large umers for them, for example [, 7, 8]. I Sectio we shall estimate variaces of weighted sum of correlated radom variales ad list some lemmas. Based o them we estalish several ew weak laws of large umers for sequeces of correlated radom variales i Sectio 3. Preiaries Let ξ,, ξ e radom variales with fiite variaces Varξ i ) <, i =,,, ad let Covξ i, ξ j ) e the covariace of ξ i ad ξ j, i, j =,,. Propositio. For ay a,, a ) R it holds that Var a i ξ i ) a i Varξ i )). Proof. Deote y σ i = Varξ i ), i =,,. By reorderig for ξ i, i =,,, we may assume Varξ i ) > for i =,, m, ad Varξ i ) = for m < i if m <. Recall that for ay a,, a ) R, Var a i ξ i ) = a i Varξ i )) + a i a j Covξ i, ξ j ). Sice Covξ i, ξ j ) σ i σ j, we have m Var a i ξ i ) = a i Varξ i )) + i<j i<j m a i a j Covξ i, ξ j )..) Let r ij deote the correlatio coefficiet Covξ i, ξ j )/σ i σ j ) etwee ξ i ad ξ j for i, j =,, m. The r ij. From.) we deduce Var a i ξ i ) = Var a i ξ i ) = m m r ij a i a j σ i σ j j= m m m ) ) a i a j σ i σ j = a i σ i = a i σ i. j= Propositio. ad Cheyshev iequality yield Propositio. Uder the assumptios of Propositio., for ay a,, a ) R ad for ay ε > it holds that ) P a i ξ i a i Eξ i ε ε Var a i ξ i ) a ε i Varξ i )).

3 Weak laws of large umers for sequeces 67 Takig a i = /, i =,,, we otai Corollary.3 For ay ε > it holds that ) P ξ i ) Eξ i ε Varξi )..) ε Moreover, if Varξ i ), ), i =,,, replacig ξ i y ξ i / Varξ i ), i =,,, ad ε y ε/ we otai ) ξ i Eξ i P Varξi ) ε ε..3) Lemma.4 Kroecker) For two sequeces of real umers a ) ad x ), if < a ad x = a coverges, the a x i. Lemma.5 i) Stolz theorem) Let x ) ad y ) e two sequeces of real umers, ad let a R {, + }. If there exists N such that x < x + for every >, ad x = + ad y y )/x x ) = a, the y /x = a. ii) Hardy-Ladau theorem) For a sequece x ) R, if x i = ad there exists a costat K > such that for every N we have either x + > K or x + < K, the x i =. i) ad ii) ca e fouded i Propositio 59 ad Exercise 44 i [5, Chapter ], respectively. 3 Weak law of large umers For a sequece of radom variales {ξ k ; k N} o Ω, F, P) with fiite variaces, Markov showed i 93 cf. [6, page 69]) that Var ξ i ) = 3.) is sufficiet for {ξ k ; k N} to oey the WLLN, i.e., for every give ε >, ) P ξ i Eξ i < ε =. 3.) I geeral, it is ot easy to check the Markov coditio 3.). For a sequece ) R +, sice Var ξ i) Varξi )) y Propositio. we otai

4 68 Yutig Lu Theorem 3. Let {ξ k ; k N} e a sequece of radom variales o Ω, F, P) with fiite variaces. If a sequece of positive umers ) satisfies Varξi ) =, 3.3) the for ay ε > it holds that P ξ i ) Eξ i < ε = ; 3.4) i particular, if Varξ) = coverges the 3.4) holds y Lemma.4. If {ξ k ; k N} is pairwise idepedet, Var ξ i) = Varξ i) ad so the Markov coditio 3.) ecomes Varξ i ) =. This is implied i the coditio 3.3) with ) = ) ecause Varξi )) = Varξi )) Varξ i ) y Jese iequality. Hece Theorem 3. with ) = ) is actually cotaied i the aove Markov assertio. Clearly, Theorem 3. ad Lemma.5i) lead to Corollary 3. Let {ξ k ; k N} e a sequece of radom variales o Ω, F, P) with fiite variaces, ad let ) R + satisfy the coditios: = + ad there exists N such that < + for every >. If Varξ ) =, 3.5) the 3.4) holds for every ε >. I particular, if Varξ ) = the 3.) holds for ay ε >. Remark 3.3 The followig two coditios are almost equivalet, Varξ ) = ad Varξi ) =.

5 Weak laws of large umers for sequeces 69 Ideed, y Lemma.5i) the former implies the latter. Coversely, suppose that the latter holds ad that there exists a costat K > such that for every N we have either Varξ + ) Varξ )) > K or Varξ + ) Varξ )) < K. The the former follows from these ad Lemma.5ii). However, it is easy to costruct a sequece satisfyig the latter oly. Cosider a sequece of radom variales {ξ k } k= with variaces Varξ k ) = for k, ad Varξ k ) = for k =, =,,. 3.6) Clearly, they do ot satisfy the first coditio. But it is easy to compute Varξ ) + + Varξ k ) if k =, = if k =, k + + ) if k. k Note that for k we have ) ) k = ) k ). It follows that Varξ ) + + Varξ k ) =. k k So Theorem 3. assures that the sequece {ξ k ; k N} satisfies the WLLN. The sequece {ξ k } k= with variaces as i 3.6) does ot satisfy the followig theorem too. Theorem 3.4 Berstei law of large umers) For a sequece of radom variales {ξ k ; k N} o Ω, F, P), suppose that all Varξ ) are positive ad uiformly ouded, ad that the correlatio coefficiets r ij = Covξ i, ξ j )/σ i σ j ) coverge to zero as i j. The it oeys the WLLN, i.e., 3.) holds for every ε >. The celerated Kolmogorov-Feller WLLN for a sequece {ξ k ; k N} of idepedet ad idetically distriuted radom variales states that k= ξ k Eξ I { ξ }) i proaility as if ad oly if P ξ > ) as cf. [3, Theorem 4.], [9, Theorem 9..4] or [, Theorem 5..3]). There also exist some recet geeralizatios of it, see [3, 7, ]. The followig result may e cosidered a geeralizatio of the sufficiecy part of Kolmogorov-Feller WLLN to sequeces of correlated ad idetically distriuted radom variales. Theorem 3.5 Let {ξ k ; k N} e a sequece of idetically distriuted radom variales, ad let ) e a sequece of positive umers with sup <. Suppose that xp ξ > x) =. 3.7) x

6 7 Yutig Lu The for every ε > it holds that P ξ i where ξ,i := ξ i I { ξi }, i =,,. ) Eξ,i < ε =, 3.8) If E ξ <, which implies 3.7), the correspodig Theorem 3.5 is a geeralizatio of Khitchi WLLN. Though our coditio that sup < is stroger tha oe i [7, )] i may cases our sequece {ξ k ; k N} is ot required to e egatively associated resp. idepedet) comparig with Theorem resp. Theorem ) i [7]. Proof of Theorem 3.5. Sice all ξ k, k N are idetically distriuted, so are ξ,i, i =,,. Let η := ξ + + ξ ad η := ξ, + + ξ,, =,,. Note that ) ) P η Eη ) > ε = P η Eη ) > ε + Pη η ) 3.9) for every ε >. For the secod term o the left had we have Pη η ) P {ξ i ξ,i }) Pξ i ξ,i ) = P ξ i > ) = P ξ > ) as 3.) y 3.7). Moreover, usig Cheyshev iequality ad Propositio. we may estimate the first term o the left had of 3.9) as follows: ) ) P η Eη ) > ε ε Varη ) Varξ ε,i ) = ) Varξ ε, ) ) Eξ,). 3.) ε From [4, Lemma..8] see also [9, Lemma 6..]) it follows that = Eξ, ) = yp ξ, > y)dy = y P ξ > y) P ξ > ) ) dy yp ξ, > y)dy yp ξ > y)dy This ad 3.) yield ) P η Eη ) > ε ) yp ξ ε > y)dy. 3.)

7 Weak laws of large umers for sequeces 7 For ay ɛ >, y 3.7) there exists y > such that yp ξ > y) < ɛ for every y y. So for ay > y we have ) yp ξ > y)dy ) y ) yp ξ > y)dy + yp ξ > y)dy ) y ) yp ξ > y)dy + ɛ y ) y + ) ɛ. < implies / as, it follows that Sice sup ) P η Eη ) > ε ε sup ) ɛ. ) yp ξ ε > y)dy Let ɛ. We otai P η Eη )/ > ε) =. This ad 3.9)- 3.) lead to the expected 3.8). There exists a more geeral versio of the Kolmogorov-Feller WLLN for triagular arrays [9, Theorem 9..5]), which icludes [4, Theorem..6]). Theorem 3.6 Cosider a triagular array sequece of radom variales o Ω, F, P), ξ,i, i =,, k, =,,. Suppose that ay two of {ξ,i } k are idepedet for every N ad that ) is a sequece of positive umers such that ad a) k P ξ,i > ), ) k E ξ,i ), where ξ,i := ξ,i I { ξ,i }. The for η := ξ, + + ξ,k ad a := k E ξ,i it holds that η a P. 3.3) It was showed i [3, page 74, Theorem 3.3] that if k = ad ξ,i = ξ i, i =,, the coditio ) i Theorem 3.6 ca e replaced y slightly weak ) k Var ξ,i ) ; ad coversely 3.3) implies a) ad ). If the idepedece coditio for radom variales i Theorem 3.6 is ot satisfied we ca otai

8 7 Yutig Lu Theorem 3.7 For a triagular array sequece of radom variales, ξ,i, i =,, k, =,,, suppose that ) is a sequece of positive umers such that ad a) k P ξ,i > ), ) k Var ξ,i ), where ξ,i := ξ,i I { ξ,i }. The for η := ξ, + + ξ,k ad a := k E ξ,i it holds that η a P. 3.4) Proof. Put η := ξ, + + ξ,k. For every give ε > we have ) ) η a P > ε η a Pη η ) + P > ε. 3.5) Overse that the coditio a) implies Pη η ) P k {ξ,i ξ,i } ) k P ξ,i > ). 3.6) Moreover, usig Cheyshev iequality, Propositio. ad the coditio ) we deduce ) η a P > ε ε Var η ) k Var ξ ε,i )). This ad 3.5)-3.6) lead to the desired 3.4). Ackowledgemets. Supported y the Traiig Program for College Studets iovatio of Chia 344) ad the NNSF 744 of Chia. I would like to express my hearty thaks to Professor Guagcu Lu for his kid guidace. Refereces [] T. K. Chadra, A. Goswami, O a extesio of the weak law of large umers of Kolmogorov ad Feller. Stoch. Aal. Appl., 34), [] K. L. Chug, A Course i Proaility Theory. 3rd ed. Academic Press,.

9 Weak laws of large umers for sequeces 73 [3] A. Cut, Proaility: A Graduate Course, d ed. Spriger-Verlag, New York, 3. [4] Rick, Durrett, Proaility: Theory ad Examples. Fourth editio. Camridge Uiversity Press,. [5] G. Klamauer, Mathematical Aalysis. Marcel Dekker, Ic., New York, 975. [6] A. N. Kolmogorov, Foudatios of the theory of proaility. Secod Elish Editio. Chelsea Pulishig Compay, New york 956. [7] V. M. Kruglov, A geeralizatio of weak law of large umers. Stoch. Aal. Appl., 9), [8] Xighui Wag, Shuhe Hu, Weak laws of large umers for arrays of depedet radom variales. Stochastics, 864), o. 5, [9] Shijia Ya, Xiufag Liu, Measure ad Proaility, i Chiese). Beijig: Beijig Normal Uiversity Press, 4. Received: Feruary, 5; Pulished: March 5, 5