J. Korea Math. Soc. 42 2005), No. 5, pp. 949 957 ALMOST SURE CONVERGENCE FOR WEIGHTED SUMS OF NEGATIVELY ORTHANT DEPENDENT RANDOM VARIABLES Mi-Hwa Ko ad Tae-Sug Kim Abstract. For weighted sum of a sequece {X, X, 1} of idetically distributed, egatively orthat depedet radom variables such that X r, r > 0, has a fiite momet geeratig fuctio, a strog law of large umbers is established. 1. Itroductio The history ad literature o the strog laws of large umbers is vast ad rich as this cocept is crucial i probability ad statistical theory. The literature o cocepts of egative depedece is much more limited but still very iterestig. Lehma[7] provided a extesive itroductory overview of various cocepts of positive ad egative depedece i the bivariate case. Negative depedece has bee particularly useful i obtaiig strog laws of large umberssee [3, 8, 9, 10]). The almost sure limitig law of weighted sums a ix i, where {X, X i, i 1} is a sequece of i.i.d. radom variables with EX = 0 ad {a i, 1 i, 1} is a array of weights, was ivestigated by may authorssee, Bai ad Cheg[1], Chow ad Lai[4]). For uiformly bouded weights {a i, 1 i, 1} ad i.i.d. radom variables X i with EX = 0, Teicher[11] obtaied 1) lim a i X i /b = 0 a.s. Received May 4, 2004. 2000 Mathematics Subject Classificatio: 60F05. Key words ad phrases: egatively orthat depedet radom variables, strog law of large umbers, idetically distributed, momet geeratig fuctio, weighted sum. This work was supported by WoKwag Uiversity Grat i 2005.
950 Mi-Hwa Ko ad Tae-Sug Kim at a rate b = 1/ log for 1 < 2, ad Chow ad Lai[4] cosidered the case of sup 1 a i ) < for some > 0. A strog law of the form 1) with more geeral ormalizig costats b was also obtaied by Cuzick[5] uder coditio sup 1 a i ) 1 < for some 1 < <. Recetly, Bai ad Cheg[1] derived the followig strog law of large umbers by cosiderig a stadard case whe X r, 0 < r, has a fiite momet geeratig fuctio: Let {X, X i, i 1} be a sequece of i.i.d. radom variables with EX = 0 ad 2) E {exph X r )} < for some h > 0 ad some r > 0 ad let {a i, 1 i } be a double array of real umbers such that, for 1 < < 3) A = lim sup A, <, A, = 1 a i. If 2) holds ad 3) holds for 0, 2), the, for 0 < 1 ad b = 1/ log ) 1/r 4) lim sup a i X i /b h 1/r A a.s., moreover, for 1 < < 2, b = 1/ log ) 1/r)+r 1)/1+r) ad EX = 0, we have 5) a i X i /b = 0 a.s. lim I this paper, we study the similar almost sure limitig behavior o the weighted sums of idetically distributed ad egatively orthat depedetnod) radom variables of the form 1) uder stroger coditio o the momet geeratig fuctio 6) E{exph X r )} < for ay h > 0 ad ay r > 0. 2. Prelimiaries This sectio will cotai some backgroud materials o egative orthat depedece which will be used i obtaiig the major strog law of large umbers i the ext sectio.
Almost sure covergece for weighted sums 951 Defiitio 2.1.Lehma[7]) Radom variables X ad Y are egatively quadrat depedetnqd) if 7) P {X x, Y y} P {X x}p {Y y} for all x, y R. A collectio of radom variables is said to be pairwise NQD if every pair of radom variables i the collectio satisfies 7). It is importat to ote that Defiitio 2.1 implies 8) P {X > x, Y > y} P {X > x}p {Y > y} for all x, y R. Moreover, it follows that 8) implies 7), ad hece, they are equivalet for pairwise NQD. Defiitio 2.2.Ebrahimi ad Ghosh[6]) The radom variables X 1, X 2, are said to be a) lower egatively orthat depedetlnod) if for each 9) P {X 1 x 1,, X x } P {X i x i } for all x 1,, x R, b) upper egatively orthat depedetunod) if for each 10) P {X 1 > x 1,, X > x } P {X i > x i } for all x 1,, x R, c) egatively orthat depedet NOD) if both 9) ad 10) hold. Remark. Ebrahimi ad Ghosh[6] showed that 9) ad 10) are ot equivalet for 3. Cosequetly, the above defiitio is eeded to defie sequeces of egatively depedet radom variables. The followig properties are listed for referece i obtaiig the mai results i the ext sectio. Detailed proofs ca be foud i [6] ad [7]. Lemma 2.3. If {X, 1} is a sequece of NOD radom variables ad {f, 1} is a sequece of Borel fuctios all of which are mootoe icreasig or all mootoe decreasig), the {f X ), 1} is a sequece of NOD radom variables. Theorem 2.4. Let X 1, X 2,, X be oegative radom variables which are upper egatively orthat depedet. The ) 11) E X i E X i )
952 Mi-Hwa Ko ad Tae-Sug Kim 3. Results Lemma 3.1. Let {X, X, 1} be a sequece of idetically distributed NQD radom variables satisfyig 6). Let {X i, 1 i, 1} be a array of rowwise NOD radom variables with EX i = 0 for 1 i ad 1, ad let {a i, 1 i, 1} be a array of positive costats. Assume that, for 1 i, some 0 < β r ad some costat C > 0 12) a i X i C X i β / log a.s. ad, for some sequece {u } of positive costats such that lim u = 0, ad some δ > 0 ad 1 < 2, 13) The 14) X i a i u X i δ /log ) 1 a.s. lim a i X i = 0 a.s. Proof. From the iequality we have e x 1 + x + 1 2 x2 e x for all x R, E[expta i X i )] 1 + 1 2 t2 a 2 ie [ X 2 i expta i X i ) ] for ay t > 0. Let ɛ > 0 ad put t = 2 log )/ɛ. It follows from 12) ad 13) ad the fact that for some τ > 0, ay fixed h > 0, there exists a costat D > 0 such that iequality x 2 De h x β ) for all x R that 15) E[expta i X i )] 1 + 1 ) 2 2 log ) 2 a 2 2 ɛ ie [ Xi 2 exp2/ɛ)log )a i X i ) ] 1 + 2 ) a ɛ 2 u log ) 3 i a i E[C X i β / log ) 2 X i δ exp2/ɛ)c X i β ]
Almost sure covergece for weighted sums 953 1 + 2 ) [ )] a ɛ 2 u log ) i 2 E exp a i ɛ C X i β 1 + 1 ) a 2 log ) i a i { 1 exp 2 log ) a } i a i for all large ad some C > 0 sice e x > 1 + x for x > 0. Next ote that, for ay t > 0, 16) ) ) E exp t a i X i = E expta i X i ) E expta i X i ) by Lemma 2.3 ad Theorem 2.4. From the Markov iequality, 15) ad 16) we obtai ) [ )] P a i X i ɛ e tɛ E exp t a i X i r 17) e tɛ 2 log e = 3/2, E expta i X i ) { 1 exp 2 log a } i a i which is summable. Sice X is are NOD accordig to Lemma 2.3, by replacig X i with X i, from the above statemet we also have ) 18) P a i X i ) ɛ 3 2 for all large. Hece, by the Borel Catelli lemma from 17) ad 18) the result 14) follows. Remark. Lemma 3.1 ca be exteded to the case where {a i } is a array of real umbers. Theorem 3.2. Let {X, X, 1} be a sequece of idetically distributed NOD radom variables with EX = 0 ad satisfyig 6) ad let {a i, 1 i, 1} be a array of positive umbers such
954 Mi-Hwa Ko ad Tae-Sug Kim that 3) holds for some 1 < 2. The, for 0 < r 1/ log ) 1/r, 19) a i X i /b 0 a.s. Proof. We first observe that +1 ad b = ) 2 E a i X i /b EX 2 a 2 i/b 2 EX 2 a i ) 2 /b 2 EX 2 A 2, 2 /b 2 0 as. It follows that a ix i /b 0 i probability. Hece, by Theorem 3.2.1 i Stout[10], it suffices to prove that a i Xi s /b 0 a.s., where {X} s is a symmetrized versio of {X }. So we eed oly to prove the result for {X } symmetric. Defie X i = X i I X i log ) 1/r) log ) 1/r I X i < log ) 1/r) + log ) 1/r I X i > log ) 1/r) ad X i = X i X i = X i log ) 1/r) I X i > log ) 1/r) + X i + log ) 1/r) I X i < log ) 1/r), where I stads for idicator fuctio. Note that both X i ad X i are NOD by Lemma 2.3 ad that 20) X i X i I X i > log ) 1/r).
Sice we have 21) Almost sure covergece for weighted sums 955 E exp X r < P =1 ) X > log 1/r < P X i > log i) 1/r i.o) = 0 by the Borel-Catelli lemma. It follows from 20) ad 21) that 22) X i X i I X i > log ) 1/r) X i I X i > log i) 1/r) <, that is, X i is bouded a.s. It follows that b 1 a i X i b 1 a i X i 23) b 1 b 1 A, max 1 i a i X i a i ) 1 X i X i /log ) 1/r 0 a.s. as. To complete the proof we will apply Lemma 3.1 to the radom variable X i ad weight b 1 a i. Note that b 1 a i X i b 1 a i log ) 1 r)/r X i r ) 1 b 1 a i log ) 1 r)/r X i r b 1 A, 1/ log ) 1 r)/r X i r = A, X i r / log
956 Mi-Hwa Ko ad Tae-Sug Kim ad X i b a i = X i A,/log ) /r A, X i /log ) /r, which satisfy coditios 12) ad 13) of Lemma 3.1. Hece, we have 24) a i X i/b 0 a.s. ad the desired result follows by 23) ad 24). Remark. If 19) holds for ay array {a i } satisfyig 3), the EX = 0 ad 6) holds. The proof is similar to that of Bai ad Cheg [1], 108 109): Suppose 19) is true for ay weights suquece satisfyig 3). Choose, for each, a 1 = = a, 1 = 0 ad a = 1/. The, by 19), we have log ) 1/r X 0 a.s., which implies that E{exph X r ) <. The followig theorem is a slight modificatio of Theorem 3.2. The theorem shows that if the ormig costat b is stroger tha that of Theorem 3.2, the coditio 6) of Theorem 3.2 ca be replaced by weaker coditio 2). Theorem 3.3. Let {X, X, 1} be a sequece of idetically distributed NOD radom variables satisfyig EX = 0 ad 2) ad let {a i, 1 i, 1} be a array of costats satisfyig 3) for some 1 < 2. The, for 0 < r < 1 ad b = 1/ log ) 1/r+β β > 0), a i X i /b 0 a.s. Proof. Defie X i = X i I X i h 1 log ) ) 1/r h 1 log ) 1/r I X i < h 1 log ) 1/r ) + h 1 log ) 1/r I X i > h 1 log ) 1/r) ad X i = X i X i for 1 i ad 1. The rest of the proof is similar to that of Theorem 3.2 ad is omitted.
Almost sure covergece for weighted sums 957 Ackowledgemets. The authors are exceptioally grateful to the referee for offerig helpful remarks ad commets that improved presetatio of the paper. Refereces [1] Z. D. Bai ad P. E. Cheg, Marcikiewicz strog laws for liear statistics, Statist. Probab. Lett. 46 2000), 105 112. [2] T. K. Chadra ad S. Ghosal, Extesios of the strog law of large umbers of Marcikiewicz ad Zygmud for depedet radom variables, Acta. Math. Hugar. 71 1996), 327 336. [3] R. Cheg ad S. Ga, Almost sure covergece of weighted sums of NA sequeces, Wuha Uiv. J. Nat. Sci. 3 1998), 11 16. [4] Y. S. Chow ad T. L. Lai, Limitig behavior of weighted sums of idepedet radom variables, A. Probab. 1 1973), 810 824. [5] J. Cuzick, A strog law for weighted sums of i.i.d. radom variables, J. Theoret. Probab. 8 1995), 625 641. [6] N. Ebrahimi ad M. Ghosh, Multivariate Negative Depedece, Comm. Statist. Theory Methods 10 1981), 307 336. [7] E. L. Lehma, Some Cocepts of Depedece, A. Statist. 43 1966), 1137 1153. [8] P. Matula, A ote o the almost sure covergece of sums of egatively depedet radom variables, Statist. Probab. Lett. 15 1992), 209 213. [9] Y. Qi, Limit theorems for sums ad maxima of pairwise egative quadrat depedet radom variables Syst. Sci. Math. Sci. 8 1995), 251 253. [10] W. F. Stout, Almost Sure Covergece, Academic Press, New York, 1974. [11] H. Teicher, Almost certai covergece i double arrays, Z. Wahrsch. Verw. Gebiete 69 1985), 331 345. Mi-Hwa Ko Statistical Research Ceter For Complex System Seoul Natioal Uiversity Seoul 151-742, Korea E-mail: kmh@srccs.su.ac.kr Tae-Sug Kim Departmet of Mathematics ad Istitute of Basic Sciece WoKwag Uiversity Jeobuk 570-749, Korea E-mail: starkim@wokwag.ac.kr