ON THE KOMLÓS RÉVÉSZ ESTIMATION PROBLEM FOR RANDOM VARIABLES WITHOUT VARIANCES. Ben-Gurion University, Israel

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1 ON THE KOMLÓS RÉVÉSZ ESTIMATION PROBLEM FOR RANDOM VARIABLES WITHOUT VARIANCES GUY COHEN Be-Gurio Uiversity, Israel Abstract. Let {X } L p P), 1 < p 2, q = p/p 1), be a sequece of martigale differeces. We prove that the Komlós Révész type weighted averages X / X q p ) 1/ X coverge a.s. ad i the L q p) p - orm, ad the limit is 0 if ad oly if =1 1/ X q p ) =. We show also that covergece eed ot hold whe we deal with a cetered ucorrelated sequece whether the series =1 1/ X 2 2 ) coverges or ot). Furthermore, for 1 < p < 2 all the results of Komlós Révész are exteded to symmetric idepedet p-stable radom variables. 1. Itroductio Let {X } be a sequece of idepedet radom variables with costat mea m ad fiite variaces. It was proved by Komlós ad Révész [12, Theorem 1] that the weighted averages X /varx ) ) 1) 1/varX ) ) coverge a.s. to m if ad oly if =1 1/varX ) ) =. It was oted there, usig Cauchy s iequality, that amog all the possible weighted averages of X 1,, X the above gives the miimal variace of error, ad this miimum is attaied oly for the weighted average 1). It was also proved i Theorem 2 there that for ay sequece of positive umbers σ for which =1 1/σ2 ) coverges there exists a sequece of cetered idepedet radom variables {ξ } L 2 P) with ξ 2 2 = σ2 such that o sequece of weighted averages of {ξ }, i particular 1), coverges to m eve) i probability. It is worth metioig that there is aother approach to the cocept of weighted averages where the weights are pre-specified idepedetly of the Date: 24 October Mathematics Subject Classificatio. Primary: 60F15, 60F25; Secodary: 60G42, 60G52, 62F12. Key words ad phrases. idepedet radom variables, martigale differeces, p- stable radom variables, weighted averages, a.s. covergece, orm covergece, cosistet estimatio of a commo mea. 1

2 2 GUY COHEN radom variables uder cosideratio. I that case we loo for coditios o the weights such that the weighted averages a.s. coverge for ay sequece of radom variables i a certai class. We metio the wor of Jamiso, Orey ad Pruitt [10] for itegrable i.i.d. radom variables exteded i the recet wor of Etemadi [7] to idetically distributed pairwise idepedet itegrable radom variables). Beyod idepedece, we metio the wor of Azuma [1] for uiformly bouded martigale differeces see also [14] for recet results ad exteded refereces). I this ote we defie Komlós Révész type weighted averages for martigale differece sequeces, eve without variaces. For {X } L p P), 1 < p 2 with dual idex q = p/p 1), they have the form X / X q p ) 1/ X q p). We prove that these weighted averages coverge a.s. ad i the correspodig orm whether the series =1 1/ X q p ) coverges or diverges. The limit is 0 if ad oly if =1 1/ X q p ) diverges. The Theorem 1 of [12] follows as a corollary. We discuss the limitatio of the theory whe we replace the idepedet radom variables by cetered ucorrelated radom variables. Fially, for p < 2 ad symmetric idepedet p-stable radom variables we obtai complete aalogs of Theorem 1 ad Theorem 2 of [12]. 2. Komlós Révész estimatio for martigale differeces Theorem 2.1. Let {X } L p P), 1 < p 2, be a sequece of martigale differeces with respect to the icreasig sequece of σ-algebras {F }. Let q = p/p 1) ad assume that X p 0 for every 1. The the weighted averages 2) X / X q p ) 1/ X q p) coverge a.s. ad i the L p -orm. The limit is 0 a.s. if ad oly if the series =1 1/ X q p ) diverges. Proof. We distiguish betwee two cases. I the first case we assume that =1 1/ X q p) diverges. We quote the Abel Dii result as it appears i Hildebradt [9]. If {d } is a sequece of positive umbers such that =1 d diverges, the for every α > 1, the series =1 d / d ) α) coverges. Now, put Y = X / X q p 1/ X q p )). By our assumptio ad by the Abel Dii theorem we obtai that Y p p = 1/ X q p 1/ X q p) ) p <. =1 =1

3 ON KOMLÓS RÉVÉSZ ESTIMATION PROBLEM 3 Usig Chow s extesio [3, Corollary 5] of the Marciiewicz-Zygmud result for martigales, we coclude that =1 Y coverges a.s. Kroecer s lemma the yields the a.s. covergece to 0 of 2). Usig Theorem 2 of Bahr ad Essee [2], we obtai p Y 2 =j p Y p p = 2 1/ X q p 1/ X q p) ) p 0. j,l =j =j Hece =1 Y coverges i the L p -orm. Usig a Baach space versio of Kroecer s lemma see [5]), the averages i 2) coverge to ecessarily) 0 i the L p -orm. Assume that =1 1/ X q p ) coverges. Put Z = X / X q p. By assumptio, =1 Z p p coverges. So by Chow s theorem the series =1 Z coverges a.s. ad the averages i 2) coverge a.s. Agai, usig the Bahr Essee result, p Z 2 =j p Z p p = 2 1/ X q p ). =j So, the series =1 Z = =1 X / X q p) coverges i the L p -orm ad so the weighted averages 2) coverge i the L p -orm ecessarily to the same a.s. limit). We will show that it is a o-zero limit. Usig Jese s iequality for coditioal expectatios e.g., [6, p. 33]), we have +1 E Z p ) +1 ) F E Z F p p = Z. Hece, So, +1 p Z =j p Z Z 1 p > 0. p 0 < Z 1 p lim Z = p Z. Hece =1 Z is o-zero ad the weighted averages 2) do ot coverge to zero, either a.s. or i orm. Corollary 2.2. Let {X } L p P), 1 < p 2, be a sequece of idepedet radom variables with E[X ] = m for every 1. Let q = p/p 1) ad assume that X m p 0 for every 1. The the weighted averages X / X m q p ) 1/ X m q p)

4 4 GUY COHEN coverge a.s. ad i the L p -orm. The limit is a.s. m if ad oly if the series =1 1/ X m q p ) diverges. Proof. Apply the previous theorem to {X m}. Remars. 1. The above corollary exteds Theorem 1 i Komlós Révész [12] to the case where oe of the radom variables has a fiite variace so for p = 2 the averages 2) are ot defied). 2. The above theorem is a geeralizatio of Komlós Révész beyod the scope of idepedece. 3. The statistical meaig of the above corollary is the followig: if you have oisy measuremets of a uow quatity m by idepedet devices, such that the imperfectio of each device is measured by the p-th orm of its deviatio which is assumed to be ow, or ca be estimated i advace for each device), the the suggested sequece of weighted averages is a strog cosistet estimator of m if the Komlós Révész type coditio holds. For p < 2, X m p is the p-th aalog of the stadard deviatio of measuremet by the -th device. The Komlós Révész type coditio meas that the sequece of p-deviatios should ot grow too fast. Example 1. A sequece of cetered idepedet radom variables {X } which for every 1 < p 2 have weighted averages 2) that coverge a.s., while their usual Cesàro averages fail to coverge a.s. We costruct a cetered idepedet sequece by the followig law: PX = ±) = 2) 1 ad PX = 0) = 1 1. Hece by costructio the series =1 P X = ) diverges ad by the Borel Catelli lemma we have lim sup X / = 1. So, the usual Cesàro averages fail to coverge a.s. O the other had, X p = p 1)/p, so =1 1/ X q p ) = =1 1/) diverges ad the weighted averages 2) coverge to 0 a.s. Example 2. For every 1 < p < 2 there exits a sequece of cetered idepedet radom variables {X } without variaces such that the series =1 1/ X q p) diverges. Tae a sequece {ξ } of cetered i.i.d. radom variables such that ξ 1 p < ad ξ 1 2 =. Put X = 1/q ξ with q = p/p 1). Clearly, the assertios of the example hold so our theorem applies, while the result of [12] caot be applied. Example 3. For every 1 < p < 2 there exits a sequece of symmetric idepedet bouded radom variables {X } such that the series =1 1/ X q p ) diverges but =1 1/ X 2 2 ) coverges. Tae a symmetric idepedet sequece {ξ } L p with ifiite variaces for which =1 1/ ξ q p) diverges. Sice ξ 2 =, for every there exists α > 0 such that E[ ξ 2 1 { ξ α }] 2. Now, defie the symmetric

5 ON KOMLÓS RÉVÉSZ ESTIMATION PROBLEM 5 idepedet sequece X = ξ 1 { ξ α }. By costructio =1 1/ X 2 2) coverges ad sice X p ξ p, the series =1 1/ X q p ) diverges. Example 4. A sequece of symmetric idepedet bouded radom variables {X } exists such that for every 1 < p < 2 the series =1 1/ X q p ) coverges but =1 1/ X 2 2 ) diverges. Defie the idepedet sequece by the followig rule PX = ± ) = 1/2. So, X p = ad sice q > 2, the sequece {X } satisfies the assertios. It is a atural questio to as whether the above theorem ad its corollary hold for other classes of radom variables. Natural classes are the cetered ucorrelated or pairwise idepedet radom variables. The extesio of Theorem 1 of [12] to the latter class was explored by Rosalsy [18]. Usig the method of N. Etemadi, he proved that if {X } is a sequece of pairwise idepedet radom variables with fiite variaces ad a costat mea m, the uder the followig three coditios i) 1/varX ) ) =, ii) lim varx ) =1 1/varX ) ) =, iii) sup X 1 <, the weighted averages 1) coverge a.s. to m. O the other had, we prove that for martigale differeces the weighted averages 2) coverge a.s. whether i) above holds or ot it is oly the idetificatio of the limit which depeds o i). We will show that for the class of ucorrelated radom variables a.s. covergece of the weighted averages 1) may fail, with or without the Komlós Révész coditio i); i the followig two examples ii) is satisfied. Example 5. A sequece of cetered ucorrelated radom variables ca be costructed such that the series =1 1/ X 2 2) coverges ad the weighted averages 2) diverge a.s. K. Tadori [19] proved the followig result: if { a } is a o-icreasig sequece for which =1 a 2 log 2 diverges, the there exists a cetered ucorrelated sequece {φ } i 0, 1) for which the orthogoal series =1 a φ diverges a.s. Hece there exists a cetered ucorrelated sequece {φ } for which the orthogoal series =1 φ / log ) ) diverges a.s. Now, put X = log φ. So, =1 1/ X 2 2 ) = =1 1/ log 2 ) ) coverges ad =1 X / X 2 2 ) = =1 φ / log ) ) diverges a.s. Hece, the weighted averages 2) fail to coverge a.s.

6 6 GUY COHEN Example 6. A sequece of cetered ucorrelated radom variables exists such that the series =1 1/ X 2 2 ) diverges ad the weighted averages 2) diverge a.s. L. Cseryá [4] has proved see also [17]) the followig result: If { a } is a o-icreasig sequece for which =1 a 2 diverges, the there exists a cetered ucorrelated sequece {φ } i 0, 1) so that lim sup a φ / log = a.s. Hece, there exists a cetered ucorrelated sequece {φ } such that lim sup φ / ) /log = a.s. Now, put X = φ. So, X / X 2 2 ) 1/ X 2 2 ) = φ / ) 1/) φ / ), log ad the weighted averages a.s. diverge. Remars. 1. Examples 5 ad 6 show that eve the covergece part of Theorem 2.1 does ot hold i geeral for cetered ucorrelated radom variables. We do ot ow if it does for cetered pairwise idepedet radom variables with fiite variaces with o additioal coditios. 2. Accordig to Tadori s wors the costructed cetered ucorrelated sequece {φ } which we refer to i Example 5 ca be tae to be uiformly bouded. The cetered ucorrelated sequece {φ } costructed by Cseryá [4] is ubouded. The questios whether oe could costruct {φ } to be real uimodular, i.e., ±1 a.s., i either costructio) are still ope. I would lie to tha professor Ferece Móricz for clarifyig this poit. Usig a result of M. Kac, affirmative aswers) to this these) questios) will imply that {X } i Example 5 ad/or Example 6 could be tae to be cetered ad pairwise idepedet. This would show that Theorem 2.1 ca ot be exteded eve to the pairwise idepedet case without the additioal assumptios) of Rosalsy ii) ad/or iii). 3. Symmetric p-stable idepedet radom variables Defiitio. A real radom variable X is called a symmetric p-stable radom variable r.v.) with parameter σ = σ p X) > 0 ad idex 0 < p < 2 if for all t R its characteristic fuctio satisfies E[expitX)] = exp σ p t p /2). It is ow that see Feller [8, XVII, 4], ad for more specific calculatio see Marcus ad Pisier [16, 1]), lim t tp P X > t) = c p σ p for some c p which depeds oly o p.

7 ON KOMLÓS RÉVÉSZ ESTIMATION PROBLEM 7 It is also ow that a p-stable radom variable might ot have a absolute p-th momet, but accordig to the above property it does have a absolute r-th momet for every r < p. Now, if {Z } are i.i.d. symmetric p-stable radom variables, the for all complex sequeces {a }, by stability we have ) 1/p. a Z D = Z 1 a p Hece, for every r < p, there exists a positive costat c p,r, which depeds oly o p ad r such that r ) 1/p. a Z = c p,r a p Lemma 3.1. Let {α } be a positive sequece with α = 1. If 1 < p < ad q = p/p 1), the for every sequece of umbers {σ } we have α p σp 1/ 1/σ q )) p 1. Proof. Usig Hölder s iequality we obtai 1 = ) p α = α σ 1/σ ) )p From ow o we always assume that 1 < p < 2. α p σp ) 1/σ q )) p 1 Theorem 3.2. Let 1 < r < p < 2. Let {Z } be symmetric p-stable idepedet radom variables. Let q = p/p 1) ad for every 1 put σ = σ p Z ). Assume that σ > 0 for every 1. The the weighted averages 3) Z /σ q ) 1/σq ) coverge i the L r -orm ad a.s. The limit is a.s. 0 if ad oly if the series =1 1/σq ) diverges. Amog all the weighted averages of {Z 1,..., Z } the weighted average 3) has the miimal L r -orm ad this miimum is attaied oly with 3). Proof. Clearly, if X is a symmetric p-stable r.v. with parameter σ, the X/σ is a symmetric p-stable r.v. with parameter 1. Therefore, the sequece {Z /σ } is a sequece of i.i.d. symmetric p-stable radom variables with commo parameter 1. Hece for r < p we have =j Z /σ q r 1/σq ) = =j 1/σ q 1 )Z /σ ) 1/σq ) r =

8 8 GUY COHEN ) c p,r =j 1/σ q ) 1/p 1/σq )) p If =1 1/σq ) diverges, the Abel Dii theorem yields covergece of the series [ =1 1/σ q )/ 1/σq )) p]. So, the right had side of *) teds to zero as mij, l). This meas that [ =1 Z /σ q)/ 1/σq ))] coverges i the L r -orm, hece i probability. By the Lévy Itô Nisio theorem see Ledoux ad Talagrad [13, Theorem 6.1]) the series coverges a.s. So, Kroecer s lemma also its Baach space versio [5]) yields the a.s. L r -orm) covergece to 0. Now, we assume that =1 1/σq ) coverges. Usig the same idea as above, for r < p, Z /σ q ) r ) 1/p = c p,r 1/σ q ) 0. j,l =j Therefore, the series l =j Z /σ q ) coverges i the L r-orm. Agai usig the Lévy Itô Nisio theorem, it coverges a.s. So do the weighted averages coverge a.s. ad i the L r -orm. The equality =j Z /σ q ) p = c p p,r r 1/σ q ) shows that the L r -limit hece the a.s. limit) is ot zero. Now for the last assertio, for ay sequece of weights w ) such that = 1, usig the Lemma above, we have, Z p = w ) σ )Z /σ ) p = c p p,r w ) σ ) p r r w) w ) c p p,r 1/σq )) p 1 = cp p,r 1/σq ) 1/σq )) p = Z /σ q ) p r 1/σq )) p. Uder the assumptio w) = 1, equality i Hölder s iequality holds oly whe w ) = 1/σ q )/ 1/σq ). Corollary 3.3. Let 1 < p < 2 ad put q = p/p 1). Let {σ } be a sequece of positive umbers such that the series =1 1/σq ) coverges. The there exists a sequece of symmetric idepedet p-stable radom variables with σ = σ p Z ) such that for ay sequece of positive weights {w ) : 1} 1 with w) = 1, the weighted averages w) Z do ot coverge to zero i probability.

9 ON KOMLÓS RÉVÉSZ ESTIMATION PROBLEM 9 Proof. Costruct a sequece of symmetric idepedet p-stable radom variables {Z } with parameters σ = σ p Z ), = 1, 2,.... By stability we have w ) Z = w ) σ )Z /σ ) D = Z 1 /σ 1 ) w ) σ ) p) 1/p. Put M = =1 1/σq ). By the lemma above, we obtai P ) > t = P Z1 /σ 1 ) w ) σ ) p) 1/p ) > t w ) Z P Z1 /σ 1 M p 1)/p > t ) > 0. Remars. 1. The above corollary is the aalog, for the p-stable case, of Theorem 2 of Komlós Révész [12]. 2. The results i this sectio could be doe for symmetric complex idepedet p-stable radom variables. ACKNOWLEDGEMENTS. I would lie to tha Michael Li for suggestig this problem ad for may helpful discussios. Refereces [1] Azuma, K. 1967). Weighted sums of certai depedet radom variables. Tôhou Math. J. 2, 19, [2] vo Bahr, B. ad Essee, C.G. Iequalities for the r-th absolute momet of a sum of radom variables, 1 r 2, A. Math. Stat ), [3] Chow, Y.S. 1965). Local covergece of martigales ad the law of large umbers, A. Math. Statist [4] Cseryá, L. 1975). O the covergece of orthogoal series, Period. Math. Hugar. 6, o. 3, [5] Derrieic, Y. ad Li, M. 2001). Fractioal Poisso equatios ad ergodic theorems for fractioal coboudaries. Israel J. Math. 123, [6] Doob, J.L. Stochastic Processes. Joh Wiley & sos, New Yor [7] Etemadi, N. 2006). Covergece of weighted averages of radom variables revisited. Proc. Amer. Math. Soc. 134, o. 9, [8] Feller, W. A itroductio to probability theory ad its applicatios. Vol. II. Joh Wiley ad Sos, Ic [9] Hildebradt, T. H. 1942). Remars o the Abel Dii theorem. Amer. Math. Mothly [10] Jamiso, B., Orey, S., ad Pruitt, W. 1965). Covergece of weighted averages of idepedet radom variables. Z. Wahrscheilicheitstheorie ud Verw. Gebiete [11] Kac. M. 1936). Sur les foctios idépedates I Propriétés géérales). Stud. Math. 6,

10 10 GUY COHEN [12] Komlós, J. ad Révész, P. 1965). O the weighted averages of idepedet radom variables. Magyar Tud. Aad. Mat. Kutató It. Közl. 9, [13] Ledoux, M. ad Talagrad, M. Probability i Baach spaces. Ergebisse der Mathemati ud ihrer Grezgebiete 3) [Results i Mathematics ad Related Areas 3)], 23. Spriger-Verlag, Berli, [14] Li, M. ad Weber, M. 2007). Weighted ergodic theorems ad strog laws of large umbers. Ergodic Theory Dyam. Systems 27, o. 2, [15] Marciiewicz, J. ad Zygmud, A. 1937). Sur les foctios idépedetes. Fud. Math. 29, curretly available at [16] Marcus, M.B. ad Pisier, G. 1984). Characterizatios of almost surely cotiuous p-stable radom Fourier series ad strogly statioary processes, Acta Math. 152, [17] Móricz, F. ad Tadori, K. 1987). Couterexamples i the theory of orthogoal series, Acta Math. Hugar. 49, o. 1-2, [18] Rosalsy, A. 1986). A strog law for weighted averages of radom variables ad the Komlós Révész estimatio problem, Calcutta Statist. Assoc. Bull. 35, o , [19] Tadori, K. 1957). Über die orthogoale Futioe. I. Acta Sci. Math. Szeged) 18, Be-Gurio Uiversity of the Negev, Israel address: guycohe@ee.bgu.ac.il