HAJEK-RENYI-TYPE INEQUALITY FOR SOME NONMONOTONIC FUNCTIONS OF ASSOCIATED RANDOM VARIABLES

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1 HAJEK-RENYI-TYPE INEQUALITY FOR SOME NONMONOTONIC FUNCTIONS OF ASSOCIATED RANDOM VARIABLES ISHA DEWAN AND B. L. S. PRAKASA RAO Received 1 April 005; Revised 6 October 005; Accepted 11 December 005 Let {Y, 1} be a sequece of omootoic fuctios of associated radom variables. We derive a Newma ad Wright 1981 type of iequality for the maximum of partial sums of the sequece {Y, 1} ad a Haje-Reyi-type iequality for omootoic fuctios of associated radom variables uder some coditios. As a applicatio, a strog law of large umbers is obtaied for omootoic fuctios of associated radom varaibles. Copyright 006 I. Dewa ad B. L. S. P. Rao. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial wor is properly cited. 1. Itroductio Let {Ω,, } aprobabilityspacead{x, 1} beasequeceofassociatedradom variables defied o it. A fiite collectio {X 1,X,...,X } is said to be associated if for every pair of fuctios hxadgxfromr to R, which are odecreasig compoetwise, Cov hx,gx 0, 1.1 wheever it is fiite, where X = X 1,X,...,X. The ifiite sequece {X, 1} is said to be associated if every fiite subfamily is associated. Associated radom variables are of cosiderable iterest i reliability studies cf. Barlow ad Proscha [1],Esary et al. [6], statistical physics cf. Newma [9, 10], ad percolatio theory cf. Cox ad Grimmet [4]. For a extesive review of several probabilistic ad statistical results for associated sequeces, see Roussas [14] ad Dewa ad Rao [5]. Newma ad Wright [1] proved a iequality for maximum of partial sums ad Praasa Rao [13] proved the Haje-Reyi-type iequality for associated radom variables. Esary et al. [6] proved that mootoic fuctios of associated radom variables are associated. Hece oe ca easily exted the above-metioed iequalities to mootoic Hidawi Publishig Corporatio Joural of Iequalities ad Applicatios Volume 006, Article ID 58317, Pages 1 8 DOI /JIA/006/58317

2 Haje-Reyi type iequality fuctios of associated radom variables. We ow geeralise the above results to some omootoic fuctios of associated radom variables. I Sectio, we discuss some prelimiaries. Two iequalities are proved for omootoic fuctios of associated radom variables i Sectio 3. Asaapplicatio,a strog law of large umbers is derived for omootoic fuctios of associated radom variables i Sectio 4.. Prelimiaries Let us discuss some defiitios ad results which will be useful i provig our mai results. Defiitio.1 Newma [11]. Let f ad f 1 be two real-valued fuctios defied o R. The f f 1 if ad oly if f 1 + f ad f 1 f are both odecreasig compoetwise. I particular, if f f 1,the f 1 will be odecreasig compoetwise. Dewa ad Rao [5] observed the followig. Remar.. Suppose that f is a real-valued fuctio defied o R.The f f 1 for some real-valued fuctio defied f 1 o R if ad oly if for x<y, f y f x f 1 y f 1 x, f x f y f 1 y f 1 x..1 It is clear that these relatios hold if ad oly if, for x<y, f y f x f1 y f 1 x.. Remar.3. If f is a Lipschitzia fuctio defied o R, that is, there exists a positive costat C such that f x f y C x y,.3 the f f, with f x = Cx..4 I geeral, if f is a Lipschitzia fuctio defied o R,the f f,where f x 1,...,x = Lip f x i, i=1 Lipf = x y f x 1,...,x f y1,..., y i=1 x i y i <. Let {X, 1} be a sequece of associated radom variables. Let i Y = f X1,X,...,.5 <, for N. ii Ỹ = f X1,X,..., iv E Y Ỹ <, E.6 iii f f,

3 I. Dewa ad B. L. S. P. Rao 3 For coveiece, we write that Y Ỹ if the coditios stated i i iv hold. The fuctios f, f are assumed to be real-valued ad deped oly o a fiite umber of X s. Let S = =1 Y, S = =1 Ỹ. Matula [8] proved the followig result which will be useful i provig our results. He used them to prove the strog law of large umbers ad the cetral limit theorem for omootoic fuctios of associated radom variables. Lemma.4. Suppose the coditios stated above i.6hold.the i Var f Var f, ii Cov f, f Var f, iii Var S Var S, iv f 1 + f + + f f 1 + f + + f, v Cov f 1 + f 1, f + f 4Cov f 1, f, vi Cov f 1 f 1, f f 4Cov f 1, f..7 For completeess, ow state the iequalities due to Newma ad Wright [1]ad Praasa Rao [13] for associated radom variables. Lemma.5 Newma ad Wright. Suppose X 1,X,...,X m are associated, mea zero, fiite variace radom variables, ad M m = maxs 1,S,...,S m,wheres = i=1 X i. The E M m Var S m..8 Remar.6. Note that if X 1,X,...,X m are associated radom variables, the X 1, X,..., X m also form a set of associated radom variables. Let Mm = max S 1, S,..., S mad M m = max S 1, S,..., S m. The M m = maxm m,mm ad M m M m +Mm so that E M m Var S m..9 Lemma.7 Praasa Rao. Let {X, 1} be a associated sequece of radom variables with VarX = σ <, 1, ad{b, 1} a positive odecreasig sequece of real umbers. The, for ay ɛ > 0, 1 P max 1 3. Mai results b i=1 Xi E X i ɛ 4ɛ Var X j 1 j Cov X j,x. b j b.10 We ow exted the Newma ad Wright s [1] result to omootoic fuctios of associated radom variables satisfyig coditios.6.

4 4 Haje-Reyi type iequality Theorem 3.1. Let Y 1,Y,...,Y m be as defied i.6 with zero-mea ad fiite variaces. Let M m = max S 1, S,..., S m. The E M m 0Var Sm. 3.1 Proof. Observe that max S = max S S E S S + E S 1 m 1 m max S S E S +max S E S. 1 m 1 m 3. Note that S E S ad S S E S are partial sums of associated radom variables each with mea zero. Hece usig the results of Newma ad Wright [1], we get that E Mm E max 1 m S [ E max S S E S + E max S E S ] 1 m 4 [ Var S m S m +Var Sm ] 1 m by Remar [ Var S m +Var Sm ] = 0Var Sm. We have used the fact that Var S = Var S S + S + S = Var S S +Var S + S +Cov S + S, S S. 3.4 Sice S + S ad S S are odecreasig fuctios of associated radom variables, it follows that Cov S + S, S S 0. Hece Var S Var S S. We ow prove a Haje-Reyi-type iequality for some omootoic fuctios of associated radom variables satisfyig coditios.6. Theorem 3.. Let {Y, 1} be sequece of omootoic fuctios of associated radom variables as defied i.6. Suppose that Y Ỹ, 1. Let{b, 1} beapositive odecreasig sequece of real umbers. The for ay ɛ > 0, 1 P max 1 b i=1 Yi E Y i ɛ 80ɛ Var Ỹ j 1 j Cov Ỹ j,ỹ. b j b 3.5

5 I. Dewa ad B. L. S. P. Rao 5 Proof. Let T = Y j EY j. Note that [ P max T ] 1 b ɛ [ T T E T T + E ] T = P max 1 b ɛ [ P max T T E T 1 b ɛ ] [ T E T + P max 1 b ɛ ] [ 16ɛ Var Ỹ j Y j Cov ] Ỹ j Y j,ỹ Y b j b [ + 16ɛ Var Ỹ j 1 j 1 j Cov ] Ỹ j,ỹ. b j b 3.6 The result follows by applyig the followig iequalities: Var Ỹ j Y j 4Var Ỹj, Cov Ỹj Y j,ỹ Y 4Cov Ỹj,Ỹ Applicatios Let C deote a geeric positive costat. Corollary 4.1. Let {Y, 1} be sequece of omootoic fuctios of associated radom variables satisfyig the coditios i.6. Assume that Var Ỹ j + 1 j < The Y j EY j coverges almost surely. Cov Ỹ j,ỹ <. 4.1 Proof. Without loss of geerality, assume that EY j = 0forallj 1. Let T = Y j ad ɛ > 0. Usig Theorem 3. is easy to see that P,m P T T m ɛ C limp N T T ɛ + P N [ Cɛ Var Ỹ j + j= m T T ɛ j < T m T ɛ Cov Ỹ j,ỹ ]. 4.

6 6 Haje-Reyi type iequality The last term teds to zero as because of 4.1. Hece the sequece of radom variables {T, 1} is Cauchy almost surely which implies that T coverges almost surely. The followig corollary proves the strog law of large umbers for omootoic fuctios of associated radom variables. Corollary 4.. Let {Y, 1} be sequece of omootoic fuctios of associated radom variables satisfyig the coditios i.6. Suppose that Var Ỹ j 1 j < Cov Ỹ j,ỹ b j b <. 4.3 The 1/b Y j EY j coverges to zero almost surely as. Proof. The proof is a immediate cosequece of Theorem 3. ad the Kroecer lemma Chug [3]. Remar 4.3. Birel [] proved a strog law of large umbers for positively depedet radom variables. Praasa Rao [13] proved a strog law of large umbers for associated sequeces as a cosequece of the Haje-Reyi-type iequality. Marciiewicz-Zygmudtype strog law of large umbers for associated radom variables, for which the secod mometis otecessarilyfiite,was studied ilouhichi [7]. Strog law of large umbers for mootoe fuctios of associated sequeces follows from these results sice mootoe fuctios of associated sequeces are associated. However Corollary 4. gives sufficiet coditios for the strog law of large umbers to hold for possibly omootoic fuctios of associated sequeces whose secod momets are fiite. For ay radom variable X ad ay costat >0, defie X = X if X, X = if X<, adx = if X>.The followig theorem is a aalogue of the three series theorem for omootoic fuctios of associated radom variables. Corollary 4.4. Let {Y, 1} be sequece of omootoic fuctios of associated radom variables. Further pose that there exists a costat >0 such that Y Ỹ satisfyig the coditios i.6ad P [ ] Y <, =1 =1 E Y <, Var Ỹj Cov Ỹj,Ỹ j <. 1 j j < The =1 Y coverges almost surely.

7 I. Dewa ad B. L. S. P. Rao 7 Corollary 4.5. Let {Y, 1} be sequece of omootoic fuctios of associated radom variables satisfyig the coditios i.6. Suppose Var Ỹ j 1 j < Let T = Y j EY j.the,foray0 <r<, [ T E Proof. Note that if ad oly if 1 [ E P b T b T b Cov Ỹ j,ỹ b j b <. 4.5 r ] <. 4.6 r ] < 4.7 r >t 1/r dt <. 4.8 The last iequality holds because of Theorem 3. ad coditio 4.5. Hece the result stated i 4.6holds. Acowledgmet The authors tha the referee for the commets ad suggestios which have led to a improved presetatio. Refereces [1] R. E. Barlow ad F. Proscha, Statistical Theory of Reliability ad Life Testig: Probability Models, Holt, Riehart ad Wisto, New Yor, [] T. Birel, A ote o the strog law of large umbers for positively depedet radom variables, Statistics & Probability Letters , o. 1, [3] K. L. Chug, A Course i Probability Theory, Academic Press, New Yor, [4] J. T. Cox ad G. Grimmett, Cetral limit theorems for associated radom variables ad the percolatio model, The Aals of Probability , o., [5] I. Dewa ad B. L. S. Praasa Rao, Asymptotic ormality for U-statistics of associated radom variables, Joural of Statistical Plaig ad Iferece , o., [6] J. Esary, F. Proscha, add. Walup, Associatio of radom variables, with applicatios, Aals of Mathematical Statistics , [7] S. Louhichi, Covergece rates i the strog law for associated radom variables, Probability ad Mathematical Statistics 0 000, o. 1, [8] P. Matula, Limit theorems for sums of omootoic fuctios of associated radom variables, Joural of Mathematical Scieces , o. 6, [9] C. M. Newma, Normal fluctuatios ad the FKG iequalities, Commuicatios i Mathematical Physics , o., [10], A geeral cetral limit theorem for FKG systems, Commuicatios i Mathematical Physics , o. 1,

8 8 Haje-Reyi type iequality [11], Asymptotic idepedece ad limit theorems for positively ad egatively depedet radom variables, Iequalities i Statistics ad Probability Licol, Neb, 198 Y. L. Tog, ed., vol. 5, Istitute of Mathematical Statistics, Califoria, 1984, pp [1] C. M. Newma ad A. L. Wright, A ivariace priciple for certai depedet sequeces, The Aals of Probability , o. 4, [13] B. L. S. Praasa Rao, Haje-Reyi-type iequality for associated sequeces, Statistics & Probability Letters 57 00, o., [14] G. G. Roussas, Positive ad egative depedece with some statistical applicatios, Asymptotics, Noparametrics, ad Time Series S. Ghosh, ed., vol. 158, Marcel Deer, New Yor, 1999, pp Isha Dewa: Theoretical Statistics ad Mathematics Uit, Idia Statistical Istitute, New Delhi , Idia address: isha@isid.isid.ac.i B. L. S. Praasa Rao: Departmet of Mathematics ad Statistics, Uiversity of Hyderabad, Hyderabad , Idia address: blsprsm@uohyd.eret.i