Convergence For Sequences Of Negatively Associated Random Variables

In this paper, we give Convergence for Identically DistributedNegatively Associated variables, Convergence for NegativelyAssociated random variables without Stationary Distribution.Firstly we prove the symmetrical NA sequence is also a NAsequence., Establish the theorem of three series for NA sequences on thenormal truncation form,Let ( X_n )_n∈N be a sequence of negatively associated random variables.If for some c>0 seriesconvergent, then is convergent a.s.In this paper, the exponential inequalities of NA sequencemere studied . The results were almost the same as thoseof the IID sequence .1.Let ( X_n )_n∈N be a sequence of negatively associated randomvariables, Let If exist positive constant g₁ , g₂…g_nand T suchthat for 0 ≤ t ≤ T let have for x ≥ GT,we have2. Let ( X_n )_n∈N be a sequence of negatively associated randomvariables with finite second momentsand E ( X n) = 0let1nn kkS X== ∑ ,2 2 2 21,nk k n kkσ EX Bσ== = ∑ ,for any1 ≤ i ≤ n , X i ≤ cBn , a. s ,,then for any ε > 0,we have2exp 1 , ( 1)( )2 2exp , ( 1)4n nc cP S Bccε ε εεε ε≥ ≤ ??????? ??? ? ??? ???? ? ?????????≤≥( ) exp arcsinn n2 2PS B hcc≥ ε ≤ ??? ? ε ??? ε?????? .where sinh a = e a ?2e?a.Secondly we give the Marcinkiewicz strong law of LargeNumbers ,Kolmogorov Strong law of Large Number ,the wittmann'sstrong law of large numbers for independent random variables isgeneralized to the case of NA random variables.1. Let ( X n )n∈ Nbe a sequence of identically distributed negativelyassociated random variables,let ∑== niSn Xi1,then for 0 < p <2,such thatS nn ?1 pnb→ 0 , a.s.necessary and sufficient conditionE X 1 P< ∞ ,and forb ∈ R;for 1 ≤ p <2, b = EX1.2. Let ( X n )n∈ Nbe a sequence of negatively associated randomvariables and E ( X n) = 0, p ≥ 1,If(1 )21p np,nn E X∞? +=∑ < ∞ then lni →m∞ S nn = 0.a . s .We investigate three equivalent sufficient conditions for weaklaws of large numbers(WLLN) for sequences of identicallydistributed NA variables(IDNA).One example is given toshow that these conditions are not equivalent to WLLN forIDNA sequences. The results illustrate the differences ofconditions for WLLN between IDNA and IID sequences .Howeverit should be noted that IDNA and IID sequences have thesame Marcinkiewicz strong laws of large numbers.Let ( X n )n∈ Nbe a sequence of identically distributed negativelyassociated random variables ,for 0 < p <2,then the following conditionsare equivalent:(i) exist sequence of real numbers {b k , k∈N},such thatm1≤ ka ≤xn S k? bkn?1p??Ρ→0, n→∞,(ii) nP ??? X1 ≥ n1p???→0, n→∞ ,(iii) max 1≤ k ≤nX kn?1p??Ρ→0, n→∞ ,We give a series of results on the complete convergence foridentically distributed negatively associated sequences, these resultshave forms very similar to that for i.i.d sequences.1. Let H (t ) >0and ψ (t ) >0 are Borel ? Cantellimeasurable functionsdefined on t >0,which have the following properties :( )( ) ( )2( )lim ,lim ,t nA If H t→∞ H t = ∞ →∞H nn= ∞is nondeacreasing for t>0,thenand then ( ) ( )1H 2t H t ≤ c,for some c1 > 0and any t > 0.(B) Ifψ ( t)Lebesgue integrable, then ( )tψ t ≥ c2 for some c2 > 0and anyt > 1.(C ) ( ) ( )for ψ t ? ∫0 t uψ u du there exixt 0 < c3 < c4 < ∞ and c>0,such that ∑( ) ( ) ∑( )?==≤ Ψ≤11413 ,ninic iψ inciψi∑i 1 Hi 2 ((i i )) ≤HcΨ 2((n n )),∞=ψ n ≥ 2.2. Let H (t )andψ (t )satisfy the properties( A )( B )(C );{X i , i∈N} be asequence of identically distributed negatively associated random variables,for anyi ∈ N,we haveP ( Xi < x) =P( X1 ?x) ,?x∈R.Let ,( ) ,1,,,1S nXSnkSnXkknin = ∑i=?==then the following conditions are equivalent:(i) ( () ( )) ( ) ( )1 11 ,( )E H X where H t is ofH tt is defined as CΨ ? < ∞?Ψinverse,,(ii) ( ) ( ) ( )∑n∞ = 2 ψ n P ???? m1≤ ka ≤xn S nk≥ εH n????< ∞ , for any ε > 0(iii) ( ) ( )∑n∞ =1 ψ n P ???? m1≤ ka ≤xn S k≥ εH n????< ∞ , for any ε > 03.Let H (t )and ψ (t )satisfy the properties ( A )--( D ), {X i , i∈N} be asequence of identically distributed negatively associated random variables,then the following conditions are equivalent:(i)' If any t > 0,we have t H(t ) ≥ c5 >0,then ΕΨ (H ?1 ( X1)) <∞,andE X 1 < ∞ , EX1= 0,(ii) ( ) ( ) ( )∑n∞ = 2 ψ n P ???? m1≤ ka ≤xn S nk≥ εH n????< ∞ , for any ε > 0(iii) ( ) ( )∑n∞ =1 ψ n P ???? m1≤ ka ≤xn S k≥ εH n????< ∞ , for any ε > 04. Especially 0 < p <2, then the following conditions areequivalent:1) Ε X1 P<∞, ΕX 1 =0 for 1 ≤ p <2,2) for any α > 1 2 and pα> 1,we have( )( )2∑n∞ =1 n pα ? Ρ m1≤ ka ≤xn S nk≥ εnα< ∞ , for any ε > 03) for any α > 1 2 and pα> 1,we have( )2∑n∞ =1 n pα ? Ρ m1≤ ka ≤xn S k≥ εnα< ∞ , for any ε > 05. Especially 0 < p <2, then the following conditions are equivalent:1) Ε [ X 1 P ln (1 +X1)] <∞ , ΕX 1 =0 for 1 ≤ p< 22) ∑n∞ =1 ln (1 + n ) Ρ ??? m1≤ ka ≤xn S n( k ) ≥ εn 1p???n< ∞ , for any ε > 03) ( )1∑n∞ =1 ln 1 + n Ρ ??? m1≤ ka ≤xn S k≥ εn p???n< ∞ ,. for any ε > 06. Especially 1 ≤ p< 2, then the following conditions areequivalent:1) Ε X1 P<∞ , ΕX 1 =0;2) ( )( )2∑n∞ =1 n p ? Ρ m1≤ ka ≤xn S nk≥ εn< ∞ , for any ε > 03) ( )2∑n∞ =1 n p ? Ρ m1≤ ka ≤xn S k≥ εn< ∞ , . for any ε > 0Finally in this article, we prove two strong laws of large number fornegatively associated random variables without stationary distribution.1. Let ( X n )n∈ Nbe a sequence of negatively associated randomvariables and ΕX n =0, n≥1, ,11= ∑≥=SXnnin i,let {a n ,n≥1}be an increasingpositive real sequence,and lni →m ∞ an=+∞,If for some p∈ [1,2] and someq∈ [1, +∞ ),we have Ε X npq< +∞ ,n ∈ Nand∑?Ε<+∞∞=????1(1)(1 )npqnpqnpnpa n aaX then →0nnaS a.s..2. Let ( X n )n∈ Nbe a sequence of negatively associated randomvariables and ΕX n =0, n≥1,let {a n ,n≥1}be an increasing positive realsequence,and lni →m ∞ an=+∞,If the following conditions hold:(i) ∑ Ε∑Ε<+∞∞=?2=11(2 4)2iijX i aiXj,(ii) 01∑Ε2 2→=niX i an,(iii)If exists constant C i>0,such that ∑Ρ><+∞=niXi Ci1( ) that and∑Ε<+∞∞=1224iCi Xiaithen Sa nn→0 a.s..We give complete convergence for sequence of NA random variables inparticular, in this paper , the exponential inequalities of NA sequence.Let ( X n )n∈ Nbe a sequence of negatively associated random variablesand EX n= 0,If exist α > 0,0 < p ≤ 2, q≥ 1, such that aq > 1, then( 1)1q ppqnn∞ n ? α?E X=∑ < ∞ and s n/ nα ?C?→ 0,( n→ ∞ ).