 Applied Mathematics, 2011, 2, 1175-1181 doi:10.4236/am.2011.29163 Published Online September 2011 (http://www.SciRP.org/journal/am) Copyright © 2011 SciRes. AM A Precise Asymptotic Behaviour of the Large Deviation Probabilities for Weighted Sums Gooty Divanji, Kokkada Vidyalaxmi Department of Statistics, Manasagangothri University of Mysore, Mysore, India E-mail: gootydivan@yahoo.co.in, vidyalaxmi.k@gmail.com Received June 11, 2011; revised July 1, 2011; accepted July 8, 2011 Abstract Let {Xn, n ≥ 1} be a sequence of independent and identically distributed positive valued random variables with a common distribution function F. When F belongs to the domain of partial attraction of a semi stable law with index , 0 <  < 1, an asymptotic behavior of the large deviation probabilities with respect to prop-erly normalized weighted sums have been studied and in support of this we obtained Chover’s form of law of iterated logarithm. Keywords: Large Deviations, Law of Iterated Logarithm, Semi-Stable Law, Domain of Partial Attraction, Weighted Sums 1. Introduction Let {Xn, n ≥1} be a sequence of independent and identi-cally distributed (i.i.d) positive valued random variables (r.v.s) with a common distribution function F. Let BV [0,1] be the set of all continuous bounded variation func-tions over [0,1]. Set nnkk=1S=X,n 1, and nnkk=1kT= fXn, where f is a member of BV[0,1]. Let {nk, k ≥ 1} be a strictly increasing subsequence of positive integers such that k+1knr1n as k  . Kruglov  established that, if there exists sequences (ak) and (bk) of real con-stants, bk   as k  , such that knkαkkSLim Pax= Gxb at all continuity points x of G, then G is necessarily a semi stable d.f with characteristic exponent, 0 <   2. When  = 2, semi-stable becomes normal. It is known that probabilities of the type nnPS>x, or either of the one sided components, are called large deviation probabilities, where {xn, n ≥ 1} is a monotone sequence of positive numbers with xn   as n   such that pnnS0x . In fact, under different conditions on sequence of r.v.s, Heyde [2-4] studied the large devia-tion problems for partial sums. In brief, for the r.v.s which are in the domain of attraction of a stable law and r.v.s which are not belong to the domain of partial attrac-tion of the normal law, Heyde  and  established the order of magnitude of the larger deviation probabilities, where as in Heyde , he obtained the precise asymp-totic behavior of large deviation probabilities for r.v.s in the domain of attraction of stable law. When r.v.s. has i.i.d symmetric stable r.v.s, Chover  obtained the law of iterated logarithm (LIL) for partial sums by normalizing in the power and for r.v.s which are in the domain of attraction of a stable law, Peng and Qi  obtained Chover’s type LIL for weighted sums, where the weights are belongs to BV[0, 1]. Many authors studied the non-trivial limit behavior for different weighted sums. See Peng and Qi  and references therein. Probability of large values plays an important role in studying non-trivial limit behavior for stable like r.v.s. As far as properly normalized partial sums of stable like r.v.s, we can use the asymptotic results of Heyde [2-4]. (See Divanji ). However the observations made by Heyde [2-4] on the large deviation probabilities implic-itly motivated us to study the large deviation probabili-ties for weighted sums. In fact, when the underlying i.i.d positive valued r.v.s are in the domain of partial attrac-tion of a semi stable law of Kruglov’s  setup, denoted as F  DP (), 0 <  < 1, a precise asymptotic behavior G. DIVANJI ET AL. 1176 of the large deviation probabilities of Heyde [2-4] can be obtained for weighted sums. In support of this can be considered for Chover’s type of non-trivial limit behav-ior for weighted sums. In the next section we present some lemmas and main result in Section 3. In the last section, we discuss the existence of Chover’s form of LIL for weighted sums. In the process i.o, a.s and s.v. mean ‘infinitely often’, ‘al-most surely’ and ‘slowly varying’ respectively. C, , k and n with or without a super script or subscript denote positive constants with k and n confined to be integers. In the sequel, observe that when  < 1, ak can always be chosen to be zero. 2. Lemmas Lemma 2.1 Let F  DP (), 0 <  < 1. Then there exists s. v. func-tion L, such that Xx1FxLim =1Lx. Lemma 2.2 Let F  DP (), 0 <  < 1 and let n1B= infx > 0: 1Fxn. Then Bn = n1/ l(n), where l is a function s. v. at . The above lemmas can be referred to Divanji and Vasudeva . Lemma 2.3 Let L be any s. v. function and let (xn) and (yn) be se-quence of real constants tending to  as n . Then for any  > 0, nnδnn nLxyLim y=Lx and nnδnnnLxyLimy= 0Lx . This lemma can be referred to Drasin and Seneta . Lemma 2.4 Let F  DP (), 0 <  < 1. Let (xn) be a monotone se-quence of real numbers tending to ∞ as n→∞ and Bn defined in Lemma 2.2. Then p11nnnBxS 0 , as n→∞. This lemma can be referred to Vasudeva and Divanji . Lemma 2.5 Let F  DP (), 0 <  < 1. Let (xn) be a monotone se-quence of real numbers tending to ∞ as n→∞ and Bn defined in lemma 2. Then p11nnnxB T0 as n → ∞, with Bn defined in Lemma 2.2. Proof Since f  BV[0,1]. Hence there exists a constant C such that f(x)≤ C and n-1k=1kk+1ffnn   C, for all n ≥ 1. Therefore nnkk=1n-1kn1k nk=1kT= fXnkk+1ffS+ f1S2Cmaxnn     kS Dividing on both sides by xnBn, we have n1k nnn nnT2C maxxB xBkS Observe that Xi’s are i.i.d posi- tive valued r.v.s which are in the domain of partial at-traction of a semi-stable law and hence k1k nnn nnSSmax xB xB n and by lemma 2.4 we have pnnnSxB. This gives pnnnT0xB , as n  . 3. Main Results Theorem 3.1 Let F  DP (), 0 <  < 1. Let (xn) be a monotone se-quence of real numbers tending to ∞ as n→∞ and Bn defined in lemma 2.2. Then nnnnnnPT xBLim= 1n PXxB. Proof To prove the assertion, it is enough to show that nnnnnnnnnnnnPT xB0< Liminfn PXxBPT xBLimSup< .n PXxB Let  > 0 and define iiniA= fX1+εxBn n and nijnnj=1, jijB=fX εxBn1, 2,,in, . Proceeding on the lines of Heyde  and Lemma 3.1 of Vasudeva , we get,  nnnnniiji=1 j=111 1PTxBPA PBPAnP APBnP A  (1) From Lemma 2.5, we have pnnnT0xB , as n   and given  > 0 with 1 – 2 > 0, we can choose N1 so Copyright © 2011 SciRes. AM G. DIVANJI ET AL.1177 nlarge such that P(Bi) > 1 – 2 for all n  N1 and for all . Further from Lemma 2.5, we see that nP(Ai)  0 as n  , so that we can choose N2 so large that n P(Ai) < , for n  N2. Thus for n  N = max (N1, N2), we obtain from (1), , this im-plies 1,2, 3,,innnPT xBnnn12PX1+εxB   nnnnnnn nnnnnnn1 2PX1+xBPT xBnPXx BnPXx BPX1+xB12 PXxB . Using Lemma 2.1, we have ααnnnnn nnαααnn nnnnnnαnnL1+ xBPTxB1 2xBnPXx BLx BxB1+εL1+ xB12LxB1+ Choose  > 0 sufficiently very small such that nnnnnL1+ xBLim =1LxB , one can find a constant C1 > 0 such that nnn1nnnPT xBLim inf>C0nPXx B. In order to complete the proof, we use truncation method. Define kkkkX,if fXxBY= n0, otherwise nn Let kkkkR=fXfYnn   k, n1,n kk=1kT= fYn and . Notice that n2,n kk=1T= Rnnn1,nnn 2,nPTxBPT >xB+PT0 . This implies 1,nn n2,nnnnnnnn nnPT xBPT0PT xB+n PXxBn PXxBn PXxB(2) Observe that 2,n1n n1PT0nPR0 =nPfXxBn , for fixed n and f is continuous BV [0,1] and it attains bounds. Hence using Lemma 1, we have, nn2,nnn nnnnαααnnααnnnnnnαnn1n PfXxBPT 0n=n PXxBnPXxBxBL1fnxB 1fLxBnxBxBL1fn1f.nLxB   (3) Using Karamata’s representation of s.v. function, one gets that   nnnnnnnnnnnn xB1fxBnεyyyynnnn 00nn xB1fnyynn xBxB xBLa11ffnn=expdy dyLxBaxBxBa1fnexpdy .axB      Since a(x)  C as x  C and (y)  0 as y  , there exists C0 > 0 and 0 < , such that nn0nnxBa1fnCaxB, (y)  0, for . nnyxBThis yield 0nn000nnxBL1fnC1Cexp log1LxB1ffnn      . (4) Substituting (4) in (3), one can find some constant C1 such that the second term in (2) becomes Copyright © 2011 SciRes. AM G. DIVANJI ET AL. 1178 -2,n1nnPT 01CfnPXx Bn. Since f  BV[0,1] and 1ff0BV[0,1], as nn  . Therefore we can find some constant C2 (>C1) such that 2,n2nnnPT 0LimC C3) such that n22kk=1422nn nnkfEYnCn xBPXxB . (7) Observe that 2nn nkm kk=1 m=1k=1km kffEY EYfEYnn n    . Now for 0 <  < 1, nnnnxBkfnkkxB 0xkfnEYEY=xdP(X < x)PXxdx Let nnkmk=1 m=122nn nnkkffEY EYnnAnx BP(XxB ), 2nkk=122nn nnkfEYnBnx BP(XxB) and nn 2xBkfnnk=1 022nn nnkfPXxdnDnx BP(Xx B) x. Notice that A  B  D. Again using Lemma 2.1, we have nnnn2xBkfnn-k=1 02- 2-nn nn2xBkfnn-k=1 nn0nn2- 2-nnkfLxxdxnDnx BLxBLxkfxdxnLxBLx B.nx B Copyright © 2011 SciRes. AM G. DIVANJI ET AL.1179 Following similar steps of (4), we can find some con-stant C5 and 0 > 0 such that 0nn50nnLx xBC1+LxB x. Hence nn0002xBkfnn--50 nnk=1 02- 2-nnkC1+fxdxx BnDnx B. Also nn000xBkfn11 +1nn001kxdx = x B f1n    0and there exists C6 (>C5) such that 02n+6nk=1nnkCf LxBnDnx Bn. Let Mn = xn Bn, where xn   and Bn   as n  . Since F DP (), 0 <  < 1, then We know that n7αnnL MCM (8) Using (8) one can find some constant C8 such that 02n+8k=12kCf nDn. Since f is Continuous BV [0,1], therefore there exists C9 such that 0n+9k=1kfnnC and hence D ≤ C9  B ≤ C9  A ≤ C9 (9) From (7) and (9), we claim that 1,nn nnnPT xB0nPXx B as n, i.e., holds. Substituting (5) and (6) in (2), we get nnnnnnPT xBLimSup< n PXxB. The proof of the theorem is completed. 4. Chover’s Form of LIL Theorem 4.1 Let F  DP (), 0 <  < 1. Then 1log log n1nnnTLimSup= ea.sB . Proof To prove the assertion, it suffices to show for any   (0, 1), that 1+nnPTBlogni.o =0 (10) and 1nnPTBlogni.o=1 (11) To prove (10), let An=1+nnTBlogn and 1+nnx= Blogn. By the above Theorem 3.1, one can find a C10 such that, n10nPACnPXx. Using Lemma 2.1, n10nnnn10 1+αnnPACnxLxLB Lx CnLBB(logn). Applying Lemma 2.3 with = 2 and using the boundedness of ,  1+ 2n11PAClogn for some C11 > 0. Consequently and (3) follows nn = 1P(A) < from the Borel-Cantelli Lemma. Define, for large k, k1kjm= minj:nβ, (12) where  > 1 and  > 0 and from the relation kkk1 k1nnn nT= TT+ T, k  1, and in order to establish (11), it is enough if we show that   (0, 1), that 1mm m kkk1knnnmPTT2Blogni.o = 1 (13) and 1mm kk-1 knn mPTBlogni.o = 0 (14) Define 1nnz = Blogn and mmmkk-1 kkn nnD= (T T)z, k11. Note that mm mmkk 1kkdnn nnTT T , k ≥ 1. By the above Theorem 3.1, one can find a constant C12 > 0 and k1 such that for all k ( k1), kk1 mkk-1kmkkk12mm nm12 mnmPDCn nPX2 zn= C n1 P(X 2z)n. Copyright © 2011 SciRes. AM G. DIVANJI ET AL. 1180 Since F  DP (), 0 <  < 1 and under Kruglov’s  setup i.e., k+1k knLim=r>1n implies that there exists  = r–1 (>1) such that k1kmmn<<1nλ for all k  k1. (15) kkk13m nPDCn PX2zm, for some C13 > 0. Now following the steps similar to those used to get an upper bound of P(An), one can find a k2 such that for all k (k2),  12k14 kPDC logn, for some C14 > 0. Hence . In view of the fact that Dk’s are 5kk=kPD = mutually independent, by applying the Borel-Cantelli Lemma, (13) is established. Observe that 1mm kk-1 k1mkmm kk1 k1mk-1nn mnnn mnPTBlognB=PTBlog n.B Again by Theorem 3.1, one can find a constant C15 and k3 such that for all k  k3, 1mm kk-1 k1k1m kknn m15 m1nmPTBlognCn P XBlogn. Again following the steps similar to those used to get an upper bound of P(An), one can find a k4 such that for all k (k4),  1k1mm kk1 kkkmnn m1531m2mn1PTBlognC nlog n By (12) we have implies δkk1mnβk+1 kkmnβnmk and (15), we have, nk+1   nk. There- fore, k+1kk 1k 1kmmmmnβnn n  k-1kk1m1n= λ , where 11=. Hence k11kkm11kk1mnn and  k1155kkm131k = kk = kmk2mmn11<.nlogn logn   k312Therefore 1mmkk-1 knn mPTBlogni.o= 0, which implies the proof of (11) follows from (13) and (14) and the proof of the theorem is completed. Another direct application of Theorem 3.1 is for the Cesàro sums of index r. Here we may write rnkrnAkf=nA , where rnΓn+r+1A=Γn+1 Γr+1. Using Ster-ling approximation, we get rrnnA=Γr+1 so that rkf1nn   k. The following result of Vasudeva  can be extended to domain of partial attraction of semi stable law and proof follows on similar lines of Theorem 2, we omit the details. Theorem 4.2 Let F  DP (), 0 <  < 1. Then 1log log n1nnnTLimSup= ea.sB , where rnnkk=1kT = 1Xn and r > 0. 5. Acknowledgements The author wishes to express his sincere thanks to U G C, New Delhi, for financial support in the form of Major Research Project (F. No: 34-156/2008 (SR)). Also ex-presses his regards to the referee for his valuable sugges-tions. 6. References  V. M. Kruglov, “On the Extension of the Class of Stable Distributions,” Theory of Probability and Its Applications, Vol. 17, 1972, pp. 685-694. doi:10.1137/1117081  C. C. 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