A Precise Asymptotic Behaviour of the Large Deviation Probabilities for Weighted Sums

Let {X n , 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 properly normalized weighted sums have been studied and in support of this we obtained Chover's form of law of iterated logarithm.


Introduction
Let {X n , n ≥1} be a sequence of independent and identically 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 functions over [0,1].Set 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  n n P S > x  , or either of the one sided components, are called large deviation probabilities, where {x n , n ≥ 1} is a monotone sequence of positive numbers with x n   as n   such that p n n S 0 x   .In fact, under different conditions on sequence of r.v.s, Heyde [2][3][4] studied the large deviation 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 attraction of the normal law, Heyde [2] and [3] established the order of magnitude of the larger deviation probabilities, where as in Heyde [3], he obtained the precise asymptotic 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 [5] 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 [6] 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 [6] 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][3][4].(See Divanji [7]).However the observations made by Heyde [2][3][4] on the large deviation probabilities implicitly motivated us to study the large deviation probabilities for weighted sums.In fact, when the underlying i.i.d positive valued r.v.s are in the domain of partial attraction of a semi stable law of Kruglov's [1] setup, denoted as F  DP (), 0 <  < 1, a precise asymptotic behavior of the large deviation probabilities of Heyde [2][3][4] can be obtained for weighted sums.In support of this can be considered for Chover's type of non-trivial limit behavior 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', 'almost 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, a k can always be chosen to be zero.

Lemma 2.2
Let F  DP (), 0 <  < 1 and let where l is a function s. v. at .The above lemmas can be referred to Divanji and Vasudeva [8].

Lemma 2.3
Let L be any s.v. function and let (x n ) and (y n ) be sequence of real constants tending to  as n .Then for This lemma can be referred to Drasin and Seneta [9].Lemma 2.4 Dividing on both sides by x n B n , we have Observe that X i 's are i.i.d positive valued r.v.s which are in the domain of partial attraction of a semi-stable law and hence

Main Results
Theorem 3.1 Let F  DP (), 0 <  < 1.Let (x n ) be a monotone sequence of real numbers tending to ∞ as n→∞ and B n defined in lemma 2.2.Then .

Proof
To prove the assertion, it is enough to show that Proceeding on the lines of Heyde [4] and Lemma 3.1 of Vasudeva [12], we get, From Lemma 2.5, we have and given  > 0 with 1 -2 > 0, we can choose N 1 so Copyright © 2011 SciRes.AM n large such that P(B i ) > 1 -2 for all n  N 1 and for all .Further from Lemma 2.5, we see that nP(A i )  0 as n  , so that we can choose N 2 so large that n P(A i ) < , for n  N 2 .Thus for n  N = max (N 1 , N 2 ), we obtain from (1), , this implies In order to complete the proof, we use truncation method. Define Observe that fixed n and f is continuous BV [0,1] and it attains bounds.Hence using Lemma 1, we have, Using Karamata's representation of s.v.function, one gets that Since a(x)  C as x  C and (y)  0 as y  , there exists C 0 > 0 and  0 < , such that Substituting ( 4) in (3), one can find some constant C 1 such that the second term in (2) becomes . Therefore we can find some constant C 2 (>C 1 ) such that Now consider the first term in the right of (2).By Tchebychev's inequality, we get By Theorem 1, on page 544, of Feller [12] and Lemma 2.1, one gets that Using similar steps of ( 4), one can find some constant C 3 such that Observe that Notice that A  B  D. Again using Lemma 2.1, we have Following similar steps of (4), we can find some constant C 5 and  0 > 0 such that there exists C 6 (>C 5 ) such that Using (8) one can find some constant C 8 such that therefore there exists C 9 such that From ( 7) and ( 9), we claim that as n, i.e., holds.
Substituting ( 5) and ( 6) in (2), we get . The proof of the theorem is completed.

Proof
To prove the assertion, it suffices to show for any   (0, 1), that and To prove (10), let Applying Lemma 2. Define, for large k, where  > 1 and  > 0 and from the relation and in order to establish (11), it is enough if we show that   (0, 1), that and By the above Theorem 3.1, one can find a constant C 12 > 0 and k 1 such that for all k ( k 1 ), Since F  DP (), 0 <  < 1 and under Kruglov's [9] setup i.e.,

 
Again by Theorem 3.1, one can find a constant C 15 and k 3 such that for all k  k 3 , Again following the steps similar to those used to get an upper bound of P(A n ), one can find a k 4 such that for all k (k 4 ), By (12) we have implies k .The following result of Vasudeva [11] 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.
a member of BV[0,1].Let {n k , k ≥ 1} be a strictly increasing subsequence of positive integers such that k  .Kruglov[1] established that, if there exists sequences (a k ) and (b k ) of real constants, b k   as k  , such that  , as n  .

1 ,
one can find a C 10 such that, steps similar to those used to get an upper bound of P(A n ), one can find a k 2 such that for all k (k 2 ), by applying the Borel-Cantelli Lemma, (13) is established.Observe that 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