Abstracts
We present an accurate description the limiting behavior of delayed sums under a non-identically distribution setup, and deduce Chover-type laws of the iterated logarithm for them. These complement and extend the results of Vasudeva and Divanji (Theory of Probability and its Applications, 37 (1992), 534-542).
stable distribution; laws of iterated logarithm; delayed sum
Apresentamos uma descrição precisa do comportamento limite de somas retardadas, e deduzimos leis do tipo Chover de logaritmo iterado para as mesmas. Isso completa e estende os resultados de Vasudeva e Divanji (Theory of Probability and its Aplications, 37 (1992), 534-542).
distribuição estável; leis do logaritmo iterado; somas retardadas
MATHEMATICAL SCIENCES
Limiting behavior of delayed sums under a non-identically distribution setup
Chen Pingyan
Department of Mathematics, Jinan University, Guangzhou, 510630, P.R. China
ABSTRACT
We present an accurate description the limiting behavior of delayed sums under a non-identically distribution setup, and deduce Chover-type laws of the iterated logarithm for them. These complement and extend the results of Vasudeva and Divanji (Theory of Probability and its Applications, 37 (1992), 534542).
Key words: stable distribution, laws of iterated logarithm, delayed sum.
RESUMO
Apresentamos uma descrição precisa do comportamento limite de somas retardadas, e deduzimos leis do tipo Chover de logaritmo iterado para as mesmas. Isso completa e estende os resultados de Vasudeva e Divanji (Theory of Probability and its Aplications, 37 (1992), 534-542).
Palavras-chave: distribuição estável, leis do logaritmo iterado, somas retardadas.
1 INTRODUCTION AND MAIN RESULTS
The distribution function F of a real valued random variable X is called stable law with exponent α(0 < α < 2), if for some σ > 0,-1 < β < 1, its characteristic function is of the form
where
If β = 0, X is a symmetric random variable. It is well-known, if F is a stable law with exponent α(0 < α < 2), we have the following tail behavior:
where c(α,σ) > 0 only depends on α and σ(cf. e.g. Feller 1971). This property will play an important role in this paper.
Let {Xn,n > 1} be a sequence of independent random variables with its partial sums . Let {an,n > 1} be a positive integer subsequence. Set Tn = Sn+an- Sn and γn = log(n/an)+log log n. The sum Tn is called a forward delayed sum (see Lai 1974). Suppose Xn's involve of two distributions F1 and F2 which are stable laws with exponents α1 and α2(0 < α1< α2 < 2). For each n > 1, let τ1(n) denote the number of random variables in the set {X1, X2,..., Xn} with distribution function F1, then τ2(n) = n-1(n) is the number of random variables with distribution function F2 in the set {X1, X2, ..., Xn}. Then (τ1(n), τ2(n)) is called the sample scheme of the sequence {Xn, n > 1}. Assume that τ1(n) = [nα1/α2] and Bn = n1/α2, where [x] is the integer part of x. By Sreehari (1970), Sn/Bn converges weakly to a composition of the two stable laws.
Let Uα1(n) be the sum of those {X1, X2,..., Xn } with distribution function F1 and Vα2(n) be the sum of those {X1, X2, ..., Xn} with distribution function F2. Then Sn = Uα1(n) + Vα2(n). One can note that in Tn there are [(n+an)α1/α2]-[nα1/α2] random variables with distribution function F1 and n+an-[(n+an)α1/α2]-(n-[nα1/α2]) random variables with distribution function F2.
The motivation of this paper is to extend and complement the results of Vasudeva and Divanji (1992). They obtained the following theorem in the special case that F1 and F2 are positive stable laws with exponents 0 < α1< α2 < 1.
THEOREM A. Let {an, n > 1} be a nondecreasing sequence with 0 < an < n and an/n non-increasing. Let F1 and F2 are positive stable law and 0 < α1< α2 < 1.
(i) If limn→ ∞ log(n/an)/loglogn = + ∞, then
(ii) If limn→ ∞log(n/an)/loglogn = 0, then
(iii) If limn→ ∞log(n/an)/log log n = s ∈ (0, + ∞), then
They only discuss the case that F1 and F2 are positive stable law with exponents 0 < α1< α2 < 1. But by their method, it is impossible to discuss the rest case. In this paper, by a new method, we will complement and extend Theorem A in three directions, namely:
(i) We will obtain more exact results.
(ii) We will discuss not only that the distributions is the positive stable laws, but also that the distributions is not necessary positive stable laws and the exponents of the stable laws in (0,2), not only in (0,1).
(iii) We will replace the restrictions 0 < an < n and an/n non-increasing of the sequence {an, n > 1} by a more general assumption lim supn→ ∞an/n < + ∞.
Recall that the kind of type law of the iterated logarithm (LIL) was first obtained by Chover (1966) for symmetric stable law, and is called Chover-type LIL. By far, some papers concern with the Chover-type LIL, for example, Chen (2002) for the weighted sums of symmetric stable law, Chen and Yu (2003) for the weighted sums of stable law without symmetric assumption, Peng and Qi (2003) for the weighted sums of law in the domain of attraction of stable law, and Chen (2004) for geometric weighted sums and Cesàro weighted sums of stable law, etc.
First we give an accurate description of the limiting behavior of Sn.
THEOREM 1.1. Let f > 0 be a nondecreasing function. Then with probability one
By Theorem 1.1, we have the following Corollary at once.
COROLLARY 1.1. For every δ > 0, we have
and
In particular
REMARK 1.1. If α1 = α2, Corollary 1.1 extends the result of Chover (1966).
THEOREM 1.2. Let {an,n > 1} be a subsequence of positive integers with limsupn→ ∞an/n < + ∞. Let f > 0 be a nondecreasing function. Then with probability one
COROLLARY 1.2. Let {an, n > 1} as Theorem 1.2. Then for every δ > 0, we have
and
In particular
COROLLARY 1.3. Let {an, n > 1} as Theorem 1.2.
(i) If limn→ ∞log(n/an)/loglogn = +∞, then
(ii) If limn→ ∞log(n/an)/loglogn = 0, then
(iii) If limn→ ∞log(n/an)/loglogn = s ∈ (0, +∞), then
COROLLARY 1.4. Let {an, n > 1} as Theorem 1.2. If α1 = α2 = a, then
REMARK 1.2. Corollary 1.4 extends the result of Zinchenko (1994).
2 PROOFS OF THE MAIN RESULTS
We need the following lemmas.
LEMMA 2.1 (see Lemma 2.1 of Chen 2004). Let f > 0 be a non-decreasing function with
then there exists a non-decreasing function g > 0 such that
LEMMA 2.2 (see Lemma 2.2 of Chen 2002). Let f > 0 be a non-decreasing function satisfying
Then there exists a non-decreasing function h > 0 such that
LEMMA 2.3 (see Lemma 3 of Chow and Lai 1973). Let {Wn, n > 1} and {Zn, n > 1 } be two sequences of random variables such that {Wi, 1 < i < n} and Zn are independent for each n > 1. Suppose Wn+ Zn → 0 a.s. and Zn → 0 in probability, then Wn → 0 a.s. and Zn → 0 a.s.
In the rest of this paper, we denote C as a generic positive number which may be different at different places, and a(n) ~ b(n) means limn→ ∞a(n)/b(n) = 1. For the sake of simplicity, we denote random variable Y1 with distribution function F1 and random variable Y2 with distribution function F2.
PROOF OF THEOREM 1.1. Assume that < ∞. First of all, we show that
Note that by (1.1), (τ1(n))-1/α1(Uτ1(n)-bτ1(n)) has the same distribution as Y1 and (τ2(n))-1/α2 (Vτ2(n)- dτ2(n)) has the same distribution as Y2, where bn = 0 if α1≠ 1 and bn = bn log n for some b ∈ (-∞, +∞) if α1 = 1, and dn = 0 if α1≠ 1 and dn = dn log n for some d ∈ (-∞, +∞) if α2 = 1. < ∞ implies that f(n) → ∞ and log n/f(n) → 0 as n→ ∞. Hence for all ε > 0
and
Hence (2.1) holds. So by standard symmetric argument (see Lemma 3.2.1 of Stout 1974), we need only to prove the result for {Xn, n > 1} symmetric.
By Lemma 2.1 of Chen (2002),
Note that
Hence
So we complete the proof of the convergent part.
Now we assume that = +∞. If
holds, then by the Borel-Cantelli lemma, we have
and note that
hence we have
Now we prove (2.2). Note that
and = +∞ implies = + ∞, so (2.2) holds.
PROOF OF THEOREM 1.2. Assume that < ∞, by Lemma 2.1, without loss of generality, wecan assume that limsupx→ ∞f(2x)/f(x) < ∞. By Theorem 1.1, we have
Note that limsupn→ ∞ < ∞, hence
Now we assume that = + ∞. Suppose
does not hold, then by Kolmogorov 0-1 law, there exists a constant c0∈ [0, ∞) such that
Hence
where h(x) is given by Lemma 2.2. It is easy to show that
i.e.
By Lemma 2.3, we have
By the Borel-Cantelli lemma
But by the same argument in the proof of Theorem 1.1, we have
This leads to a contradiction, so we complete the proof.
PROOF OF COROLLARY 1.3. By Theorem 1.2, we have
= 0 a.s. ∀ δ > 0 and
Hence we have
P(|Tn| >Bn (log n) (1+δ)/α1, i.o) = 0, ∀ δ > 0 and P(|Tn| > Bn (log n)1/α1, i.o.) = 1,
where P(An, i.o.) = P(limsupn→ ∞An) and An is a sequence of events. So we have
and
(i) If limn®¥log(n/an)/loglogn = ¥, then
and
hence we have
(ii) If limn®¥log(n/an)/loglogn = 0, then
and
hence we have
(iii) If limn®¥log(n/an)/loglogn = s Î (0,¥), then
and
hence we have
ACKNOWLEDGMENTS
Research supported by the National Natural Science Foundation of China. The author is thankful to the referee for her/his helpful remarks which improved the presentation of the paper.
Manuscript received on July 28, 2007; accepted for publication on June 9, 2008; presented by DJAIRO G. FIGUEIREDO
AMS Classification: 60F05, 60F15
E-mail: tchenpy@jnu.edu.cn
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Publication Dates
-
Publication in this collection
25 Nov 2008 -
Date of issue
Dec 2008
History
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Accepted
09 June 2008 -
Received
28 July 2007