Limiting behavior of delayed sums under a non-identically distribution setup

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), 534–542).

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 {X_n,n > 1} be a sequence of independent random variables with its partial sums . Let {a_n,n > 1} be a positive integer subsequence. Set T_n = S_n+an- S_n and γ_n = log(n/a_n)+log log n. The sum T_n is called a forward delayed sum (see Lai 1974). Suppose X_n'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,..., X_n} 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, ..., X_n}. Then (τ₁(n), τ₂(n)) is called the sample scheme of the sequence {X_n, n > 1}. Assume that τ₁(n) = [n^α1/α2] and B_n = n^1/α2, where [x] is the integer part of x. By Sreehari (1970), S_n/B_n converges weakly to a composition of the two stable laws.

Let U_α1(n) be the sum of those {X₁, X₂,..., X_n } with distribution function F₁ and V_α2(n) be the sum of those {X₁, X₂, ..., X_n} with distribution function F₂. Then S_n = U_α1(n) + V_α2(n). One can note that in T_n there are [(n+a_n)^α1/α2]-[n^α1/α2] random variables with distribution function F₁ and n+a_n-[(n+a_n)^α1/α2]-(n-[n^α1/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 {a_n, n > 1} be a nondecreasing sequence with 0 < a_n < n and a_n/n non-increasing. Let F₁ and F₂ are positive stable law and 0 < α₁< α₂ < 1.

(i) If lim_{n→ ∞} log(n/a_n)/loglogn = + ∞, then

(ii) If lim_{n→ ∞}log(n/a_n)/loglogn = 0, then

(iii) If lim_{n→ ∞}log(n/a_n)/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 < a_n < n and a_n/n non-increasing of the sequence {a_n, n > 1} by a more general assumption lim sup_{n→ ∞}a_n/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 S_n.

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 {a_n,n > 1} be a subsequence of positive integers with limsup_{n→ ∞}a_n/n < + ∞. Let f > 0 be a nondecreasing function. Then with probability one

COROLLARY 1.2. Let {a_n, n > 1} as Theorem 1.2. Then for every δ > 0, we have

and

In particular

COROLLARY 1.3. Let {a_n, n > 1} as Theorem 1.2.

(i) If lim_{n→ ∞}log(n/a_n)/loglogn = +∞, then

(ii) If lim_{n→ ∞}log(n/a_n)/loglogn = 0, then

(iii) If lim_{n→ ∞}log(n/a_n)/loglogn = s ∈ (0, +∞), then

COROLLARY 1.4. Let {a_n, 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 {W_n, n > 1} and {Z_n, n > 1 } be two sequences of random variables such that {W_i, 1 < i < n} and Z_n are independent for each n > 1. Suppose W_n+ Z_n → 0 a.s. and Z_n → 0 in probability, then W_n → 0 a.s. and Z_n → 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 lim_{n→ ∞}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(U_τ1(n)-b_τ1(n)) has the same distribution as Y₁ and (τ₂(n))^-1/α2 (V_τ2(n)- d_τ2(n)) has the same distribution as Y₂, where b_n = 0 if α₁≠ 1 and b_n = bn log n for some b ∈ (-∞, +∞) if α₁ = 1, and d_n = 0 if α₁≠ 1 and d_n = 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 {X_n, 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 limsup_{x→ ∞}f(2x)/f(x) < ∞. By Theorem 1.1, we have

Note that limsup_{n→ ∞} < ∞, 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(A_n, i.o.) = P(limsup_{n→ ∞}A_n) and A_n is a sequence of events. So we have

and

(i) If lim_n®¥log(n/a_n)/loglogn = ¥, then

and

hence we have

(ii) If lim_n®¥log(n/a_n)/loglogn = 0, then

and

hence we have

(iii) If lim_n®¥log(n/a_n)/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: [e-mail]

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Publication Dates

• Publication in this collection
  25 Nov 2008
• Date of issue
  Dec 2008

History

• Accepted
  09 June 2008
• Received
  28 July 2007