Abstract
Let X1, X2,... be a strictly stationary and negatively associated sequence of random variables with mean zero and positive, finite variance, set Sn = X1+ ... + Xn, Mn = max1 < k < n |Sk|. Under appropriate moment conditions, we obtain precise rates in law of the logarithm for the moment convergence of Sn and Mn.
the rates; complete moment convergence; negatively associated sequences
The rates in complete moment convergence for negatively associated sequences
Yuexu Zhao* * Corresponding author. ; Zheyong Qiu; Chunguo Zhang; Yehua Zhao
Department of mathematics, Hangzhou Dianzi University, Hangzhou 310018, China. E-mail: yxzhao@hdu.edu.cn
ABSTRACT
Let X1, X2,... be a strictly stationary and negatively associated sequence of random variables with mean zero and positive, finite variance, set Sn = X1+ ... + Xn, Mn = max1 < k < n |Sk|. Under appropriate moment conditions, we obtain precise rates in law of the logarithm for the moment convergence of Sn and Mn.
Mathematical subject classification: Primary: 60F15; Secondary: 60G50.
Key words: the rates, complete moment convergence, negatively associated sequences.
1 Introduction and main results
A finite family of random variables, X1, X2,..., Xn, is said to be negatively associated if, for every pair of disjoint subsets T1 and T2 of {1,2,..., n},
Cov ( f1( Xi , i ∈ T1 ), f2 (Xj , j ∈ T2)) < 0,
whenever 1 and 2 are coordinatewise increasing and the covariance exists. An infinite family is negatively associated if every finite subfamily is negatively associated. This definition was introduced by Alam and Saxena [1] and Joag-Dev and Proschan [7], and has found many applications in percolation theory, multivariate statistical analysis and reliability theory [2].
Under appropriate conditions, lots of results have been obtained for negatively associated sequences, the central limit theorem (CLT) [13], probability inequalities [15, 17], weak convergence [19, 20], almost sure convergence [12], law of the iterated logarithm (LIL) [16] and complete convergence [8, 9], precise rates [5, 21, 22].
Let X1, X2,... be a sequence of independent and identically distributed (i.i.d.) random variables. Gut and Spataru [6] obtained a result below.
Theorem A. Suppose that EX1 = 0 and = σ2 < ∞. Then, for 0< δ <1,
where N is a standard normal random variable.
Liu and Lin [11] proved the following theorem for i.i.d. random variables.
Theorem B. Suppose that
for 0 < α < 1. Then
Conversely, if (1.3) is true, then (1.2) holds.
The purpose of the present paper is to investigate precise asymptotics in complete moment convergence, our results not only extend (1.3) to negatively associated sequences, but give a maximal analog of (1.3) and other versions. To formulate our results,we need some extra notation. Let X1, X2,... be strictlystationary and negatively associated random variables, EX1 = 0, < ∞, , set Sn = X1+ ... + Xn, Mn = max1 < k < n |Sk|, write log for the natural logarithm, log x = loge(x ∨ e) , [z] denotes the integer part of z, C stands for a positive constant whose value may be different fromline to line. Our results read as follows.
Theorem 1.1.Iffor any δ > 2, then we have
and
Conversely, if (1.5) is true, then . Where Γ ( · ) is the Gamma function.
Theorem1.2. If
for any δ > 0, then we have
and
Conversely, if (1.7) is true, then .
Without loss of generality, throughout the paper, we will suppose that σ 2 = 1.
2 Proof of Theorem 1.1
In order to verify this result, we first give three elementary but useful lemmas.
Lemma 2.1. [13].Let {Xn: n > 1} be a strictly stationary and negatively associated sequence of random variables with mean zero and
then
Lemma 2.2. [18].Let {Xn: n> 1} be strictly stationary and negatively associated sequence of random variables, EX1 = µ, 0 < Var X1 = σ2 < ∞ and , set Sm = Xk, write
Wn(t) =(Sm+ (nt – m) Xm+1– ntµ) , m < n < m + 1, 0 < t < T.
Then
where {W(t) : t > 0} is a standard Wiener process and C[0, T] is the usual C space on [0, T].
Lemma 2.3. [10].Let {Xn: n > 1} be a negatively associated sequence ofrandom variables with mean zero, < ∞, set Sn = Xk, = . Then, for any a > 0 and b > 0, we have
Observe the following formula.
Similarly, one can obtain the corresponding equation for Mn. In the rest of this section, we give the following propositions.
Proposition 2.1. We have
and
where N is a standard normal random variable and {W(t) : t > 0} is a standard Wiener process.
Proposition 2.2. Under the conditions of Theorem 1.1, we have
and
Remark 2.1. The proofs of Propositions 2.1 and 2.2 are very standard, so we omit them.
Proposition 2.3. Under the conditions of Theorem 1.1, we have
and
Proof. It follows that
Using the following result of Billingsley [3].
Proposition 2.4. Under the conditions of Theorem 1.1, we have
and
Proof. We only prove (2.12), let H (∈) = [exp(M/∈δ)], M > 4, 0 < ∈ < 1/4, δ > 2, denote Δn = supx |P (|Sn| > ) – P(|N| > x)|, we have
We first estimate Σ1, it follows that
The estimate of Σ3 is easy. By Toeplitz's lemma, one can complete the proof of term Σ4. As to Σ5, taking , b = a/m in Lemma 2.3, it turns out that
An easy calculation leads to
by Toeplitz's lemma, we have lim∈↓0∈δ-2 Σ6 = 0. Turn to Σ7, it follows that
applying Toeplitz's lemma again, we have lim∈↓0∈δ-2Σ7 = 0. Now let usconsider Σ2, notice that
Applying Markov's inequality, one can complete the estimate of Σ8. By Lemma 2.3, taking m > δ/2, it turns out
Proof of Theorem 1.1. Combining Propositions 2.1 ~ 2.4, we have
Then, the proof of (1.4) follows from (2.17) and (2.18), similarly, one can obtain (1.5).
For the sufficient part, using the standard method, we first show that
It is easy to see that
furthermore, we have
Recalling (1.5) and (2.4), which yields
Hence, we have
So (2.19) follows from (2.22) and (2.23). We next show that
By the result of Joag-Dev et al. [7], it turns out that
by (2.19), for sufficiently large n, we have
By (2.26) and using elementary inequality, we have
Finally, the sufficient part follows from (2.27) together with
3 Proof of Theorem 1.2
According to the proof of Theorem 1.1, we only give the main ideas of the proofs of (1.6) and (1.7). Observe (2.4) and (2.5), it is natural to give the following Propositions.
Proposition 3.1. We have
and
Proof. By a careful calculation, it follows that
Then, along the same lines as those of the proof of Proposition 2.2, we have
With the help of Billingsley's result, one can complete the proof of (3.2).
Proposition 3.2. Under the conditions of Theorem 1.2, we have
and
Proof. Recalling the proof of Proposition 2.3, it is easy to get
The following proof is similar to that of Proposition 2.4, so we have
The moment condition (log|X1|)δ < ∞ is used as follows, note the corresponding part of Σ9, we have
we then complete the proof of (3.5). Finally, observe that
along the same proof lines of (3.5), one can complete the proof of (3.6).
Proof of Theorem 1.2. By virtue of Propositions 3.1 and 3.2, one can obtain (1.6) and (1.7). With the help of (2.27), the sufficient part is obvious.
Received:11/II/09.
Accepted:19/II/09.
#CAM-55/09.
Supported by the Natural Science Foundation of Department of Education of Zhejiang Province(Y200804716)
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Publication Dates
-
Publication in this collection
19 Mar 2010 -
Date of issue
2010
History
-
Received
11 Feb 2009 -
Accepted
19 Feb 2009