The rates in complete moment convergence for negatively associated sequences

The rates in complete moment convergence for negatively associated sequences

Yuexu Zhao^*; Zheyong Qiu; Chunguo Zhang; Yehua Zhao

Department of mathematics, Hangzhou Dianzi University, Hangzhou 310018, China. E-mail: [e-mail]

ABSTRACT

Let X₁, X₂,... be a strictly stationary and negatively associated sequence of random variables with mean zero and positive, finite variance, set S_n = X₁+ ... + X_n, M_n = max_{1 < k < n} |S_k|. Under appropriate moment conditions, we obtain precise rates in law of the logarithm for the moment convergence of S_n and M_n.

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,..., X_n, is said to be negatively associated if, for every pair of disjoint subsets T₁ and T₂ of {1,2,..., n},

Cov ( f₁( X_i , i ∈ T₁ ), f₂ (X_j , j ∈ T₂)) < 0,

whenever ƒ₁ and ƒ₂ 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 S_n = X₁+ ... + X_n, M_n = max_{1 < k < n} |S_k|, write log for the natural logarithm, log x = log_e(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 {X_n: n > 1} be a strictly stationary and negatively associated sequence of random variables with mean zero and

then

Lemma 2.2. [18].Let {X_n: n> 1} be strictly stationary and negatively associated sequence of random variables, EX₁ = µ, 0 < Var X₁ = σ² < ∞ and , set S_m = X_k, write

W_n(t) =(S_m+ (nt – m) X_m+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 {X_n: n > 1} be a negatively associated sequence ofrandom variables with mean zero, < ∞, set S_n = X_k, = . Then, for any a > 0 and b > 0, we have

Observe the following formula.

Similarly, one can obtain the corresponding equation for M_n. 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].

one can obtain (2.11).

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 Δ_n = sup_x |P (|S_n| > ) – 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

we have lim _∈↓0∈^δ-2Σ₉ = 0. The proof of (2.12) is now complete.

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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Corresponding author.

Publication Dates

• Publication in this collection
  19 Mar 2010
• Date of issue
  2010

History

• Received
  11 Feb 2009
• Accepted
  19 Feb 2009