Convergence rates in the law of large numbers for long-range dependent linear processes.

Baum and Katz (Trans. Am. Math. Soc. 120:108-123, 1965) obtained convergence rates in the Marcinkiewicz-Zygmund law of large numbers. Their result has already been extended to the short-range dependent linear processes by many authors. In this paper, we extend the result of Baum and Katz to the long-range dependent linear processes. As a corollary, we obtain convergence rates in the Marcinkiewicz-Zygmund law of large numbers for short-range dependent linear processes.

Introduction

There are many literature works concerning the convergence rates in the Marcinkiewicz-Zygmund law of large numbers. One can refer to Alf [2], Alsmeyer [3], Baum and Katz [1], Heyde and Rohatgi [4], Hu and Weber [5], Rohatgi [6], and so on.

Baum and Katz [1] obtained the following convergence rates in the Marcinkiewicz-Zygmund law of large numbers.

Theorem 1.1

Baum and Katz [1]

Let , and be a sequence of independent and identically distributed (i.i.d.) random variables. Then and imply

When , the cases of and have already been proved by Hsu and Robbins [7] and Katz [8], respectively.

Let be a sequence of i.i.d. random variables and be a sequence of real numbers. Here and in the following, denotes the set of all integers. Then is called a linear process or an infinite order moving average process if is defined by If , then has short memory or is short-range dependent. If , then has long memory or is long-range dependent (see Chapter 3 in Giraitis et al. [9]).

In the short-range dependent case, Koopmans [10] showed that if has the moment generating function, then the strong law of large numbers for the linear process holds with exponential convergence rate. Hanson and Koopmans [11] generalized this result to a class of linear processes of independent but non-identically distributed random variables and to arbitrary subsequences of . Li et al. [12] extended Katz [8] theorem to the setting of short-range dependent linear processes.

Theorem 1.2

Li et al. [12]

Let . Let be an absolutely summable sequence of real numbers. Suppose that is the linear process of a sequence of i.i.d. random variables with mean zero and . Then

Note that Theorem 1.2 corresponds to Theorem 1.1 with . Zhang [13] extended Theorem 1.1 with to the short-range dependent linear process of a sequence of identically distributed φ-mixing random variables. Since independent random variables are also φ-mixing, it follows by Zhang [13] theorem that Theorem 1.2 also holds for .

In this paper, we obtain convergence rates in the Marcinkiewicz-Zygmund law of large numbers for long-range dependent linear processes of i.i.d. random variables. For convenience of notation, let where . In the long-range dependent case, Characiejus and Račkauskas [14] obtained the convergence rate in the Marcinkiewicz-Zygmund law of large numbers for the linear process which is slightly different from (1.1) and defined by where if .

Theorem 1.3

Characiejus and Račkauskas [14]

Let be defined as above and . Let be a sequence of real numbers such that where if . Assume that If and , then

The above theorem shows a convergence rate in the Marcinkiewicz-Zygmund weak law of large numbers with the norming sequence .

We now compare Theorem 1.3 with Theorem 1.1. Since Theorem 1.3 deals with only the case , it is interesting to prove that Theorem 1.3 holds for the case . When , Theorem 1.1 requires a finite pth moment condition, but Theorem 1.3 requires more than finite pth moment. To apply Theorem 1.3, it is necessary to estimate . If is an absolutely summable sequence, then we have, by the result of Burton and Dehling [15] (see also Lemma 2.4), that for any and hence (1.3) holds with replaced by . However, for the long-range dependent case, it is not easy to estimate .

In this paper, we extend Theorem 1.1 to the long-range dependent linear processes. As a corollary, we obtain a long-range dependent setting of Theorem 1.2. Further, we propose a method to estimate for the long-range dependent case.

Throughout this paper, C denotes a positive constant which may vary at each occurrence. For events A and B, denotes the indicator function of the event A, and .

Convergence of long-range dependent linear processes

In this section, we extend Theorem 1.1 to the long-range dependent linear processes. To prove the main results, we need the following lemmas. The first one is the von Bahr-Esseen inequality (see von Bahr and Esseen [16]). The second is known as Fuk-Nagaev inequality (see Corollary 1.8 in Nagaev [17]).

Lemma 2.1

Let be a sequence of independent random variables with and for some . Then, for all , where is a positive constant depending only on t.

Lemma 2.2

Let be a sequence of independent random variables with . Then, for any and ,

The following lemma is well known and can be easily proved by using a standard method.

Lemma 2.3

Let and ζ be a random variable. Then the following statements hold.

If , then .

If , then .

If , then .

If , then .

The following lemma is useful to estimate when the sequence is absolutely summable. However, it is not applicable to the long-range dependent case.

Lemma 2.4

Burton and Dehling [15]

Let be an absolutely convergent series of real numbers with . Then, for any , where .

We now state and prove our main results. The first theorem treats the case .

Theorem 2.1

Let and . Let be a sequence of real numbers with Suppose that is the linear process of a sequence of i.i.d. random variables with mean zero and . Furthermore, assume that one of the following conditions holds.

If , then

If , then and Then

Proof

(1) For each , we have and hence, By the Markov inequality, Lemmas 2.1 and 2.3, we have Thus the first series on the right-hand side of (2.1) converges.

Similarly, by the Markov inequality, Lemmas 2.1 and 2.3, we have Hence the second series on the right-hand side of (2.1) also converges.

(2) For each , we have and hence,

By the Markov inequality, Lemmas 2.1 and 2.3, we have Thus the first series on the right-hand side of (2.2) converges.

We next prove that the second series on the right-hand side of (2.2) converges. We have by Lemma 2.2 that for , Hence it is enough to show that two series on the right-hand side of (2.3) converge.

If we take , then we have by Lemma 2.3 that Hence the first series on the right-hand side of (2.3) converges.

Finally, we show that the second series on the right-hand side of (2.3) converges. Since , we have that which implies that  □

The next theorem treats the case .

Theorem 2.2

Let . Let be a sequence of real numbers with Suppose that is the linear process of a sequence of i.i.d. random variables with mean zero and . Furthermore, assume that and Then

The proof is similar to that of Theorem 2.1(1). We proceed with two cases and .

For the case , we have by Lemmas 2.1 and 2.3 that As in the proof of Theorem 2.1(1), we have that

For the case , we rewrite as If , then . It follows by Lemma 2.4 that as . Hence The rest of the proof is the same as that of the previous case and is omitted. □

The following corollary extends Theorem 1.1 to the short-range dependent linear processes.

Corollary 2.1

Let , , and . Let be an absolutely summable sequence of real numbers. Suppose that is the linear process of a sequence of i.i.d. random variables with mean zero and . Then

We first note that If , then we take θ such that . Then By Lemma 2.4, for any , there exist positive constants and independent of n such that Then all conditions on in Theorems 2.1 and 2.2 are easily satisfied. Hence the proof follows from Theorems 2.1 and 2.2. □

Remark 2.1

In Corollary 2.1, the case (i.e., and ) is not considered. In fact, Corollary 2.1 does not hold for this case (see Sung [18]).

An estimation of for the long-range dependent case

As we have seen in Sections 1 and 2, it is easy to estimate for the short-range dependent case. In this section, we propose a method to estimate for the long-range dependent case. It is not easy to estimate when the sequence is not absolutely summable. For simplicity, we will consider non-increasing sequences of positive numbers. For the finiteness of , without loss of generality, it is necessary to assume that if and .

Lemma 3.1

Let . Let be a non-increasing sequence of positive real numbers satisfying if and . Then

Since if and , we get that Similarly, Thus the proof is completed. □

The following lemma can be found in Martikainen [19].

Lemma 3.2

Martikainen [19]

Let be a non-decreasing sequence of positive real numbers. Then

Similarly, we can obtain a counterpart of Lemma 3.2.

Lemma 3.3

Let be a non-decreasing sequence of positive real numbers. Then

The proof is similar to that of Lemma 3.2 and is omitted. □

Using Lemmas 3.2 and 3.3, we have the following lemma.

Lemma 3.4

Let and let be a sequence of positive real numbers satisfying , , and Then the following statements hold:

.

.

The proof of (i) follows from Lemma 3.2. The proof of (ii) follows from Lemma 3.3. □

Now we present a method to estimate for the long-range dependent case.

Theorem 3.1

Let , and let be a sequence of positive real numbers satisfying the same conditions as in Lemma  3.4. Then there exist positive constants and independent of n such that where if .

By the condition , we have , which implies . The upper bound of follows by Lemmas 3.1 and 3.4. For the lower bound, we have by that It follows that for all large n Since , Hence the lower bound follows from Lemma 3.1. □

Finally, we give two long-range dependent linear processes.

Example 3.1

Let if and if . Then the series diverges, but converges if . Observe that If , then By Lemma 3.1, for any , there exist positive constants and independent of n such that Let be the long-range dependent linear process of a sequence of i.i.d. random variables with mean zero and , where and . Then all conditions of Theorem 2.1 are easily satisfied. By Theorem 2.1,

Example 3.2

Let . Let if and if , where . Then the series diverges, but converges if . Since , we have by Theorem 3.1 that Let be the long-range dependent linear process of a sequence of i.i.d. random variables with mean zero and . Take θ such that . Then all conditions of Theorem 2.2 are easily satisfied. By Theorem 2.2,