Lobachevskii Journal of Mathematics

http://ljm.ksu.ru
ISSN 1818-9962

Vol. 26, 2007, 17–25 \begin{matrix}c \\ ◯ \end{matrix} K. Budsaba, P. Chen, A. Volodin

Kamon Budsaba, Pingyan Chen, Andrei Volodin

LIMITING BEHAVIOUR OF MOVING AVERAGE PROCESSES BASED ON A SEQUENCE OF ρ^{−} MIXING

AND NEGATIVELY ASSOCIATED RANDOM VARIABLES

(submitted by D. Kh. Mushtari)

Abstract. Let {Y_{i}, −∞ < i < ∞} be a doubly infinite sequence of identically distributed ρ^{−}-mixing or negatively associated random variables, {a_{i}, −∞ < i < ∞} a sequence of real numbers. In this paper, we prove the rate of convergence and strong law of large numbers for the partial sums of moving average processes {∑_{i=−∞}^{∞} a_{i}Y_{i+n}, n ≥ 1} under some moment conditions.

1. Preliminaries

Let {Y_{i}, −∞ < i < +∞} be a doubly infinite sequence of identically distributed random variables and {a_{i}, −∞ < i < +∞} be an absolutely summable sequence of real numbers. Next, let

X_{n} = ∑_{i=−∞}^{∞} a_{i}Y_{i+n}, n ≥ 1

2000 Mathematical Subject Classification. 60F15. Key words and phrases. Moving average process, Kolmogorov and

Marcinkiewicz-Zygmund strong law of large numbers, Rate of complete convergence, ρ^{−}-mixing, ρ^{∗}-mixing, Negative association.

The work of A. Volodin is supported by a grant from the Natural Sciences and Engineering Research Council of Canada.

be the moving average process based on the sequence {Y_{i}, −∞ < i < +∞}. As usual, we denote S_{n} = ∑_{k=1}^{n} X_{k}, n ≥ 1, the sequence of partial sums.

Under the assumption that {Y_{i}, −∞ < i < +∞} is a sequence of independent identically distributed random variables, many limiting results have been obtained for the moving average process {X_{n}, n ≥ 1}. For example, Ibragimov (1962) established the central limit theorem, Burton and Dehling (1990) obtained a large deviation principle, and Li, Rao, and Wang (1992) obtained the complete convergence result for {X_{n}, n ≥ 1}.

Certainly, even if {Y_{i}, −∞ < i < +∞} is the sequence of independent identically distributed random variables, the moving average random variables {X_{n}, n ≥ 1} are dependent. This kind of dependence is called weak dependence. The partial sums of weakly dependent random variables {X_{n}, n ≥ 1} have similar limiting behaviour properties in comparison with the limiting properties of independent identically distributed random variables.

Very few results for a moving average process based on a dependent sequence are known. In this paper, we provide two results on the limiting behaviour of a moving average process based on a ρ^{−}-mixing and negatively associated sequences.

Let {Y_{i}, −∞ < i < ∞} be a sequence of random variables defined on a probability space (Ω, F, P). For a set of integer numbers T denote σ-algebra F(T) = σ(Y_{i}, i ∈ T) and as usual, for a σ-algebra F we denote by L^{2}(F) the class of all F-measurable random variables with the finite second moment.

For two sets S and T of real numbers we denote

dist(S, T) = inf{|s − t|, s ∈ S, t ∈ T }.

The following definition was introduced in Zang and Wang (1999). A sequence of random variables {Y_{i}, −∞ < i < ∞} is called ρ^{−}-mixing if

ρ^{−}(s) = sup{ρ(S, T); S, T are sets of integers , dist(S, T) ≥ s} → 0

as s → ∞, where

ρ^{−}(S, T) = max {0, sup (Corr [f(Y_{i}, i ∈ S), g(Y_{j}, j ∈ T)])} ,

where supremum is taken over all coordinatewise increasing real functions f on R^{S} and g on R^{T} and by Corr(·, ·) we denote the classical correlation coefficient.

Next, a sequence of random variables {Y_{i}, −∞ < i < ∞} is called ρ^{∗}-mixing if for some integer s ≥ 1

ρ(s)^{∗} = sup sup{Corr(X, Y) : X ∈ L^{2}(F_{S}), Y ∈ L^{2}(F_{T})} < 1,

where the first sup is taken over all pairs of nonempty finite sets S, T of integers, such that dist(S, T) ≥ s. The notion of ρ^{∗}-mixing seems to be similar to the notion of ρ-mixing, but Bryc and Smolenski (1993) showed that they are quite different from each other.

Recall that a finite family of random variables {Y_{i}, 1 ≤ i ≤ n} is said to be negatively associated, if for any disjoint finite subsets S and T of integers and any real coordinatewise nondecreasing functions f on R^{S} and g on R^{T},

Cov (f(Y_{i}, i ∈ S), g(Y_{j}, j ∈ T)) ≤ 0

whenever the covariance exists. This concept was studied in Joag-Dev and Proschan (1983).

It is easy to see that {Y_{i}, −∞ < i < ∞} is negatively associated if and only if ρ^{−}(s) = 0 for all s ≥ 1 and ρ^{−}(s) ≤ ρ^{∗}(s). Hence the notion of ρ^{−}-mixing is weaker than both notions of negative association and ρ^{∗}-mixing.

We also need the following simple statement (see Property P2 in Wang and Lu (2006))

Property of ρ^{−}-mixing random variables. Let {Y_{n}, n ≥ 1} be a sequence of ρ^{−}-mixing random variables. If {f_{n}, n ≥ 1} is a sequence of real functions all of which are monotone nondecreasing (or all monotone nonincreasing), then {f_{n}(Y_{n}), n ≥ 1} is a sequence of ρ^{−}-mixing random variables.

Note that Property P2 in Wang and Lu (2006) is stated only for increasing functions. It is simple to see that this property remains true for nondecreasing functions, too. The statement for nonincreasing functions follows for the observation that if a function f_{n} is nonincreasing, then the function −f_{n} is nondecreasing.

Recall that a measurable function l is said to be slowly varying if for each λ > 0

lim_{x→∞} \frac{l(λx)}{l(x)} = 1.

We refer to Seneta (1976) for other equivalent definitions and for detailed and comprehensive study of properties of such functions.

In the following, C will represent a positive constants although its value may change from one appearance to the next.

The following result was proved in Budsaba, Chen, and Volodin (2007).

Theorem BCV. Let l(x) be a positive slowly varying function and 1 ≤ p < 2, r ≥ 1, pr ≠ 1. Suppose {Y_{i}, −∞ < i < ∞} is a sequence of identically distributed and ρ^{−}-mixing random variables and {X_{n}, n ≥ 1} is the moving average process based on the sequence {Y_{i}, −∞ < i < ∞}. Then EY_{1} = 0 and E|Y_{1}|^{rp}l(|Y_{1}|^{p}) < ∞ imply that for all ε > 0

∑_{n=1}^{∞} n^{r−2}l(n)P (max_{k≤n} |∑_{j=1}^{k} X_{j}| ≥ εn^{1/p}) < ∞.

In particular, the assumptions EY_{1} = 0 and E|Y_{1}|^{p} < ∞, 1 < p < 2 imply Marcinkiewicz-Zygmund strong law of large numbers

n^{−1/p} ∑_{k=1}^{n} X_{k} → 0 a.s. as n → ∞.

Next, in Concluding Remark 4, Budsaba, Chen, and Volodin (2007) mentioned that the case p = r = 1 is not treated in Theorem BCV and that the authors believe that the result can be proved under the additional assumption ∑_{i=−∞}^{∞} |a_{i}|^{s} < ∞ for some 0 < s < 1.

In this paper it is shown that this suggestion is true. Namely, we prove the following result.

Theorem 1. Let l(x) be a positive slowly varying function and suppose that {Y_{i}, −∞ < i < ∞} is a sequence of identically distributed ρ^{−}-mixing random variables. Let {X_{n}, n ≥ 1} be the moving average process based on the sequence {Y_{i}, −∞ < i < ∞}. Let moreover {a_{i}, −∞ < i < ∞} be a sequence of real numbers with ∑_{i=−∞}^{∞} |a_{i}|^{s} < ∞ for some 0 < s < 1. Then EY_{1} = 0 and E|Y_{1}|l(|Y_{1}|) < ∞ imply that for all ε > 0

∑_{n=1}^{∞} n^{−1}l(n)P { max_{1≤k≤n} | ∑_{j=1}^{k} X_{j}| > εn} < ∞.

In particular, EY_{1} = 0 implies the following Kolmogorov strong law of large numbers

n^{−1} ∑_{k=1}^{n} X_{k} → 0 a.s. as n → ∞.

Proof. Let Y_{nj}^{(1)} = −nI(Y_{j} < −n) + Y_{i}I(|Y_{j}| ≤ n) + nI(Y_{j} > n), and Y_{nj}^{(2)} = Y_{j} − Y_{nj}^{(1)}. Then by Property of ρ^{−}-mixing random variables, for

LIMITING BEHAVIOUR 21

any n ≥ 1, {Y_{nj}^{(1)}, −∞ < j < ∞} and {Y_{nj}^{(2)}, −∞ < j < ∞} are sequences of ρ^{−}-mixing random variables. Note that

∑_{k=1}^{n} X_{k} = ∑_{k=1}^{n} ∑_{i=−∞}^{∞} a_{i}Y_{i+k} = ∑_{i=−∞}^{∞} a_{i} ∑_{j=i+1}^{i+n} Y_{j}

and

n^{−1} max_{1≤k≤n} |E ∑_{i=−∞}^{∞} a_{i} ∑_{j=i+1}^{i+k} Y_{nj}^{(1)}| ≤ n^{−1} ∑_{i=−∞}^{∞} |a_{i}| max_{1≤k≤n} |E ∑_{j=i+1}^{i+k} Y_{nj}^{(1)}|

≤ Cn^{−1}n(|EY_{1}I(|Y_{1}| ≤ n)| + nP {|Y_{1}| > n}) = CE|Y_{1}|I(|Y_{1}| > n) + CnP {|Y_{1}| > n} → 0,

as n → ∞. Hence for n large enough

n^{−1} max_{1≤k≤n} | ∑_{i=−∞}^{∞} a_{i} ∑_{j=i+1}^{i+k} Y_{nj}^{(1)}| < ε/4.

Therefore it is enough to prove that

I_{1} := ∑_{n=1}^{∞} n^{−1}l(n)P { max_{1≤k≤n} | ∑_{i=−∞}^{∞} a_{i} ∑_{j=i+1}^{i+k}(Y_{nj}^{(1)} − EY_{nj}^{(1)})| > εn/4} < ∞

and

I_{2} := ∑_{n=1}^{∞} n^{−1}l(n)P ( max_{1≤k≤n} | ∑_{i=−∞}^{∞} a_{i} ∑_{j=i+1}^{i+k} Y_{nj}^{(2)}| > εn/2) < ∞.

For I_{2}, by Markov inequality, we have

I_{2} ≤ C ∑_{n=1}^{∞} n^{−1−s}l(n)E max_{1≤k≤n} | ∑_{i=−∞}^{∞} a_{i} ∑_{j=i+1}^{i+k} Y_{nj}^{(2)}|^{s}

≤ C ∑_{n=1}^{∞} n^{−s}l(n)(E|Y_{1}|^{s}I(|Y_{1}| > n) + n^{s}P {|Y_{1}| > n})

≤ C ∑_{n=1}^{∞} n^{−s}l(n)E|Y_{1}|^{s}I(|Y_{1}| > n)

= ∑_{n=1}^{∞} n^{−s}l(n) ∑_{m=n}^{∞} E|Y_{1}|^{s}P (m < |Y_{1}| ≤ m + 1)

22 K. BUDSABA, P. CHEN, A. VOLODIN

= ∑_{m=1}^{∞} E|Y_{1}|^{s}P (m < |Y_{1}| ≤ m + 1) ∑_{n=1}^{m} n^{−s}l(n)

≤ C ∑_{m=1}^{∞} m^{1−s}l(m)E|Y_{1}|^{s}P (m < |Y_{1}| ≤ m + 1)

≤ CE|Y_{1}|l(|Y_{1}|) < ∞.

For I_{1}, by Markov and Hölder inequalities, we have

I_{1} ≤ C ∑_{n=1}^{∞} n^{−1}l(n)n^{−2}E max_{1≤k≤n} | ∑_{i=−∞}^{∞} a_{i} ∑_{j=i+1}^{i+k}(Y_{nj}^{(1)} − EY_{nj}^{(1)})|^{2}

≤ C ∑_{n=1}^{∞} n^{−1}l(n)n^{−2}E ( ∑_{i=−∞}^{∞} |a_{i}| max_{1≤k≤n} | ∑_{j=i+1}^{i+k}(Y_{nj}^{(1)} − EY_{nj}^{(1)})|)^{2}

≤ C ∑_{n=1}^{∞} n^{−3}l(n)( ∑_{i=−∞}^{∞} |a_{i}|) ∑_{i=−∞}^{∞} |a_{i}|E max_{1≤k≤n} | ∑_{j=i+1}^{i+k}(Y_{nj}^{(1)} − EY_{nj}^{(1)})|^{2}

≤ C ∑_{n=1}^{∞} n^{−2}l(n)(E|Y_{1}|^{2}I(|Y_{1}| ≤ n) + n^{2}P {|Y_{1}| > n})

≤ C ∑_{n=1}^{∞} n^{−2}l(n)E|Y_{1}|^{2}I(|Y_{1}| ≤ n) + CE|Y_{1}|l(|Y_{1}|)

= C ∑_{n=1}^{∞} n^{−2}l(n) ∑_{m=1}^{n} E|Y_{1}|^{2}I(m − 1 < |Y_{1}| ≤ m) + CE|Y_{1}|l(|Y_{1}|)

= C ∑_{m=1}^{∞} E|Y_{1}|^{2}I(m − 1 < |Y_{1}| ≤ m) ∑_{n=m}^{∞} n^{−2}l(n) + CE|Y_{1}|l(|Y_{1}|)

≤ C ∑_{m=1}^{∞} m^{−1}l(m)E|Y_{1}|^{2}I(m − 1 < |Y_{1}| ≤ m) + CE|Y_{1}|l(|Y_{1}|)

≤ C ∑_{m=1}^{∞} E|Y_{1}|l(|Y_{1}|)I(m − 1 < |Y_{1}| ≤ m) + CE|Y_{1}|l(|Y_{1}|)

≤ CE|Y_{1}|l(|Y_{1}|) < ∞. Now we show the almost sure convergence. By the first part of Theo-

rem, EY_{1} = 0 (and hence E|Y_{1}| < ∞) implies

∑_{n=1}^{∞} n^{−1}P { max_{1≤k≤n} |S_{n}| ≥ εn^{1/p}} < ∞, for all ε > 0.

Therefore

∞ > ∑_{n=1}^{∞} n^{−1}P { max_{1≤m≤n} |S_{m}| > εn}

= ∑_{k=1}^{∞} ∑_{n=2^{k−1}}^{2^{k}} n^{−1}P { max_{1≤m≤n} |S_{m}| > εn}

≥ 1/2 ∑_{k=1}^{∞} P { max_{1≤m≤2^{k−1}} |S_{m}| > ε2^{k}}.

By Borel-Cantelli lemma

2^{−k} max_{1≤m≤2^{k}} |S_{m}| → 0 almost surely

which implies that S_{n}/n → 0 almost surely.□

The second theorem treats the case when the sequence {a_{i}, −∞ < i < +∞} is not absolutely summable. Theorem 2. Let 1 < q ≤ 2. Assume that there exists s, 1 < s < q, such that

∑_{i=−∞}^{+∞} |a_{i}|^{s} < ∞.

Suppose {Y_{i}, −∞ < i < ∞} is a sequence of identically distributed negatively associated random variables and {X_{n}, n ≥ 1} is the moving average process based on the sequence {Y_{i}, −∞ < i < ∞}. Then for all p, 0 < p < \frac{sq}{sq−q+s}, the assumptions EY_{1} = 0 and E|Y_{1}|^{q} < ∞ imply Marcinkiewicz-Zygmund strong law of large numbers

n^{−1/p} ∑_{k=1}^{n} X_{k} → 0 a.s. as n → ∞.

Remark. Note that the assumption 1 < s < q ≤ 2 implies that p < 2.

Proof. Without loss of generality, we assume that a_{i} ≥ 0 for all i. Let {b_{n}, n ≥ 1} be sequence of positive numbers that will be specified later. By Theorem 2 (1.6) of Shao (2000), we have

E |∑_{k=1}^{n} b_{k}X_{k}|^{q} = E | ∑_{i=−∞}^{+∞} (∑_{k=1}^{n} b_{k}a_{i−k}) Y_{i}|^{q} ≤ C ∑_{i=−∞}^{+∞} (∑_{k=1}^{n} b_{k}a_{i−k})^{q} .

24 K. BUDSABA, P. CHEN, A. VOLODIN

An application of Hölder and then Jensen inequalities yields

|∑_{k=1}^{n} b_{k}a_{i−k}|^{q} ≤ n^{t(1−1/s)} (∑_{k=1}^{n} b_{k}^{s}a_{i−k}^{s})^{q/s}

≤ n^{q(1−1/s)} ∑_{k=1}^{n} a_{i−k}^{s}b_{k}^{q} ( ∑_{i=−∞}^{+∞} a_{i}^{s})^{q/s−1} .

Hence

E |∑_{k=1}^{n} b_{k}X_{k}|^{q} ≤ C ( ∑_{i=−∞}^{+∞} a_{i}^{s})^{q/s} n^{q(1−1/s)} ∑_{k=1}^{n} b_{k}^{q} = Cn^{q(1−1/s)} ∑_{k=1}^{n} b_{k}^{q}.

By Theorem 3.3 of Móricz, Serfling, and Stout (1982), we have the following maximal inequality

E ( max_{1≤m≤n} |∑_{k=1}^{m} b_{k}X_{k}|)^{q} ≤ C log_{2}^{q}(2n)n^{q(1−1/s)} ∑_{k=1}^{n} b_{k}^{q}.

This maximal inequality implies the almost sure convergence of the series ∑_{k=1}^{∞} b_{k}X_{k} as soon as ∑_{k=1}^{∞} b_{k}^{q}k^{q(1−1/s)} log_{2}^{q}(2k) < ∞ (see for instance, Loève (1978) Section 36.1). The application of Kronecker lemma with b_{k} = k^{−1/p} concludes the proof of Theorem 2. □

References

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[7] Loève M. Probability Theory II, 4 edition, Spring-Verlag, New York, 1978.

[8] Móricz, F. A., Serfling, R. J., Stout, W. F. Moment and probability bounds with quasisuperadditive structure for the maximum partial sum. Ann. Probab. 10 (1982), 1032-1040.

[9] Seneta. E. Regularly varying functions, Lecture Notes in Math. 508 (1976) Springer-Berlin.

LIMITING BEHAVIOUR 25

[10] Shao, Q.M. A comparison theorem on inequalities between negatively associated and independent random variables, J. Theor. Probab. 13 (2000), 343-56.

[11] Wang, J.F. and Lu, F.B. Inequalities of maximum of partial sums and weak convergence for a class of weak dependent random variables. Acta Math. Sin. (Engl. Ser.) 22 (2006), 693-700

Department of Mathematics and Statistics, Thammasat University, Rangsit Center, Pathumthani, 12121, Thailand E-mail address: kamon@mathstat.sci.tu.ac.th

Department of Mathematics, Jinan University, Guangzhou, 510630, China E-mail address: chenpingyan@yahoo.com.cn

Department of Mathematics and Statistics, University of Regina, Regina, Saskatchewan, Canada, S4S 0A2 E-mail address: andrei@math.uregina.ca

Received March 18, 2007