1 Introduction
Due to the estimation of least squares regression coefficients in linear regression and nonparametric curve estimation, it is very interesting and meaningful to study the limit behaviors for the weighted sums of random variables.
We recall the concept of \(\rho^{*}\)-mixing random variables.
A number of limit results for
\(\rho^{*}\)-mixing sequences of random variables have been established by many authors. We refer to Bradley [
3] for the central limit theorem, Bryc and Smolenski [
4], Peligrad and Gut [
5], and Utev and Peligrad [
6] for the moment inequalities, and Sung [
1] for the complete convergence of weighted sums.
Special cases for weighted sums have been studied by Bai and Cheng [
7], Chen et al. [
8], Choi and Sung [
9], Chow [
10], Cuzick [
11], Sung [
12], Thrum [
13], and others. In this paper, we focus on the array weights
\(\{a_{nk}, 1\le k\le n, n\ge1\}\) of real numbers satisfying
$$ \sup_{n\geq1}n^{-1}\sum^{n}_{k=1}|a_{nk}|^{\alpha}< \infty $$
(1.1)
for some
\(\alpha>0\). In fact, under condition (
1.1), many authors have studied the limit behaviors for the weighted sums of random variables.
Let
\(\{X, X_{n}, n\geq1\}\) be a sequence of independent and identically distributed random variables. When
\(\alpha=2\), Chow [
10] showed that the Kolmogorov strong law of large numbers
$$ n^{-1}\sum^{n}_{k=1}a_{nk}X_{k} \rightarrow0 \quad \mbox{a.s.} $$
(1.2)
holds if
\(EX=0\) and
\(EX^{2}<\infty\). Cuzick [
11] generalized Chow’s result by showing that (
1.2) also holds if
\(EX=0\) and
\(E|X|^{\beta}<\infty\) for
\(\beta>0\) with
\(1/\alpha+1/\beta=1\). Bai and Cheng [
7] proved that the Marcinkiewicz–Zygmund strong law of large numbers
$$ n^{-1/p}\sum^{n}_{k=1}a_{nk}X_{k} \rightarrow0 \quad \mbox{a.s.} $$
(1.3)
holds if
\(EX=0\) and
\(E|X|^{\beta}<\infty\), where
\(1\leq p<2\) and
\(1/\alpha +1/\beta=1/p\). Chen and Gan [
14] showed that if
\(0< p<1\) and
\(E|X|^{\beta}<\infty\), then (
1.3) still holds without the independent assumption.
Under condition (
1.1), a convergence rate in the strong law of large numbers is also discussed. Chen [
15] showed that
$$ \sum^{\infty}_{n=1}n^{r-2}P\Biggl\{ \max_{1\leq m\leq n} \Biggl\vert \sum^{m}_{k=1}a_{nk}X_{k} \Biggr\vert >\varepsilon n^{1/p}\Biggr\} < \infty,\quad \forall \varepsilon>0, $$
(1.4)
if
\(\{X, X_{n}, n\geq1\}\) is a sequence of identically distributed negatively associated (NA) random variables with
\(EX=0\) and
\(E|X|^{(r-1)\beta}<\infty\), where
\(r>1\),
\(1\leq p<2\),
\(1/\alpha+1/\beta=1/p\), and
\(\alpha< rp\). The main tool used in Chen [
15] is the exponential inequality for NA random variables (see Theorem 3 in Shao [
16]). Sung [
1] proved (
1.4) for a sequence of identically distributed
\(\rho^{*}\)-mixing random variables with
\(EX=0\) and
\(E|X|^{rp}<\infty\), where
\(\alpha>rp\), by using the Rosenthal moment inequality. Since the Rosenthal moment inequality for NA has been established by Shao [
16], it is easy to see that Sung’s result also holds for NA random variables. However, for
\(\rho^{*}\)-mixing random variables, we do not know whether the corresponding exponential inequality holds or not, and so the method of Chen [
15] does not work for
\(\rho^{*}\)-mixing random variables. On the other hand, the method of Sung [
1] is complex and not applicable to the case
\(\alpha\leq rp\).
In this paper, we show that (
1.4) holds for a sequence of identically distributed
\(\rho^{*}\)-mixing random variables with suitable moment conditions. The moment conditions for the cases
\(\alpha< rp\) and
\(\alpha>rp\) are optimal. The moment conditions for
\(\alpha=rp\) are nearly optimal. Although the main tool is the Rosenthal moment inequality for
\(\rho ^{*}\)-mixing random variables, our method is simpler than that of Sung [
1] even in the case
\(\alpha>rp\).
We also extend (
1.4) to complete moment convergence, that is, we provide moment conditions under which
$$ \sum^{\infty}_{n=1}n^{r-2-q/p} E \Biggl( \max_{1\leq m\leq n} \Biggl\vert \sum^{m}_{k=1}a_{nk}X_{k} \Biggr\vert -\varepsilon n^{1/p} \Biggr)_{+}^{q}< \infty,\quad \forall\varepsilon>0, $$
(1.5)
where
\(q>0\).
Note that if (
1.5) holds for some
\(q>0\), then (
1.4) also holds. The proof is well known.
Throughout this paper, C always stands for a positive constant that may differ from one place to another. For events A and B, we denote \(I(A, B)=I(A\cap B)\), where \(I(A)\) is the indicator function of an event A.
3 Main results
We first present complete convergence for weighted sums of \(\rho ^{*}\)-mixing random variables.
Now we extend Theorem
3.1 to complete moment convergence.
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