Abstract
In this paper, we extend some known results about complete convergence and establish the complete convergence and complete moment convergence for randomly weighted sums of martingale difference sequence. Our results can generalize some conclusions related to Hsu–Robbins–Erdös strong laws and Baum–Katz type theorems for martingales.
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1 Introduction
The complete convergence plays a key role in the development of probability theory, especially in establishing the rate of convergence. Hsu and Robbins [1] introduced the concept of complete convergence as follows. A sequence \(\{X_{n}, n\ge1\}\) is said to converge completely to C if
where C is a constant. By the Borel–Cantelli lemma, it follows that \(X_{n}\to C\) almost surely as \(n\to\infty\). If \(\{X_{n}, n\ge1\}\) is independent and identically distributed (i.i.d.) random variables, the converse is true.
Suppose that \(\{X_{n}, n\ge1\}\) is a random variable sequence defined on the fixed probability space \((\Omega, \mathcal{F}, P)\). Denote \(S_{n}=\sum_{i=1}^{n}X_{i}\), \(S_{0}=0\), \(\log x=\log(\max\{e, x\})\), \(x^{+}=xI(x\ge 0)\), and \(\mathcal{F}_{0}=\{\Omega, \emptyset\}\). Let \(\{\mathcal {F}_{n}, n\ge1\}\) be an increasing sequence of σ fields with \(\mathcal{F}_{n}\subset\mathcal{F}\) for each \(n\ge1\). If \(X_{n}\) is \(\mathcal{F}_{n}\) measurable for each \(n\ge1\), then σ fields \(\{ \mathcal{F}_{n}, n\ge1\}\) are thought to be adapted to the random variable sequence \(\{X_{n}, n\ge1\}\) and \(\{X_{n}, \mathcal{F}_{n}, n\ge1\} \) is thought to be an adapted stochastic sequence. The following theorem is a generalization of some known results.
Theorem 1.1
(Hsu–Robbins–Erdös strong law [1, 2])
Let \(\{X_{n}, n\ge1\}\) be a sequence of independent and identically distributed random variables. Assume that \(EX_{n}=0\) and set \(S_{n}=\sum_{i=1}^{n}X_{i}\), \(n\ge1\). Then \(EX_{n}^{2}<\infty\) is equivalent to the condition that
In probability theory, Hsu–Robbins–Erdös strong law as a basic theorem has been extended in several directions by some authors. The following theorem is given by Baum and Katz [3] to establish a rate of convergence.
Theorem 1.2
(Baum and Katz strong law)
Let \(\alpha>1/2\), \(\alpha p>1\), and let \(\{X_{n}, n\ge1\}\) be a sequence of independent and identically distributed random variables. Assume that \(EX_{n}=0\) if \(\alpha\le1\), and set \(S_{n}=\sum_{i=1}^{n}X_{i}\), \(n\ge1\). Then \(E|X_{n}|^{p}<\infty\) is equivalent to the condition that
and also equivalent to the condition that
Motivated by the above results for i.i.d. random variables, many authors have studied them for dependent cases. The case for weighted sums of extended negatively dependent (END) random variable sequence was investigated by Shen et al. [4]. Miao et al. [5] improved some known results and studied the Baum–Katz type convergence rate in the Marcinkiewicz–Zygmund strong law for martingales. Chen et al. [6] also gave some extended results for the sequence of martingale difference.
The aims of the present paper are to extend the results on complete convergence for the sequence of martingale difference. The following definitions will be used frequently in this paper.
Definition 1.1
If \(\{X_{n}, \mathcal{F}_{n}, n\ge1\}\) is an adapted stochastic sequence with
and \(E|X_{n}|<\infty\) for each \(n\ge1\), then the sequence \(\{X_{n}, \mathcal{F}_{n}, n\ge1\}\) is called a martingale difference sequence.
Definition 1.2
A real-valued function \(l(x)\), positive and measurable on \((0, \infty )\), is said to be slowly varying if
for each \(\lambda>0\).
Definition 1.3
A sequence \(\{X_{n}, n\ge1\}\) of random variables is said to be stochastically dominated by a random variable X if there exists a positive constant C, such that
for all \(x\ge0\) and \(n\ge1\).
Now let us recall some known results for complete convergence of martingales.
Theorem 1.3
Let \(\{X_{n}, \mathcal{F}_{n}, n\ge1\}\) be an \(L^{p}\)-bounded martingale difference sequence. If \(0<1/\alpha<2<p\) or \(1< p<2\), \(1\le1/\alpha\le p\), then
If \(p=\alpha=1\), the martingale difference sequence satisfies
then (1.5) holds.
Wang and Hu [9] further studied the Baum–Katz type theorem for the maximal partial sum of martingale difference sequence.
Theorem 1.4
[9] Let \(\{X_{n}, \mathcal{F}_{n}, n\ge1\}\) be a sequence of martingale difference, which is stochastically dominated by a random variable X. Let \(l(x)>0\) be a slowly varying function as \(x\to\infty\). Let \(\alpha>1/2\), \(p\ge1\) and \(\alpha p\ge1\). When \(p\ge2\), we further assume that
for some \(q>\frac{2(\alpha p-1)}{2\alpha-1}\). If
then for any \(\epsilon>0\),
Yang et al. [10] generalized the results of Stoica [7, 8] and Wang et al. [11] for the nonweighted sums of martingale difference sequence to the case of randomly weighted sums.
Theorem 1.5
[10] Let \(\{X_{n}, \mathcal{F}_{n}, n\ge1\}\) be a martingale difference sequence stochastically dominated by a nonnegative random variable X with \(EX^{p}<\infty\). Assume that \(\{A_{n}, n\ge1\}\) is a random sequence, and it is independent of \(\{X_{n}, n\ge1\}\). Denote \(\mathcal {G}_{0}=\{\emptyset, \Omega\}\) and \(\mathcal{G}_{n}=\sigma(X_{1},\ldots, X_{n})\), \(n\ge1\). Let \(\alpha>1/2\), \(1< p<2\), and \(1\le\alpha<2\). If
then
If \(\alpha>1/2\), \(p\ge2\), and for some \(q>\frac{2(\alpha p-1)}{2\alpha -1}\), we assume that
Let
then (1.9) holds.
If \(\alpha>0\) and \(p=1\), the martingale difference sequence is stochastically dominated by a nonnegative random variable X with \(E[X\log(1+X)]<\infty\), and (1.8) holds, then
We shall study the complete convergence and complete moment convergence for randomly weighted sums of martingale difference sequence. The paper is organized as follows. The next section is devoted to the descriptions of our main results, and their proofs will be given in Sect. 3. Throughout the paper, we use the constant C to denote a universal real number that is not necessarily the same in each appearance.
2 Main results
Theorem 2.1
Let \(\{X_{n}, \mathcal{F}_{n}, n\ge1\}\) be a sequence of martingale difference, which is stochastically dominated by a random variable X. Let \(l(x)>0\) be a slowly varying function as \(x\to\infty\). Suppose that \(\{b_{n}, n\ge1\}\) and \(\{c_{n}, n\ge1\}\) are sequences of positive constants such that, for \(p>1\), \(\alpha>0\), \(\alpha p\ge1\), and some \(q\ge\max\{2, p\}\),
and
where \(c_{n}\to\infty\) as \(n\to\infty\). Assume that \(\{A_{n}, n\ge1\}\) is a random sequence independent of \(\{ X_{n}, n\ge1\}\) such that
If
then for any \(\epsilon>0\),
Corollary 2.1
Under the conditions of Theorem 2.1, we take \(b_{n}=n^{\alpha p-2}\), \(c_{n}=n^{\alpha}\) for \(\alpha>1/2\), \(p>1\), and \(\alpha p\ge1\). If
then for any \(\epsilon>0\),
Remark 2.1
Obviously, (2.6) can be checked by Theorem 2.1 and Lemma 3.4. Under the conditions of Corollary 2.1, if we take \(A_{i}\equiv1\), \(i\ge1\), then we have (1.7), i.e., the conclusion of Wang and Hu [9] holds for \(p>1\). On the other hand, if we take \(l(x)\equiv1\), then we can get Remark 2.1 in Yang et al. [10]. So our results can imply these known results.
Example 2.1
Under the conditions of Theorem 2.1, we take \(b_{n}=n^{r-2}\), \(l(n)=\log n\), and \(c_{n}=n^{r/p}\) for \(p>1\) and \(r>p\). If
then for any \(\epsilon>0\),
Example 2.2
Under the conditions of Theorem 2.1, we take \(b_{n}=\frac{\log n}{n}\), \(l(n)=\log n\), and \(c_{n}=(n\log n)^{\frac{1}{p}}\) for \(1< p\le2\). If
then for any \(\epsilon>0\),
Theorem 2.2
Let \(\{X_{n}, \mathcal{F}_{n}, n\ge1\}\) be a sequence of martingale difference, which is stochastically dominated by a random variable X. Let \(l(x)>0\) be a slowly varying function as \(x\to\infty\). Suppose that \(\{b_{n}, n\ge1\}\) and \(\{c_{n}, n\ge1\}\) are sequences of positive constants such that, for \(p>1\), \(\alpha>0\), \(\alpha p\ge1\), and some \(q\ge\max\{2, p\}\),
and
where \(c_{n}\to\infty\) as \(n\to\infty\). Assume that \(\{A_{n}, n\ge1\}\) is a random sequence independent of \(\{ X_{n}, n\ge1\}\) such that
If
then for any \(\epsilon>0\),
Theorem 2.3
Let \(\{X_{n}, \mathcal{F}_{n}, n\ge1\}\) be a sequence of martingale difference, which is stochastically dominated by a random variable X. Let \(l(x)>0\) be a slowly varying function as \(x\to\infty\). Suppose that \(\{b_{n}, n\ge1\}\) and \(\{c_{n}, n\ge1\}\) are sequences of positive constants such that, for \(p=1\), \(\alpha>0\), \(\alpha p\ge1\), and some \(q\ge2\),
where \(c_{n}\to\infty\) as \(n\to\infty\). Assume that \(\{A_{n}, n\ge1\}\) is a random sequence independent of \(\{ X_{n}, n\ge1\}\) and satisfying (2.9). If
then we have formula (2.11).
Corollary 2.2
Under the conditions of Theorem 2.2 for \(p>1\), we take \(b_{n}=n^{\alpha p-2-\alpha}\), \(c_{n}=n^{\alpha}\) for \(\alpha>1/2\) and \(\alpha p\ge1\). If
then for any \(\epsilon>0\),
Under the conditions of Theorem 2.3 for \(p=1\), if we take \(b_{n}=n^{-2}\), \(c_{n}=n^{\alpha}\) for \(\alpha>0\). Then, for any \(\epsilon >0\), we have
Remark 2.2
Obviously, (2.14) and (2.15) can be checked by Theorem 2.2, Theorem 2.3, and Lemma 3.4. Under the conditions of Corollary 2.2, if we take \(l(x)\equiv1\), then we have (1.9) and (1.11), i.e., the results of Yang et al. [10] can be generalized by our conclusions. On the other hand, if we take \(A_{i}\equiv1\), \(i\ge1\), then we can get Theorem 3.3 and Theorem 3.4 in Wang and Hu [9]. Hence, our conclusions can extend these known results.
Example 2.3
Under the conditions of Theorem 2.2, we take \(b_{n}=n^{r-2-r/p}\), \(l(n)=\log n\), and \(c_{n}=n^{r/p}\) for \(p>1\) and \(r>p\). If
then for any \(\epsilon>0\),
Example 2.4
Under the conditions of Theorem 2.2, we take \(b_{n}=\frac{(\log n)^{1-1/p}}{n^{1+1/p}}\), \(l(n)= (\log n)^{1-1/p}\) and \(c_{n}=(n\log n)^{\frac {1}{p}}\) for \(1< p\le2\). If
then for any \(\epsilon>0\),
Remark 2.3
If the conditions of Theorem 2.2 or Theorem 2.3 hold, then for any \(\epsilon>0\), we can get
In fact, it can be checked that for any \(\epsilon>0\),
Remark 2.4
If \(A_{n}=a_{n}\), \(n\ge1\) is non-random (the case of constant weighted), then we can get the results of Theorems 2.1–2.3 for the non-random weighted sums of martingale difference sequence.
3 Proofs for the main results
Throughout this section, we use the constant C to denote a generic real number that is not necessarily the same in each appearance.
3.1 Several lemmas
To prove the main results of the paper, we need to recall the following lemmas.
Lemma 3.1
([12])
If \(\{X_{i}, \mathcal{F}_{i}, 1\le i\le n\}\) is a sequence of martingale difference and \(q>0\), then there exists a constant C depending only on p such that
Lemma 3.2
Let \(\{X_{n}, n\ge1\}\) be a sequence of random variables, which is stochastically dominated by a random variable X. Then, for any \(a>0\) and \(b>0\), the following two statements hold:
and
Lemma 3.3
[16] Let \(\{Y_{n}, n\ge1\}\) and \(\{Z_{n}, n\ge1\}\) be sequences of random variables. Then, for any \(q>1\), \(\epsilon>0\), and \(a>0\),
Lemma 3.4
([17])
If \(l(x)>0\) is a slowly varying function as \(x\to\infty\), then
-
(1)
\(\lim_{x\to\infty}\frac{l(tx)}{l(x)}=1\) for each \(t>0\); \(\lim_{x\to\infty}\frac{l(x+u)}{l(x)}=1\) for each \(u>0\);
-
(2)
\(\lim_{k\to\infty}\sup_{2^{k}\le x<2^{k+1}}\frac {l(x)}{l(2^{k})}=1\);
-
(3)
\(\lim_{x\to\infty}x^{\delta}l(x)=\infty\), \(\lim_{x\to\infty }x^{-\delta} l(x)=0\) for each \(\delta>0\);
-
(4)
\(C_{1}2^{kr}l(\epsilon2^{k})\le\sum_{j=1}^{k}2^{jr}l(\epsilon2^{j})\le C_{2}2^{kr}l(\epsilon2^{k})\) for every \(r>0\), \(\epsilon>0\), positive integer k, and some \(C_{1}>0\), \(C_{2}>0\);
-
(5)
\(C_{1}2^{kr}l(\epsilon2^{k})\le\sum_{j=k}^{\infty}2^{jr}l(\epsilon 2^{j})\le C_{2}2^{kr}l(\epsilon2^{k})\) for every \(r<0\), \(\epsilon>0\), positive integer k, and some \(C_{1}>0\), \(C_{2}>0\).
3.2 Proof of Theorem 2.1
For fixed \(n\ge1\), denote
Since
we have
To prove (2.5), it is enough to show \(H<\infty, I<\infty\), and \(J<\infty\). Obviously, it follows from Hölder’s inequality, Lyapunov’s inequality, and (2.3) that
By the fact that \(\{A_{n}, n\ge1\}\) is independent of \(\{X_{n}, n\ge1\}\), it is easy to check by Markov’s inequality, Lemma 3.2, (3.2), (2.1), and (2.4) that
For I, since \(\{X_{n}, \mathcal{F}_{n}, n\ge1\}\) is a sequence of martingale difference, we can see that \(\{X_{n}, \mathcal{G}_{n}, n\ge1\} \) is also a sequence of martingale difference. Combining with the fact that \(\{A_{n}, n\ge1\}\) is independent of \(\{X_{n}, n\ge1\}\), we have
Consequently, by Markov’s inequality and the proof of (3.3), we have
Next, we shall show that \(J<\infty\). Let \(X_{ni}=X_{i}I(|X_{i}|\le c_{n})\) and \(\hat{Y}_{ni}=a_{i}X_{ni}-E(a_{i}X_{ni}|\mathcal{G}_{i-1})\). It can be found that for fixed real numbers \(a_{1}, \ldots, a_{n}\), \(\{\hat {Y}_{ni}, \mathcal{G}_{i}, 1\le i\le n\}\) is also a sequence of martingale difference. Note that \(\{A_{1}, \ldots, A_{n}\}\) is independent of \(\{X_{n1}, \ldots, X_{nn}\}\). So, by Markov’s inequality and Lemma 3.1, we have
For \(J_{1}\), we have by \(C_{r}\)-inequality, Lemma 3.2 with \(b=c_{n}\), and (2.3) that
For \(J_{11}\), we have by (2.1) and (2.4) that
By the proof of (3.3), it follows
Furthermore, by Hölder’s inequality and (2.3), for any \(1< p\le2\), we have
Obviously, for \(1\le i\le n\), it has
Combining (3.9) and (2.2), we obtain that
By (3.1) and (3.3)–(3.11), we can get (2.5). This completes the proof of Theorem 2.1.
3.3 Proof of Theorem 2.2
As the proof of Theorem 2.1,
where \(Y_{ni}=A_{i}X_{i}I(|X_{i}|\le c_{n})-E[A_{i}X_{i}I(|X_{i}|\le c_{n})|\mathcal {G}_{i-1}]\), \(i=1, 2, \ldots \) . By Lemma 3.3 with \(a=c_{n}\), we have
To prove (2.11), it is enough to show \(H_{1}<\infty\), \(H_{2}<\infty \), and \(H_{3}<\infty\).
Since \(\{A_{n}, n\ge1\}\) is independent of \(\{X_{n}, n\ge1\}\), we have by Lemma 3.2, (3.2), (2.7), and (2.10) that
For \(H_{2}\), by a similar proof of (3.4), we have \(E(A_{n}X_{n}|\mathcal{G}_{n-1})=0\) a.s., \(n\ge1\). Combining with (3.13), we get
Next, from a similar proof of Theorem 2.1 (see (3.5)), we turn to prove \(H_{3}<\infty\).
For \(H_{31}\), by \(C_{r}\)-inequality, Lemma 3.2, and (2.9), we have
From the condition \(q>\max\{2, p\}\), (2.7), and (2.10), we get
By the proof of (3.13), it follows
For \(H_{32}\), from a similar proof of Theorem 2.1 (see (3.9)–(3.11)), combining (3.10), (3.9), and (2.8), we get
Therefore, we can get (2.11) by (3.12)–(3.19). This completes the proof of Theorem 2.2.
3.4 Proof of Theorem 2.3
By a similar proof of Theorem 2.2, we take \(q=2\). It is enough to prove \(H_{1}<\infty\), \(H_{2}<\infty\), and \(H_{3}<\infty\). Combining (3.13) with conditions (2.12), (2.13), we have
By the proof of (3.14) and (3.20), we get
For \(H_{3}\), from a similar proof of Theorem 2.1 (see (3.5) for \(q=2\)), we have
Then, according to (2.12) and (3.20), we can get
Hence, the desired result follows from (3.20)–(3.22). This completes the proof of Theorem 2.3.
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This work was supported by the National Natural Science Foundation of China (11471104).
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Ma, H., Sun, Y. Complete convergence and complete moment convergence for randomly weighted sums of martingale difference sequence. J Inequal Appl 2018, 173 (2018). https://doi.org/10.1186/s13660-018-1770-3
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DOI: https://doi.org/10.1186/s13660-018-1770-3