The equivalent conditions of complete convergence are established for
weighted sums of
Let
Obviously
A sequence of random variables is said to be a
Note that if
The concept of
A sequence
In view of the Borel-Cantelli lemma, this implies that
Let
In many stochastic models, the assumption of independence among random variables is not plausible. So it is necessary to extend the concept of independence to dependence cases. Peligrad and Gut [
Let
Inspired by Theorem 2.1 of Kuczmaszewska [
Throughout this paper, the symbol
Now we state our main results of this paper. The proofs will be given in Section
Let
When proving the limit theorem of
When
Let
Corollary
An and Yuan [
Let
The following lemmas are useful for the proof of the main results.
Suppose
If
Let
Then for any
Let
Since
Note that
Obviously, by Lemma
Combining with the Cauchy-Schwarz inequality and (
Now, we substitute (
Consequently, we prove our main results.
First, we prove that
Note that
Thus, without loss of generality, we may assume that
For fixed
Firstly, we show that
If
If
If
Note that, if
Hence, by Kronecker lemma and (
From (
It is easy to check that for all
Therefore, in order to prove (ii), we only need to prove that
By (
That is, (
Since
When
Taking
When
Similarly to the proof of inequality (
Now, we prove the converse. To prove that (ii) implies (i), it suffices to show that
Noting that
Combining with the condition of
Thus, for sufficiently large
Therefore, by applying Lemma
Taking
It follows from Borel-Cantelli lemma that
Hence,
For all positive integers
The proof of Corollary
The author declares that there is no conflict of interests regarding the publication of this paper.
The author is grateful to the editor and the anonymous referees for their valuable comments and some helpful suggestions that improved the clarity and readability of the paper. The research is supported by the National Natural Science Foundation of China (11226200) and the Natural Science Research Project of Anhui Province (KJ2013Z265).