We investigate the complete moment convergence of double-indexed weighted sums of martingale differences. Then it is easy to obtain the Marcinkiewicz-Zygmund-type strong law of large numbers of double-indexed weighted sums of martingale differences. Moreover, the convergence of double-indexed weighted sums of martingale differences is presented in mean square. On the other hand, we give the application to study the convergence of the state observers of linear-time-invariant systems and present the convergence with probability one and in mean square.

Hsu and Robbins [

Let

Many authors have extended Theorem

In this paper, we study the moment complete convergence of double-indexed weighted sums of martingale differences. Then it is easy to obtain the Marcinkiewicz-Zygmund-type strong law of large numbers of double-indexed weighted sums of martingale differences. Moreover, the convergence of double-indexed weighted sums of martingale differences is presented in mean square. For the details, see Theorem

Recall that the sequence

Throughout the paper, let

The following lemmas are useful for the proofs of the main results.

If

Let

Let

First, we give the complete moment convergence of double-indexed weighted sums of martingale differences.

Let

Taking

Let

Next, we investigate the convergence in mean square.

Let

Wang et al. [

Let

By Lemma

Next, we turn to prove

If

Since

In this section, we give the applications of Corollary

For

In our setup, the output

We would like to estimate the state

It is obvious that state estimation will not be possible if the system is not observable. Also, in this paper,

The system (

For both

Suppose that

Since the second term is known, it will be denoted by

Define

Then, (

Suppose that

Denote

As a result, for any

Under Assumption

For some

We can investigate the convergence of double-indexed summations of random variables form

Recently, Wang et al. [

As an application of Corollary

Let Assumptions

As an application to Theorem

Let

If we assume that, for each

It can be seen that

On the other hand, by Assumption

For

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work is supported by the NNSF of China (11171001, 11201001, and 11326172), Natural Science Foundation of Anhui Province (1208085QA03 and 1408085QA02), Higher Education Talent Revitalization Project of Anhui Province (2013SQRL005ZD), Academic and Technology Leaders to Introduction Projects of Anhui University, and Doctoral Research Start-up Funds Projects of Anhui University.