#### Abstract

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.

#### 1. Introduction

Hsu and Robbins [1] introduced the concept of complete convergence; that is, a sequence of random variables is said to converge completely to a constant if for all . By Borel-Cantelli lemma, it follows that almost surely (a.s.). The converse is true if is independent. But the converse cannot always be true for the dependent case. Hsu and Robbins [1] obtained that the sequence of arithmetic means of independent and identically distributed (i.i.d.) random variables converges completely to the expected value if the variance of the summands is finite. Erdös [2] proved the converse. The result of Hsu-Robbins-Erdös is a fundamental theorem in probability theory, and it has been generalized and extended in several directions by many authors. Baum and Katz [3] gave the following generalization to establish a rate of convergence in the sense of Marcinkiewicz-Zygmund-type strong law of large numbers.

Theorem 1. Let , , and be a sequence of i.i.d. random variables. Assume that if . Then the following statements are equivalent:(i);(ii) for all .

Many authors have extended Theorem 1 to the martingale differences. For example, Yu [4] obtained the complete convergence for weighted sums of martingale differences; Ghosal and Chandra [5] gave the complete convergence of martingale arrays; Stoica [6, 7] investigated the Baum-Katz-Nagaev-type results for martingale differences and the rate of convergence in the strong law of large numbers for martingale differences; Wang et al. [8] also studied the complete convergence and complete moment convergence for martingale differences, which generalized some results of Stoica [6, 7]; Yang et al. [9] obtained the complete convergence for the moving average process of martingale differences and so forth. For other works about convergence analysis, one can refer to Gut [10], Chen et al. [11], Sung [1214], Sung and Volodin [15], Hu et al. [16], and the references therein.

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 5, Corollary 6, and Theorem 7 in Section 2. On the other hand, we give the applications of Corollary 6 and Theorem 7 to study the convergence of the state observers of linear-time-invariant systems and present their convergence with probability one and in mean square, respectively (see Theorems 11 and 12 in Section 3).

Recall that the sequence is stochastically dominated by a nonnegative random variable if for some positive constant and for all .

Throughout the paper, let , be the indicator function of set , and let denote some positive constants not depending on , which may be different in various places.

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

Lemma 2 (cf. Hall and Heyde [17, Theorem 2.11]). If are martingale differences and , then there exists a constant depending only on such that

Lemma 3 (cf. Sung [12, Lemma 2.4]). Let and be sequences of random variables. Then for any , , , and , one has

Lemma 4 (cf. Wang et al. [8, Lemma 2.2]). Let be a sequence of random variables stochastically dominated by a nonnegative random variable . Then for any , , and , the following two statements hold: Consequently, . Here , , and are positive constants.

#### 2. The Convergence of Double-Indexed Weighted Sums of Martingale Differences

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

Theorem 5. Let , , and be martingale differences stochastically dominated by a nonnegative random variable with . Let be a triangular array of real numbers. For some , we assume that and Then for every ,

Taking and for in Theorem 5, we have the following result.

Corollary 6. Let , be martingale differences stochastically dominated by a nonnegative random variable with . Let be a triangular array of real numbers. For some , one assumes that and (4) holds true. Then for every , In particular, one has

Next, we investigate the convergence in mean square.

Theorem 7. Let and be martingale differences stochastically dominated by a nonnegative random variable with . Let be a triangular array of real numbers and Then, one has where is a positive constant.

Remark 8. Wang et al. [8] obtained the complete convergence and complete moment convergence for nonweighted martingale differences, which generalized some results of Stoica [6, 7]. In this paper, we study the complete moment convergence of double-indexed weighted sums of martingale differences. So we extend the results of Wang et al. [8] and Stoica [6, 7] to the case of double-indexed weighted sums of martingale differences. On the other hand, we give the applications of Corollary 6 and Theorem 7 to study the convergence of the state observers of linear-time-invariant systems and present the convergence with probability one and in mean square, respectively (see Theorems 11 and 12 in Section 3).

Proof of Theorem 5. Let , . It can be found that ,.
By Lemma 3 with , for any , we obtain that For , it is easy to see that . Consequently, for any , we get by Hölder’s inequality and (4) that So, it can be checked by Markov’s inequality, Lemma 4, (11), and () that Since are martingale differences, by the martingale property and the proof of (12), one has that
Next, we turn to prove under conditions of Theorem 5. It can be seen that are also martingale differences. So, by Markov’s inequality, (10), and Lemma 2 with , it can be found that Obviously, it follows that Combining (11) with , we obtain that following from the fact that . Meanwhile, by inequality, Lemma 4, and (4), By the conditions and , we have that , which implies . So, we obtain by that By the proof of (12), one has that Thus, by (15)–(20), we have that . So, it completes the proof of (5).

Proof of Corollary 6. If and , then one has . So as an application of Theorem 5, one gets (6) immediately. On the other hand, it can be seen that So by (5) and (21) with , we have for every that It follows from Borel-Cantelli lemma that So, (7) holds.

Proof of Theorem 7. Since are martingale differences, it can be found by Lemmas 2 and 4 and (8) that Consequently, (9) holds true.

#### 3. Applications to the Convergence of the State Observers of Linear-Time-Invariant Systems

In this section, we give the applications of Corollary 6 and Theorem 7 to study the convergence of the state observers of linear-time-invariant systems.

For , consider an MISO (multi-input-single-output) linear-time-invariant system where , , and are known system matrices, and for , is the control input, is the state, and is the system output. The initial state is unknown. We are interested in estimation of , from some limited observations on .

In our setup, the output is only measured at a sequence of sampling time instants with measured values , and noise

We would like to estimate the state from information on , , and . In practical systems, the irregular sampling sequences can be generated by different means such as randomized sampling, event-triggered sampling, and signal quantization.

It is obvious that state estimation will not be possible if the system is not observable. Also, in this paper, is assumed to be martingale difference. We give the following assumption.

Assumption 9. The system (25) is observable; that is, the observability matrix has full rank.

For both and , the solution to system (25) can be expressed as

Suppose that is a sequence of sampling times. For , we have

Since the second term is known, it will be denoted by . This leads to the observations

Define

Then, (30) can be written as

Suppose that is full rank, which will be established later. Then, a least-squares estimate of is given by Here, denotes the transpose of . From (32) and (33), the estimation error for at sampling time is for some . For convergence analysis, one must consider a typical entry in . By the Cayley Hamilton theorem (see Ogata [18]), the matrix exponential can be expressed by a polynomial function of of order at most , where the time functions can be derived by the Lagrange-Hermite interpolation method (see Ogata [18]). This implies that where and is the observability matrix.

Denote Then which implies that

As a result, for any , one has

Under Assumption 9, exists. Convergence results will be established by the following two sufficient conditions: and , for some . So we need the following persistent excitation (PE) condition, which was used by Wang et al. [19] and Thanh et al. [20].

Assumption 10. For some , where is the small eigenvalue of for a suitable symmetric .

We can investigate the convergence of double-indexed summations of random variables form for some . Here, is a triangular array of real numbers and is a sequence of martingale differences. It can be seen that (42) is a special case of (7) in Corollary 6. The th component of takes the form where is a triangular array of real numbers. The convergence analysis of (43) for is a special case of (42) or (7) in Corollary 6.

Recently, Wang et al. [19] investigated the convergence analysis of the state observers of linear-time-invariant systems under -mixing sampling. Thanh et al. [20] studied the convergence analysis of double-indexed and randomly weighted sums of -mixing sequence and gave its application to state observers. For more related works, one can refer to [1823] and the references therein.

As an application of Corollary 6 to the observers and state estimation, we obtain the following theorem.

Theorem 11. Let Assumptions 9 and 10 hold. Let and be martingale differences stochastically dominated by a nonnegative random variable with . Suppose that for any , one has and where . Then Consequently,

As an application to Theorem 7, we get the following result.

Theorem 12. Let and Assumptions 9 and 10 hold. Assume that are martingales differences stochastically dominated by a nonnegative random variable with . For , it is supposed that Then

Remark 13. If we assume that, for each , is uniformly bounded, then we can find that condition (44) holds for any . On the other hand, similar to Theorems 11 and 12, Wang et al. [19] also obtained the convergence of the state observers with probability one and in mean square under -mixing sampling (see Theorems 4 and 5 of Wang et al. [19]). So Theorems 11 and 12 generalize the results of Wang et al. [19] to the case of martingale differences.

Proof of Theorem 11. It can be seen that To prove (45), it suffices to look at the th component of For any , by and (44), we can obtain (45) from Corollary 6 with , in (43), and .
On the other hand, by Assumption 9, exists, and by (41) in Assumption 10, exists and where is the largest eigenvalue. Together with and (45), it follows (46).

Proof of Theorem 12. For , by (47), (8) holds. Applying Theorem 7 with , , and , we obtain that for a typical term in (49), Together with (49), (53), and (55), we obtain that where is a positive constant. Lastly, by (56), (48) holds true.

#### Conflict of Interests

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

#### Acknowledgments

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.