The Convergence of Double-Indexed Weighted Sums of Martingale Differences and Its Application

and Applied Analysis 3 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 X ni = X i 1(|X i | ≤ n α ), 1 ≤ i ≤ n. It can be found that a ni X i = a ni X i 1(|X i | > n α ) + [a ni X ni − a ni E(X ni | F i−1 )] + a ni E(X ni | F i−1 ), 1 ≤ i ≤ n. By Lemma 3 with a = n, for any q > 1, we obtain that


Introduction
Hsu and Robbins [1] introduced the concept of complete convergence; that is, a sequence of random variables {  ,  ≥ 1} is said to converge completely to a constant  if ∑ ∞ =1 (|  − | ≥ ) < ∞ for all  > 0. By Borel-Cantelli lemma, it follows that   →  almost surely (a.s.).The converse is true if {  ,  ≥ 1} 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  > 1/2,  > 1, and {  ,  ≥ 1} be a sequence of i.i.d.random variables.Assume that  1 = 0 if  ≤ 1.Then the following statements are equivalent: 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 [12][13][14], 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 lineartime-invariant systems and present their convergence with probability one and in mean square, respectively (see Theorems 11 and 12 in Section 3).
The following lemmas are useful for the proofs of the main results.

The Convergence of Double-Indexed Weighted Sums of Martingale Differences
First, we give the complete moment convergence of doubleindexed weighted sums of martingale differences.
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.

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.
In our setup, the output () is only measured at a sequence of sampling time instants {  } with measured values (  ), and noise    (  ) =  (  ) −   . (26) 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  >  0 and  <  0 , the solution to system (25) can be expressed as Suppose that {  , 1 ≤  ≤ } is a sequence of sampling times.For   ≤   , we have Since the second term is known, it will be denoted by Then, (30) can be written as Suppose that Φ  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 1/2 <  < 1.For convergence analysis, one must consider a typical entry in (1/  )Φ     .By the Cayley Hamilton theorem (see Ogata [18]), the matrix exponential can be expressed by a polynomial function of  of order at most  0 − 1, where the time functions   () can be derived by the Lagrange-Hermite interpolation method (see Ogata [18]).This implies that where   (  −   ) = [ 1 (  −   ), . . .,   0 (  −   )] and   is the observability matrix.Denote Then which implies that As a result, for any  > 0, one has Under Assumption 9,  −1 0 exists.Convergence results will be established by the following two sufficient conditions: (1/  )Ψ     → 0 and (1/  )Ψ   Ψ  ≥ , for some  > 0. 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 1/2 <  < 1, where  min () is the small eigenvalue of  for a suitable symmetric .
We can investigate the convergence of double-indexed summations of random variables form for some 1/2 <  < 1.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 (1/  )Ψ     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 [18][19][20][21][22][23] and the references therein.
As an application of Corollary 6 to the observers and state estimation, we obtain the following theorem.
Proof of Corollary 6.If  = 2 and  = 2/, then one has  = 2.So as an application of Theorem 5, one gets (6) immediately.On the other hand, it can be seen that