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

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 {𝑋_𝑛,𝑛≥1} is said to converge completely to a constant 𝐶 if ∑^∞_𝑛=1𝑃(|𝑋_𝑛−𝐶|≥𝜀)<∞ for all 𝜀>0. By Borel-Cantelli lemma, it follows that 𝑋_𝑛→𝐶 almost surely (a.s.). The converse is true if {𝑋_𝑛,𝑛≥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 𝛼>1/2, 𝛼𝑝>1, and {𝑋_𝑛,𝑛≥1} be a sequence of i.i.d. random variables. Assume that 𝐸𝑋₁=0 if 𝛼≤1. Then the following statements are equivalent:(i)𝐸|𝑋₁|^𝑝<∞;(ii)∑^∞_𝑛=1𝑛^𝛼𝑝−2𝑃(max_{1≤𝑘≤𝑛}|∑^𝑘_𝑖=1𝑋_𝑖|>𝜀𝑛^𝛼)<∞ for all 𝜀>0.

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 [12–14], 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 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 {𝑋_𝑛,𝑛≥1} is stochastically dominated by a nonnegative random variable 𝑋 if sup_𝑛≥1𝑃(|𝑋_𝑛|>𝑡)≤𝐾𝑃(𝑋>𝑡) for some positive constant 𝐾 and for all 𝑡≥0.

Throughout the paper, let ℱ₀={∅,Ω}, 𝟙(𝐵) be the indicator function of set 𝐵𝑥⁺=𝑥𝟙(𝑥≥0), and let 𝐾,𝐾₁,𝐾₂,… denote some positive constants not depending on 𝑛, 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 {𝑋_𝑖,ℱ_𝑖,1≤𝑖≤𝑛} are martingale differences and 𝑝>0, then there exists a constant 𝐾 depending only on 𝑝 such that 𝐸max_{1≤𝑘≤𝑛}^𝑘_𝑖=1𝑋_𝑖^𝑝≤𝐾𝐸^𝑛_𝑖=1𝐸𝑋²_𝑖∣ℱ_𝑖−1^𝑝/2+𝐸max_{1≤𝑖≤𝑛}𝑋_𝑖^𝑝,𝑛≥1. (1)

Lemma 3 (cf. Sung [12, Lemma 2.4]). Let {𝑋_𝑛,𝑛≥1} and {𝑌_𝑛,𝑛≥1} be sequences of random variables. Then for any 𝑛≥1, 𝑞>1, 𝜀>0, and 𝑎>0, one has 𝐸max_{1≤𝑗≤𝑛}^𝑗_𝑖=1(𝑋_𝑖+𝑌_𝑖)−𝜀𝑎⁺≤1𝜀^𝑞+11𝑞−1𝑎^𝑞−1𝐸max_{1≤𝑗≤𝑛}^𝑗_𝑖=1𝑋_𝑖^𝑞+𝐸max_{1≤𝑗≤𝑛}^𝑗_𝑖=1𝑌_𝑖. (2)

Lemma 4 (cf. Wang et al. [8, Lemma 2.2]). Let {𝑋_𝑛,𝑛≥1} be a sequence of random variables stochastically dominated by a nonnegative random variable 𝑋. Then for any 𝑛≥1, 𝑎>0, and 𝑏>0, the following two statements hold: 𝐸𝑋_𝑛^𝑎𝟙𝑋_𝑛≤𝑏≤𝐾₁𝐸𝑋^𝑎𝟙(𝑋≤𝑏)+𝑏^𝑎𝑃(,𝐸𝑋𝑋>𝑏)_𝑛𝑎𝟙𝑋_𝑛>𝑏≤𝐾₂𝐸𝑋^𝑎𝟙(𝑋>𝑏). (3) Consequently, 𝐸|𝑋_𝑛|^𝑎≤𝐾₃𝐸𝑋^𝑎. Here 𝐾₁, 𝐾₂, and 𝐾₃ 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 𝛼>1/2, 𝑝≥2, and {𝑋_𝑛,ℱ_𝑛,𝑛≥1} be martingale differences stochastically dominated by a nonnegative random variable 𝑋 with 𝐸𝑋^𝑝<∞. Let {𝑎_𝑛𝑖,1≤𝑖≤𝑛,𝑛≥1} be a triangular array of real numbers. For some 𝑞>2(𝛼𝑝−1)/(2𝛼−1), we assume that 𝐸[sup_𝑛≥1𝐸(𝑋²_𝑛∣ℱ_𝑛−1)]^𝑞/2<∞ and ^𝑛_𝑖=1𝑎_𝑛𝑖^𝑞=𝑂(𝑛). (4) Then for every 𝜀>0, ^∞_𝑛=1𝑛^{𝛼𝑝−2−𝛼}𝐸max_{1≤𝑘≤𝑛}^𝑘_𝑖=1𝑎_𝑛𝑖𝑋_𝑖−𝜀𝑛^𝛼+<∞. (5)

Taking 𝑝=2𝑙 and 𝛼=2/𝑝 for 1≤𝑙<2 in Theorem 5, we have the following result.

Corollary 6. Let 1≤𝑙<2, {𝑋_𝑛,ℱ_𝑛,𝑛≥1} be martingale differences stochastically dominated by a nonnegative random variable 𝑋 with 𝐸𝑋^2𝑙<∞. Let {𝑎_𝑛𝑖,1≤𝑖≤𝑛,𝑛≥1} be a triangular array of real numbers. For some 𝑞>2𝑙/(2−𝑙), one assumes that 𝐸[sup_𝑛≥1𝐸(𝑋²_𝑛∣ℱ_𝑛−1)]^𝑞/2<∞ and (4) holds true. Then for every 𝜀>0, ^∞_𝑛=1𝑛^−1/𝑙𝐸max_{1≤𝑘≤𝑛}^𝑘_𝑖=1𝑎_𝑛𝑖𝑋_𝑖−𝜀𝑛^1/𝑙+<∞. (6) In particular, one has lim_𝑛→∞1𝑛^𝑛1/𝑙_𝑖=1𝑎_𝑛𝑖𝑋_𝑖=0,𝑎.𝑠. (7)

Next, we investigate the convergence in mean square.

Theorem 7. Let 𝑟>1/2 and {𝑋_𝑛,ℱ_𝑛,𝑛≥1} be martingale differences stochastically dominated by a nonnegative random variable 𝑋 with 𝐸𝑋²<∞. Let {𝑎_𝑛𝑖,1≤𝑖≤𝑛,𝑛≥1} be a triangular array of real numbers and ^𝑛_𝑖=1𝑎²_𝑛𝑖=𝑂(𝑛). (8) Then, one has 𝑛^2𝑟−1𝐸1𝑛^𝑟𝑛_𝑖=1𝑎_𝑛𝑖𝑋_𝑖²≤𝐾,𝑛≥1, (9) where 𝐾 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 𝑋_𝑛𝑖=𝑋_𝑖𝟙(|𝑋_𝑖|≤𝑛^𝛼), 1≤𝑖≤𝑛. It can be found that 𝑎_𝑛𝑖𝑋_𝑖=𝑎_𝑛𝑖𝑋_𝑖𝟙(|𝑋_𝑖|>𝑛^𝛼)+[𝑎_𝑛𝑖𝑋_𝑛𝑖−𝑎_𝑛𝑖𝐸(𝑋_𝑛𝑖∣ℱ_𝑖−1)]+𝑎_𝑛𝑖𝐸(𝑋_𝑛𝑖∣ℱ_𝑖−1),1≤𝑖≤𝑛.
By Lemma 3 with 𝑎=𝑛^𝛼, for any 𝑞>1, we obtain that ^∞_𝑛=1𝑛^{𝛼𝑝−2−𝛼}𝐸max_{1≤𝑘≤𝑛}^𝑘_𝑖=1𝑎_𝑛𝑖𝑋_𝑖−𝜀𝑛^𝛼+≤𝐾₁^∞_𝑛=1𝑛^{𝛼𝑝−2−𝑞𝛼}×𝐸max_{1≤𝑘≤𝑛}^𝑘_𝑖=1𝑎_𝑛𝑖𝑋_𝑛𝑖−𝑎_𝑛𝑖𝐸𝑋_𝑛𝑖∣ℱ_𝑖−1^𝑞+^∞_𝑛=1𝑛^{𝛼𝑝−2−𝛼}𝐸max_{1≤𝑘≤𝑛}^𝑘_𝑖=1𝑎_𝑛𝑖𝑋_𝑖𝟙𝑋_𝑖>𝑛^𝛼+^∞_𝑛=1𝑛^{𝛼𝑝−2−𝛼}𝐸max_{1≤𝑘≤𝑛}^𝑘_𝑖=1𝑎_𝑛𝑖𝐸𝑋_𝑛𝑖∣ℱ_𝑖−1∶=𝐻₁+𝐻₂+𝐻₃. (10) For 𝑝≥2, it is easy to see that 𝑞>2(𝛼𝑝−1)/(2𝛼−1)≥2. Consequently, for any 1≤𝑠≤2, we get by Hölder’s inequality and (4) that ^𝑛_𝑖=1𝑎_𝑛𝑖^𝑠≤^𝑛_𝑖=1𝑎_𝑛𝑖^{𝑞𝑠/𝑞𝑛}_𝑖=11^{1−𝑠/𝑞}=𝑂(𝑛). (11) So, it can be checked by Markov’s inequality, Lemma 4, (11), and 𝐸𝑋^𝑝<∞ (𝑝≥2) that 𝐻₂≤^∞_𝑛=1𝑛^{𝑛𝛼𝑝−2−𝛼}_𝑖=1𝑎_𝑛𝑖𝐸𝑋_𝑖𝟙𝑋_𝑖>𝑛^𝛼≤𝐾₁^∞_𝑛=1𝑛^{𝛼𝑝−1−𝛼}𝐸𝑋𝟙𝑋>𝑛^𝛼=𝐾₁^∞_𝑛=1𝑛^{∞𝛼𝑝−1−𝛼}_𝑚=𝑛𝐸𝑋𝟙𝑚^𝛼<𝑋≤(𝑚+1)^𝛼=𝐾₁^∞_𝑚=1𝐸𝑋𝟙𝑚^𝛼<𝑋≤(𝑚+1)^𝛼𝑚_𝑛=1𝑛^{𝛼𝑝−1−𝛼}≤𝐾₂^∞_𝑚=1𝑚^{𝛼𝑝−𝛼}𝐸𝑋𝟙𝑚^𝛼<𝑋≤(𝑚+1)^𝛼≤𝐾₂𝐸𝑋^𝑝<∞. (12) Since {𝑋_𝑖,ℱ_𝑖,1≤𝑖≤𝑛} are martingale differences, by the martingale property and the proof of (12), one has that 𝐻₃=^∞_𝑛=1𝑛^{𝛼𝑝−2−𝛼}𝐸max_{1≤𝑘≤𝑛}^𝑘_𝑖=1𝑎_𝑛𝑖𝐸𝑋_𝑖𝟙𝑋_𝑖≤𝑛^𝛼∣ℱ_𝑖−1=^∞_𝑛=1𝑛^{𝛼𝑝−2−𝛼}𝐸max_{1≤𝑘≤𝑛}^𝑘_𝑖=1𝑎_𝑛𝑖𝐸𝑋_𝑖𝟙𝑋_𝑖>𝑛^𝛼∣ℱ_𝑖−1≤𝐾₁^∞_𝑛=1𝑛^{𝑛𝛼𝑝−2−𝛼}_𝑖=1𝑎_𝑛𝑖𝐸𝑋_𝑖𝟙𝑋_𝑖>𝑛^𝛼≤𝐾₂^∞_𝑛=1𝑛^{𝛼𝑝−1−𝛼}𝐸𝑋𝟙𝑋>𝑛^𝛼≤𝐾₃𝐸𝑋^𝑝<∞. (13)
Next, we turn to prove 𝐻₁<∞ under conditions of Theorem 5. It can be seen that 𝑎_𝑛𝑖𝑋_𝑛𝑖−𝑎_𝑛𝑖𝐸𝑋_𝑛𝑖∣ℱ_𝑖−1,ℱ_𝑖,1≤𝑖≤𝑛 (14) are also martingale differences. So, by Markov’s inequality, (10), and Lemma 2 with 𝑝=𝑞, it can be found that 𝐻₁=𝐾₁^∞_𝑛=1𝑛^{𝛼𝑝−2−𝑞𝛼}𝐸max_{1≤𝑘≤𝑛}^𝑘_𝑖=1𝑎_𝑛𝑖𝑋_𝑛𝑖−𝑎_𝑛𝑖𝐸×𝑋_𝑛𝑖∣ℱ_𝑖−1^𝑞≤𝐾₂^∞_𝑛=1𝑛^{𝛼𝑝−2−𝑞𝛼}𝐸^𝑛_𝑖=1𝐸𝑎_𝑛𝑖𝑋_𝑛𝑖−𝑎_𝑛𝑖𝐸×𝑋_𝑛𝑖∣ℱ_𝑖−1²∣ℱ_𝑖−1^𝑞/2+𝐾₃^∞_𝑛=1𝑛^{𝑛𝛼𝑝−2−𝑞𝛼}_𝑖=1𝐸𝑎_𝑛𝑖𝑋_𝑛𝑖−𝑎_𝑛𝑖𝐸𝑋_𝑛𝑖∣ℱ_𝑖−1^𝑞=∶𝐾₂𝐻₁₁+𝐾₃𝐻₁₂. (15) Obviously, it follows that 𝐸𝑎_𝑛𝑖𝑋_𝑛𝑖−𝐸𝑎_𝑛𝑖𝑋_𝑛𝑖∣ℱ_𝑖−1²∣ℱ_𝑖−1=𝐸𝑎²_𝑛𝑖𝑋²_𝑖𝟙𝑋_𝑖≤𝑛^𝛼∣ℱ_𝑖−1−𝐸𝑎_𝑛𝑖𝑋_𝑖𝟙𝑋_𝑖≤𝑛^𝛼∣ℱ_𝑖−1²≤𝑎²_𝑛𝑖𝐸𝑋²_𝑖𝟙𝑋_𝑖≤𝑛^𝛼∣ℱ_𝑖−1≤𝑎²_𝑛𝑖𝐸𝑋²_𝑖∣ℱ_𝑖−1,a.s.,1≤𝑖≤𝑛. (16) Combining (11) with 𝐸[sup_𝑖≥1𝐸(𝑋²_𝑖∣ℱ_𝑖−1)]^𝑞/2<∞, we obtain that 𝐻₁₁≤^∞_𝑛=1𝑛^{𝛼𝑝−2−𝑞𝛼𝑛}_𝑖=1𝑎²_𝑛𝑖^𝑞/2𝐸sup_𝑖≥1𝐸𝑋²_𝑖∣ℱ_𝑖−1^𝑞/2≤𝐾₄^∞_𝑛=1𝑛^{𝛼𝑝−2−𝑞𝛼+𝑞/2}<∞, (17) following from the fact that 𝑞>2(𝛼𝑝−1)/(2𝛼−1). Meanwhile, by 𝐶_𝑟 inequality, Lemma 4, and (4), 𝐻₁₂≤𝐾₅^∞_𝑛=1𝑛^{𝑛𝛼𝑝−2−𝑞𝛼}_𝑖=1𝑎_𝑛𝑖^𝑞𝐸𝑋_𝑖^𝑞𝟙𝑋_𝑖≤𝑛^𝛼≤𝐾₆^∞_𝑛=1𝑛^{𝛼𝑝−1−𝑞𝛼}𝐸𝑋^𝑞𝟙𝑋≤𝑛^𝛼+𝐾₇^∞_𝑛=1𝑛^𝛼𝑝−1𝑃𝑋>𝑛^𝛼≤𝐾₆^∞_𝑛=1𝑛^{𝛼𝑝−1−𝑞𝛼}𝐸𝑋^𝑞𝟙𝑋≤𝑛^𝛼+𝐾₇^∞_𝑛=1𝑛^{𝛼𝑝−1−𝛼}𝐸𝑋𝟙𝑋>𝑛^𝛼=∶𝐾₆𝐻^∗₁₁+𝐾₇𝐻^∗₁₂. (18) By the conditions 𝑝≥2 and 𝛼>1/2, we have that 2(𝛼𝑝−1)/(2𝛼−1)−𝑝≥0, which implies 𝑞>𝑝. So, we obtain by 𝐸𝑋^𝑝<∞ that 𝐻^∗₁₁=^∞_𝑛=1𝑛^{𝑛𝛼𝑝−1−𝑞𝛼}_𝑖=1𝐸𝑋^𝑞𝟙(𝑖−1)^𝛼<𝑋≤𝑖^𝛼=^∞_𝑖=1𝐸𝑋^𝑞𝟙(𝑖−1)^𝛼<𝑋≤𝑖^𝛼∞_𝑛=𝑖𝑛^{𝛼𝑝−1−𝑞𝛼}≤𝐾₈^∞_𝑖=1𝐸𝑋^𝑝𝑋^𝑞−𝑝𝟙(𝑖−1)^𝛼<𝑋≤𝑖^𝛼𝑖^{𝛼𝑝−𝑞𝛼}≤𝐾₈𝐸𝑋^𝑝<∞. (19) By the proof of (12), one has that 𝐻^∗₁₂=^∞_𝑛=1𝑛^{𝛼𝑝−1−𝛼}𝐸𝑋𝟙𝑋>𝑛^𝛼≤𝐾₉𝐸𝑋^𝑝<∞. (20) Thus, by (15)–(20), we have that 𝐻₁<∞. So, it completes the proof of (5).

Proof of Corollary 6. If 𝑝=2𝑙 and 𝛼=2/𝑝, then one has 𝛼𝑝=2. So as an application of Theorem 5, one gets (6) immediately. On the other hand, it can be seen that ^∞_𝑛=1𝑛^{𝛼𝑝−2−𝛼}𝐸max_{1≤𝑘≤𝑛}^𝑘_𝑖=1𝑎_𝑛𝑖𝑋_𝑖−𝜀𝑛^𝛼+=^∞_𝑛=1𝑛^{𝛼𝑝−2−𝛼∞}₀𝑃max_{1≤𝑘≤𝑛}^𝑘_𝑖=1𝑎_𝑛𝑖𝑋_𝑖−𝜀𝑛^𝛼≥>𝑡𝑑𝑡_∞𝑛=1𝑛^{𝛼𝑝−2−𝛼𝜀𝑛𝛼}₀𝑃max_{1≤𝑘≤𝑛}^𝑘_𝑖=1𝑎_𝑛𝑖𝑋_𝑖−𝜀𝑛^𝛼>𝑡𝑑𝑡≥𝜀^∞_𝑛=1𝑛^𝛼𝑝−2𝑃max_{1≤𝑘≤𝑛}^𝑘_𝑖=1𝑎_𝑛𝑖𝑋_𝑖>2𝜀𝑛^𝛼. (21) So by (5) and (21) with 𝛼𝑝=2, we have for every 𝜀>0 that ^∞_𝑛=1𝑃max_{1≤𝑘≤𝑛}^𝑘_𝑖=1𝑎_𝑛𝑖𝑋_𝑖>𝜀𝑛^1/𝑙<∞. (22) It follows from Borel-Cantelli lemma that lim_𝑛→∞1𝑛^𝑛1/𝑙_𝑖=1𝑎_𝑛𝑖𝑋_𝑖=0,a.s. (23) So, (7) holds.

Proof of Theorem 7. Since {𝑎_𝑛𝑖𝑋_𝑖,ℱ_𝑖,1≤𝑖≤𝑛} are martingale differences, it can be found by Lemmas 2 and 4 and (8) that 𝐸1𝑛^𝑟𝑛_𝑖=1𝑎_𝑛𝑖𝑋_𝑖²=1𝑛^2𝑟𝐸^𝑛_𝑖=1𝑎_𝑛𝑖𝑋_𝑖²≤𝐾₁𝑛^𝑛2𝑟_𝑖=1𝑎²_𝑛𝑖𝐸𝑋²_𝑖≤𝐾₂𝑛^2𝑟𝐸𝑋^2𝑛_𝑖=1𝑎²_𝑛𝑖≤𝐾₃𝑛^2𝑟−1,𝑛≥1. (24) 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 𝑡≥0, consider an MISO (multi-input-single-output) linear-time-invariant system 𝑥̇(𝑡)(𝑡)(𝑡),𝑦(𝑡)(𝑡),=𝐴𝑥+𝐵𝑢=𝐶𝑥 (25) where 𝐴∈𝑅^𝑚0×𝑚0, 𝐵∈𝑅^𝑚0×𝑚1, and 𝐶∈𝑅^1×𝑚0 are known system matrices, and for 𝑡≥0, 𝑢(𝑡)∈𝑅^𝑚1 is the control input, 𝑥(𝑡)∈𝑅^𝑚0 is the state, and 𝑦(𝑡)∈𝑅 is the system output. The initial state 𝑥(0) is unknown. We are interested in estimation of 𝑥(𝑡), from some limited observations on 𝑦(𝑡).

In our setup, the output 𝑦(𝑡) is only measured at a sequence of sampling time instants {𝑡_𝑖} with measured values 𝛾(𝑡_𝑖), and noise 𝑑_𝑖𝛾𝑡_𝑖=𝑦𝑡_𝑖−𝑑_𝑖. (26)

We would like to estimate the state 𝑥(𝑡) from information on 𝑢(𝑡), {𝑡_𝑖}, and {𝛾(𝑡_𝑖)}. In practical systems, the irregular sampling sequences {𝛾(𝑡_𝑖)} 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, 𝑑_𝑘 is assumed to be martingale difference. We give the following assumption.

Assumption 9. The system (25) is observable; that is, the observability matrix 𝑊^′_𝑜=𝐶^′,(𝐶𝐴)^′,…,𝐶𝐴^𝑚0−1′ (27) has full rank.

For both 𝑡>𝑡₀ and 𝑡<𝑡₀, the solution to system (25) can be expressed as 𝑥(𝑡)=𝑒^{𝐴(𝑡−𝑡0)}𝑥𝑡₀+^𝑡_𝑡0𝑒^{𝐴(𝑡−𝜏)}𝐵𝑢(𝜏)𝑑𝜏. (28)

Suppose that {𝑡_𝑖,1≤𝑖≤𝑛} is a sequence of sampling times. For 𝑡_𝑖≤𝑡_𝑛, we have 𝛾𝑡_𝑖+𝑑_𝑖=𝑦𝑡_𝑖=𝐶𝑒^{𝐴(𝑡𝑖−𝑡𝑛)}𝑥𝑡_𝑛+𝐶^𝑡𝑖_𝑡𝑛𝑒^{𝐴(𝑡𝑖−𝜏)}𝐵𝑢(𝜏)𝑑𝜏. (29)

Since the second term is known, it will be denoted by 𝑣(𝑡_𝑖,𝑡_𝑛)=𝐶∫^𝑡𝑖_𝑡𝑛𝑒^{𝐴(𝑡𝑖−𝜏)}𝐵𝑢(𝜏)𝑑𝜏. This leads to the observations 𝐶𝑒^{𝐴(𝑡𝑖−𝑡𝑛)}𝑥𝑡_𝑛=𝛾𝑡_𝑖−𝑣𝑡_𝑖,𝑡_𝑛+𝑑_𝑖,1≤𝑖≤𝑛. (30)

Define Φ_𝑛=𝐶𝑒^{𝐴(𝑡1−𝑡𝑛)}⋮𝐶𝑒^{𝐴(𝑡𝑛−1−𝑡𝑛)}𝐶,Γ_𝑛=𝛾𝑡₁⋮𝛾𝑡_𝑛−1𝛾𝑡_𝑛,𝑉_𝑛=𝑣𝑡₁,𝑡_𝑛⋮𝑣𝑡_𝑛−1,𝑡_𝑛0,𝐷_𝑛=𝑑₁⋮𝑑_𝑛−1𝑑_𝑛. (31)

Then, (30) can be written as Φ_𝑛𝑥𝑡_𝑛=Γ_𝑛−𝑉_𝑛+𝐷_𝑛. (32)

Suppose that Φ_𝑛 is full rank, which will be established later. Then, a least-squares estimate of 𝑥(𝑡_𝑛) is given by 𝑥𝑡_𝑛=Φ^′_𝑛Φ_𝑛⁻¹Φ^′_𝑛Γ_𝑛−𝑉_𝑛. (33) Here, 𝐺^′ denotes the transpose of 𝐺. From (32) and (33), the estimation error for 𝑥(𝑡_𝑛) at sampling time 𝑡_𝑛 is 𝑒𝑡_𝑛=𝑥𝑡_𝑛−𝑥𝑡_𝑛=Φ^′_𝑛Φ_𝑛⁻¹Φ^′_𝑛𝐷_𝑛=1𝑛^𝑟Φ^′_𝑛Φ_𝑛⁻¹1𝑛^𝑟Φ^′_𝑛𝐷_𝑛 (34) for some 1/2<𝑟<1. For convergence analysis, one must consider a typical entry in (1/𝑛^𝑟)Φ^′_𝑛𝐷_𝑛. By the Cayley Hamilton theorem (see Ogata [18]), the matrix exponential can be expressed by a polynomial function of 𝐴 of order at most 𝑚₀−1, 𝑒^𝐴𝑡=𝛼₁(𝑡)𝐼+⋯+𝛼_𝑚0(𝑡)𝐴^𝑚0−1, (35) where the time functions 𝛼_𝑖(𝑡) can be derived by the Lagrange-Hermite interpolation method (see Ogata [18]). This implies that 𝐶𝑒^{𝐴(𝑡𝑖−𝑡𝑛)}=𝛼₁𝑡_𝑖−𝑡_𝑛,…,𝛼_𝑚0𝑡_𝑖−𝑡_𝑛𝐶⋮𝐶𝐴𝐶𝐴^𝑚0−1=𝜑^′𝑡_𝑖−𝑡_𝑛𝑊_𝑜, (36) where 𝜑^′(𝑡_𝑖−𝑡_𝑛)=[𝛼₁(𝑡_𝑖−𝑡_𝑛),…,𝛼_𝑚0(𝑡_𝑖−𝑡_𝑛)] and 𝑊_𝑜 is the observability matrix.

Denote Ψ_𝑛=𝜑^′𝑡₁−𝑡_𝑛⋮𝜑^′(0). (37) Then Φ_𝑛=Ψ_𝑛𝑊_𝑜, (38) which implies that 1𝑛^𝑟Φ^′_𝑛Φ_𝑛=𝑊^′_𝑜1𝑛^𝑟Ψ^′_𝑛Ψ_𝑛𝑊_𝑜,1𝑛^𝑟Φ^′_𝑛𝐷_𝑛=1𝑛^𝑟𝑊^′_𝑜Ψ^′_𝑛𝐷_𝑛. (39)

As a result, for any 𝑟>0, one has 𝑒𝑡_𝑛=1𝑛^𝑟Φ^′_𝑛Φ_𝑛⁻¹1𝑛^𝑟Φ^′_𝑛𝐷_𝑛=𝑊_𝑜⁻¹1𝑛^𝑟Ψ^′_𝑛Ψ_𝑛⁻¹1𝑛^𝑟Ψ^′_𝑛𝐷_𝑛. (40)

Under Assumption 9, 𝑊₀⁻¹ exists. Convergence results will be established by the following two sufficient conditions: (1/𝑛^𝑟)Ψ^′_𝑛𝐷_𝑛→0 and (1/𝑛^𝑟)Ψ^′_𝑛Ψ_𝑛≥𝛽𝐼, for some 𝛽>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<𝑟<1, 𝛽=inf_𝑛≥1𝜎_min1𝑛^𝑟Ψ^′_𝑛Ψ_𝑛>0, (41) where 𝜎_min(𝐻) is the small eigenvalue of 𝐻 for a suitable symmetric 𝐻.

We can investigate the convergence of double-indexed summations of random variables form 1𝑛^𝑟𝑛_𝑖=1𝑎_𝑛𝑖𝑑_𝑖 (42) for some 1/2<𝑟<1. Here, {𝑎_𝑛𝑖} is a triangular array of real numbers and {𝑑_𝑖} is a sequence of martingale differences. It can be seen that (42) is a special case of (7) in Corollary 6. The 𝑗th component of (1/𝑛^𝑟)Ψ^′_𝑛𝐷_𝑛 takes the form 1𝑛^𝑟𝑛_𝑖=1𝛼_𝑗𝑡_𝑖−𝑡_𝑛𝑑_𝑖, (43) where {𝛼_𝑗(𝑡_𝑖−𝑡_𝑛)} is a triangular array of real numbers. The convergence analysis of (43) for 𝑒(𝑡_𝑛) 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 𝜌^∗-mixing sampling. Thanh et al. [20] studied the convergence analysis of double-indexed and randomly weighted sums of 𝜌^∗-mixing sequence and gave its application to state observers. For more related works, one can refer to [18–23] 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 1/2<𝑟<1 and {𝑑_𝑛,ℱ_𝑛,𝑛≥1} be martingale differences stochastically dominated by a nonnegative random variable 𝑑 with 𝐸𝑑^2/𝑟<∞. Suppose that for any 𝑞>2/(2𝑟−1), one has 𝐸[sup_𝑛≥1𝐸(𝑑²_𝑛∣ℱ_𝑛−1)]^𝑞/2<∞ and ^𝑛_𝑖=1𝛼_𝑗(𝑡_𝑖−𝑡_𝑛)^𝑞=𝑂(𝑛), (44) where 1≤𝑗≤𝑚₀. Then 1𝑛^𝑟Ψ^′_𝑛𝐷_𝑛⟶0,𝑎.𝑠. (45) Consequently, 𝑒𝑡_𝑛⟶0,𝑎.𝑠. (46)

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

Theorem 12. Let 1/2<𝑟<1 and Assumptions 9 and 10 hold. Assume that {𝑑_𝑛,ℱ_𝑛,𝑛≥1} are martingales differences stochastically dominated by a nonnegative random variable 𝑑 with 𝐸𝑑²<∞. For 1≤𝑗≤𝑚₀, it is supposed that ^𝑛_𝑖=1𝛼²_𝑗𝑡_𝑖−𝑡_𝑛=𝑂(𝑛). (47) Then 𝜁=sup_𝑛≥1𝑛^2𝑟−1𝐸𝑒^′𝑡_𝑛𝑒𝑡_𝑛<∞. (48)

Remark 13. If we assume that, for each 1≤𝑖≤𝑛, {𝜑(𝑡_𝑖−𝑡_𝑛)} is uniformly bounded, then we can find that condition (44) holds for any 𝑞. 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 𝜌^∗-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 1𝑛^𝑟Ψ^′_𝑛𝐷_𝑛=1𝑛^𝑟𝑛_𝑖=1𝛼₁𝑡_𝑖−𝑡_𝑛𝑑_𝑖⋮1𝑛^𝑟𝑛_𝑖=1𝛼_𝑚0𝑡_𝑖−𝑡_𝑛𝑑_𝑖. (49) To prove (45), it suffices to look at the 𝑗th component 1𝑛^𝑟𝑛_𝑖=1𝛼_𝑗𝑡_𝑖−𝑡_𝑛𝑑_𝑖 (50) of 1𝑛^𝑟Ψ^′_𝑛𝐷_𝑛. (51) For any 𝑞>2/(2𝑟−1), by 𝐸[sup_𝑛≥1𝐸(𝑑²_𝑛∣ℱ_𝑛−1)]^𝑞/2<∞ and (44), we can obtain (45) from Corollary 6 with 𝑙=1/𝑟, 𝑎_𝑛𝑖=𝛼_𝑗(𝑡_𝑖−𝑡_𝑛) in (43), and 𝑋_𝑛=𝑑_𝑛.
On the other hand, by Assumption 9, 𝑊₀⁻¹ exists, and by (41) in Assumption 10, ((1/𝑛^𝑟)Ψ^′_𝑛Ψ_𝑛)⁻¹ exists and 𝜎_max1𝑛^𝑟Ψ^′_𝑛Ψ_𝑛⁻¹≤1𝛽, (52) where 𝜎_max(⋅) is the largest eigenvalue. Together with 𝑒𝑡_𝑛=𝑊_𝑜⁻¹1𝑛^𝑟Ψ^′_𝑛Ψ_𝑛⁻¹1𝑛^𝑟Ψ^′_𝑛𝐷_𝑛 (53) and (45), it follows (46).

Proof of Theorem 12. For 1≤𝑗≤𝑚₀, by (47), (8) holds. Applying Theorem 7 with 𝑎_𝑛𝑖=𝛼_𝑗(𝑡_𝑖−𝑡_𝑛), 𝑋_𝑛=𝑑_𝑛, and 1/2<𝑟<1, we obtain that for a typical term 1𝑛^𝑟𝑛_𝑖=1𝛼_𝑗𝑡_𝑖−𝑡_𝑛𝑑_𝑖 (54) in (49), 𝑛^2𝑟−1𝐸1𝑛^𝑟𝑛_𝑖=1𝛼_𝑗(𝑡_𝑖−𝑡_𝑛)𝑑_𝑖²≤𝐾₁,𝑛≥1. (55) Together with (49), (53), and (55), we obtain that 𝑛^2𝑟−1𝐸𝑒^′𝑡_𝑛𝑒𝑡_𝑛≤𝑚₀𝐾₂<∞, (56) where 𝐾₂ 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.