AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 893906 10.1155/2014/893906 893906 Research Article The Convergence of Double-Indexed Weighted Sums of Martingale Differences and Its Application Yang Wenzhi http://orcid.org/0000-0002-1520-2161 Wang Xinghui Li Xiaoqin Hu Shuhe Ivanov Ivan School of Mathematical Science Anhui University Hefei 230039 China ahu.edu.cn 2014 2432014 2014 23 01 2014 19 02 2014 24 3 2014 2014 Copyright © 2014 Wenzhi Yang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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  introduced the concept of complete convergence; that is, a sequence of random variables {Xn,n1} is said to converge completely to a constant C if n=1P(|Xn-C|ε)< for all ε>0. By Borel-Cantelli lemma, it follows that XnC almost surely (a.s.). The converse is true if {Xn,n1} is independent. But the converse cannot always be true for the dependent case. Hsu and Robbins  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  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  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 {Xn,n1} be a sequence of i.i.d. random variables. Assume that EX1=0 if α1. Then the following statements are equivalent:

E|X1|p<;

n=1nαp-2P(max1kn|i=1kXi|>εnα)< for all ε>0.

Many authors have extended Theorem 1 to the martingale differences. For example, Yu  obtained the complete convergence for weighted sums of martingale differences; Ghosal and Chandra  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.  also studied the complete convergence and complete moment convergence for martingale differences, which generalized some results of Stoica [6, 7]; Yang et al.  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 , Chen et al. , Sung , Sung and Volodin , Hu et al. , 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 {Xn,n1} is stochastically dominated by a nonnegative random variable X if supn1P(|Xn|>t)KP(X>t) for some positive constant K and for all t0.

Throughout the paper, let 0={,Ω}, 1(B) be the indicator function of set Bx+=x1(x0), 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 [<xref ref-type="bibr" rid="B17">17</xref>, Theorem 2.11]).

If {Xi,i,1in} are martingale differences and p>0, then there exists a constant K depending only on p such that (1)E(max1kn|i=1kXi|p)K{E(i=1nE(Xi2i-1))p/2+E(max1in|Xi|p)},mmmmmmmmmmmmmmmmmmmmmmmmn1.

Lemma 3 (cf. Sung [<xref ref-type="bibr" rid="B12">12</xref>, Lemma 2.4]).

Let {Xn,n1} and {Yn,n1} be sequences of random variables. Then for any n1, q>1, ε>0, and a>0, one has (2)E(max1jn|i=1j(Xi+Yi)|-εa)+(1εq+1q-1)1aq-1E(max1jn|i=1jXi|q)+E(max1jn|i=1jYi|).

Lemma 4 (cf. Wang et al. [<xref ref-type="bibr" rid="B8">8</xref>, Lemma 2.2]).

Let {Xn,n1} be a sequence of random variables stochastically dominated by a nonnegative random variable X. Then for any n1, a>0, and b>0, the following two statements hold: (3)E[|Xn|a1(|Xn|b)]K1{E[Xa1(Xb)]+baP(X>b)},E[|Xn|a1(|Xn|>b)]K2E[Xa1(X>b)]. Consequently, E|Xn|aK3EXa. 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, p2, and {Xn,n,n1} be martingale differences stochastically dominated by a nonnegative random variable X with EXp<. Let {ani,1in,n1} be a triangular array of real numbers. For some q>2(αp-1)/(2α-1), we assume that E[supn1E(Xn2n-1)]q/2< and (4)i=1n|ani|q=O(n). Then for every ε>0, (5)n=1nαp-2-αE(max1kn|i=1kaniXi|-εnα)+<.

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

Corollary 6.

Let 1l<2, {Xn,n,n1} be martingale differences stochastically dominated by a nonnegative random variable X with EX2l<. Let {ani,1in,n1} be a triangular array of real numbers. For some q>2l/(2-l), one assumes that E[supn1E(Xn2n-1)]q/2< and (4) holds true. Then for every ε>0, (6)n=1n-1/lE(max1kn|i=1kaniXi|-εn1/l)+<. In particular, one has (7)limn1n1/li=1naniXi=0,a.s.

Next, we investigate the convergence in mean square.

Theorem 7.

Let r>1/2 and {Xn,n,n1} be martingale differences stochastically dominated by a nonnegative random variable X with EX2<. Let {ani,1in,n1} be a triangular array of real numbers and (8)i=1nani2=O(n). Then, one has (9)n2r-1E(1nri=1naniXi)2K,n1, where K is a positive constant.

Remark 8.

Wang et al.  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.  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 <xref ref-type="statement" rid="thm2.1">5</xref>.

Let Xni=Xi1(|Xi|nα), 1in. It can be found that aniXi=aniXi1(|Xi|>nα)+[aniXni-aniE(Xnii-1)]+aniE(Xnii-1),1in.

By Lemma 3 with a=nα, for any q>1, we obtain that (10)n=1nαp-2-αE(max1kn|i=1kaniXi|-εnα)+K1n=1nαp-2-qα×E(max1kn|i=1k[aniXni-aniE(Xnii-1)]|q)+n=1nαp-2-αE(max1kn|i=1kaniXi1(X|Xi|>nα)|)+n=1nαp-2-αE(max1kn|i=1kaniE(Xnii-1)|):=H1+H2+H3. For p2, it is easy to see that q>2(αp-1)/(2α-1)2. Consequently, for any 1s2, we get by Hölder’s inequality and (4) that (11)i=1n|ani|s(i=1n|ani|q)s/q(i=1n1)1-s/q=O(n). So, it can be checked by Markov’s inequality, Lemma 4, (11), and EXp< (p2) that (12)H2n=1nαp-2-αi=1n|ani|E[|Xi|1(|Xi|>nα)]K1n=1nαp-1-αE[X1(X>nα)]=K1n=1nαp-1-αm=nE[X1(mα<X(m+1)α)]=K1m=1E[X1(mα<X(m+1)α)]n=1mnαp-1-αK2m=1mαp-αE[X1(mα<X(m+1)α)]K2EXp<. Since {Xi,i,1in} are martingale differences, by the martingale property and the proof of (12), one has that (13)H3=n=1nαp-2-αE(max1kn|i=1kaniE[Xi1(|Xi|nα)i-1]|)=n=1nαp-2-αE(max1kn|i=1kaniE[Xi1(|Xi|>nα)i-1]|)K1n=1nαp-2-αi=1n|ani|E[|Xi|1(|Xi|>nα)]K2n=1nαp-1-αE[X1(X>nα)]K3EXp<.

Next, we turn to prove H1< under conditions of Theorem 5. It can be seen that (14){[aniXni-aniE(Xnii-1)],i,1in} are also martingale differences. So, by Markov’s inequality, (10), and Lemma 2 with p=q, it can be found that (15)H1=K1n=1nαp-2-qαE(max1kn|i=1k[aniXni-aniEVVVVVVVVVvvvvvvvvvvvvvV×(Xnii-1)]i=1k|qi=1k)K2n=1nαp-2-qαE(i=1nE{[aniXni-aniEVsssssssssssssssssssvvvvvsss×(Xnii-1)]2i-1}i=1n)q/2+K3n=1nαp-2-qαi=1nE|aniXni-aniE(Xnii-1)|q=:K2H11+K3H12. Obviously, it follows that (16)E{[aniXni-E(aniXnii-1)]2i-1}=E[ani2Xi21(|Xi|nα)i-1]-[E(aniXi1(|Xi|nα)i-1)]2ani2E[Xi21(|Xi|nα)i-1]ani2E(Xi2i-1),a.s.,1in. Combining (11) with E[supi1E(Xi2i-1)]q/2<, we obtain that (17)H11n=1nαp-2-qα(i=1nani2)q/2E(supi1supi1E(Xi2i-1))q/2K4n=1nαp-2-qα+q/2<, following from the fact that q>2(αp-1)/(2α-1). Meanwhile, by Cr inequality, Lemma 4, and (4), (18)H12K5n=1nαp-2-qαi=1n|ani|qE[|Xi|q1(|Xi|nα)]K6n=1nαp-1-qαE[Xq1(Xnα)]+K7n=1nαp-1P(X>nα)K6n=1nαp-1-qαE[Xq1(Xnα)]+K7n=1nαp-1-αE[X1(X>nα)]=:K6H11*+K7H12*. By the conditions p2 and α>1/2, we have that 2(αp-1)/(2α-1)-p0, which implies q>p. So, we obtain by EXp< that (19)H11*=n=1nαp-1-qαi=1nE[Xq1((i-1)α<Xiα)]=i=1E[Xq1((i-1)α<XiαX)]n=inαp-1-qαK8i=1E[XpXq-p1((i-1)α<Xiα)]iαp-qαK8EXp<. By the proof of (12), one has that (20)H12*=n=1nαp-1-αE[X1(X>nα)]K9EXp<. Thus, by (15)–(20), we have that H1<. So, it completes the proof of (5).

Proof of Corollary <xref ref-type="statement" rid="coro2.1">6</xref>.

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 (21)n=1nαp-2-αE(max1kn|i=1kaniXi|-εnα)+=n=1nαp-2-α0P(max1kn|i=1kaniXi|-εnα>t)dtn=1nαp-2-α0εnαP(max1kn|i=1kaniXi|-εnα>t)dtεn=1nαp-2P(max1kn|i=1kaniXi|>2εnα). So by (5) and (21) with αp=2, we have for every ε>0 that (22)n=1P(max1kn|i=1kaniXi|>εn1/l)<. It follows from Borel-Cantelli lemma that (23)limn1n1/li=1naniXi=0,a.s. So, (7) holds.

Proof of Theorem <xref ref-type="statement" rid="thm2.2">7</xref>.

Since {aniXi,i,1in} are martingale differences, it can be found by Lemmas 2 and 4 and (8) that (24)E(1nri=1naniXi)2=1n2rE(i=1naniXi)2K1n2ri=1nani2EXi2K2n2rEX2i=1nani2K3n2r-1,n1. 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 t0, consider an MISO (multi-input-single-output) linear-time-invariant system (25)x˙(t)=Ax(t)+Bu(t),y(t)=Cx(t), where ARm0×m0, BRm0×m1, and CR1×m0 are known system matrices, and for t0, u(t)Rm1 is the control input, x(t)Rm0 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 {ti} with measured values γ(ti), and noise di(26)γ(ti)=y(ti)-di.

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

Assumption 9.

The system (25) is observable; that is, the observability matrix (27)Wo=[C,(CA),,(CAm0-1)] has full rank.

For both t>t0 and t<t0, the solution to system (25) can be expressed as (28)x(t)=eA(t-t0)x(t0)+t0teA(t-τ)Bu(τ)dτ.

Suppose that {ti,1in} is a sequence of sampling times. For titn, we have (29)γ(ti)+di=y(ti)=CeA(ti-tn)x(tn)+CtntieA(ti-τ)Bu(τ)dτ.

Since the second term is known, it will be denoted by v(ti,tn)=CtntieA(ti-τ)Bu(τ)dτ. This leads to the observations (30)CeA(ti-tn)x(tn)=γ(ti)-v(ti,tn)+di,1in.

Define (31)Φn=[CeA(t1-tn)CeA(tn-1-tn)C],Γn=[γ(t1)γ(tn-1)γ(tn)],Vn=[v(t1,tn)v(tn-1,tn)0],Dn=[d1dn-1dn].

Then, (30) can be written as (32)Φnx(tn)=Γn-Vn+Dn.

Suppose that Φn is full rank, which will be established later. Then, a least-squares estimate of x(tn) is given by (33)x^(tn)=(ΦnΦn)-1Φn(Γn-Vn). Here, G denotes the transpose of G. From (32) and (33), the estimation error for x(tn) at sampling time tn is (34)e(tn)=x^(tn)-x(tn)=(ΦnΦn)-1ΦnDn=(1nrΦnΦn)-11nrΦnDn for some 1/2<r<1. For convergence analysis, one must consider a typical entry in (1/nr)ΦnDn. By the Cayley Hamilton theorem (see Ogata ), the matrix exponential can be expressed by a polynomial function of A of order at most m0-1, (35)eAt=α1(t)I++αm0(t)Am0-1, where the time functions αi(t) can be derived by the Lagrange-Hermite interpolation method (see Ogata ). This implies that (36)CeA(ti-tn)=[α1(ti-tn),,αm0(ti-tn)][CCACAm0-1]=φ(ti-tn)Wo, where φ(ti-tn)=[α1(ti-tn),,αm0(ti-tn)] and Wo is the observability matrix.

Denote (37)Ψn=[φ(t1-tn)φ(0)]. Then (38)Φn=ΨnWo, which implies that (39)1nrΦnΦn=Wo1nrΨnΨnWo,1nrΦnDn=1nrWoΨnDn.

As a result, for any r>0, one has (40)e(tn)=(1nrΦnΦn)-11nrΦnDn=Wo-1(1nrΨnΨn)-11nrΨnDn.

Under Assumption 9, W0-1 exists. Convergence results will be established by the following two sufficient conditions: (1/nr)ΨnDn0 and (1/nr)ΨnΨnβI, for some β>0. So we need the following persistent excitation (PE) condition, which was used by Wang et al.  and Thanh et al. .

Assumption 10.

For some 1/2<r<1, (41)β=infn1σmin(1nrΨnΨn)>0, where σmin(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 (42)1nri=1nanidi for some 1/2<r<1. Here, {ani} is a triangular array of real numbers and {di} 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/nr)ΨnDn takes the form (43)1nri=1nαj(ti-tn)di, where {αj(ti-tn)} is a triangular array of real numbers. The convergence analysis of (43) for e(tn) is a special case of (42) or (7) in Corollary 6.

Recently, Wang et al.  investigated the convergence analysis of the state observers of linear-time-invariant systems under ρ*-mixing sampling. Thanh et al.  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  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 {dn,n,n1} be martingale differences stochastically dominated by a nonnegative random variable d with Ed2/r<. Suppose that for any q>2/(2r-1), one has E[supn1E(dn2n-1)]q/2< and (44)i=1n|αj(ti-tn)|q=O(n), where 1jm0. Then (45)1nr||ΨnDn||0,a.s. Consequently, (46)e(tn)0,a.s.

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 {dn,n,n1} are martingales differences stochastically dominated by a nonnegative random variable d with Ed2<. For 1jm0, it is supposed that (47)i=1nαj2(ti-tn)=O(n). Then (48)ζ=supn1n2r-1Ee(tn)e(tn)<.

Remark 13.

If we assume that, for each 1in, {φ(ti-tn)} 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.  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. ). So Theorems 11 and 12 generalize the results of Wang et al.  to the case of martingale differences.

Proof of Theorem <xref ref-type="statement" rid="thm3.1">11</xref>.

It can be seen that (49)1nrΨnDn=[  1nri=1nα1(ti-tn)di        1nri=1nαm0(ti-tn)di]. To prove (45), it suffices to look at the jth component (50)1nri=1nαj(ti-tn)di of (51)1nrΨnDn. For any q>2/(2r-1), by E[supn1E(dn2n-1)]q/2< and (44), we can obtain (45) from Corollary 6 with l=1/r, ani=αj(ti-tn) in (43), and Xn=dn.

On the other hand, by Assumption 9, W0-1 exists, and by (41) in Assumption 10, ((1/nr)ΨnΨn)-1 exists and (52)σmax((1nrΨnΨn)-1)1β, where σmax(·) is the largest eigenvalue. Together with (53)e(tn)=Wo-1(1nrΨnΨn)-11nrΨnDn and (45), it follows (46).

Proof of Theorem <xref ref-type="statement" rid="thm3.2">12</xref>.

For 1jm0, by (47), (8) holds. Applying Theorem 7 with ani=αj(ti-tn), Xn=dn, and 1/2<r<1, we obtain that for a typical term (54)1nri=1nαj(ti-tn)di in (49), (55)n2r-1E(1nri=1nαj(ti-tn)di)2K1,n1. Together with (49), (53), and (55), we obtain that (56)n2r-1Ee(tn)e(tn)m0K2<, 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.

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