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

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Wenzhi Yang ◽  
Xinghui Wang ◽  
Xiaoqin Li ◽  
Shuhe Hu
Keyword(s):  
Linear Time ◽  
Weighted Sums ◽  
Complete Moment Convergence ◽  
Mean Square ◽  
Martingale Differences ◽  
Strong Law ◽  
Linear Time Invariant ◽  
Moment Convergence ◽  
State Observers ◽  
Linear Time Invariant Systems

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.

Complete moment convergence for the dependent linear processes with application to the state observers of linear-time-invariant systems

2020 ◽  
Vol 65 (4) ◽  
pp. 725-745
Author(s):  
Chao Lu ◽  
Chao Lu ◽  
Xuejun J Wang ◽  
Xuejun J Wang ◽  
Yi Wu ◽  
...  
Keyword(s):  
Linear Time ◽  
The State ◽  
Complete Moment Convergence ◽  
Linear Processes ◽  
Linear Time Invariant ◽  
Moment Convergence ◽  
State Observers ◽  
Linear Time Invariant Systems ◽  
Time Invariant

Пусть $X_t=\sum_{j=-\infty}^{\infty}A_j\varepsilon_{t-j}$ - зависимый линейный процесс, где $\{\varepsilon_n, n\in \mathbf{Z}\}$ - последовательность $m$-обобщенных отрицательно зависимых ($m$-END) случайных величин с нулевым средним, которая стохастически доминируется случайной величиной $\varepsilon$, и пусть $\{A_n, n\in \mathbf{Z}\}$ - другая последовательность случайных величин с нулевым средним, обладающая свойством $m$-END. При подходящих условиях установлена полная моментная сходимость для зависимых линейных процессов. В частности, приведены достаточные условия полной моментной сходимости. В качестве приложения исследуется сходимость наблюдателей состояния для линейных стационарных систем.

scholarly journals Complete Moment Convergence for Sung’s Type Weighted Sums ofB-Valued Random Elements

2016 ◽  
Vol 2016 ◽  
pp. 1-8
Author(s):  
Wei Li ◽  
Pingyan Chen ◽  
Soo Hak Sung
Keyword(s):  
Law Of Large Numbers ◽  
Sufficient Conditions ◽  
Weighted Sums ◽  
Complete Moment Convergence ◽  
Real Numbers ◽  
Strong Law ◽  
Moment Convergence ◽  
Large Numbers ◽  
Random Elements ◽  
Necessary And Sufficient

Letp≥1/αand1/2<α≤1.Let{X,Xn,  n≥1}be a sequence of independent and identically distributedB-valued random elements and let{ani,  1≤i≤n,  n≥1}be an array of real numbers satisfying∑i=1naniq=O(n)for someq>p.We give necessary and sufficient conditions for complete moment convergence of the form∑n=1∞n(p-v)α-2E∑i=1naniXi-εnα+v<∞,  ∀ε>0, where0<v<p.A strong law of large numbers for weighted sums of independentB-valued random elements is also obtained.

scholarly journals Complete moment convergence for weighted sums of negatively orthant dependent random variables

Filomat ◽  
2017 ◽  
Vol 31 (5) ◽  
pp. 1195-1206 ◽  
Author(s):  
Xuejun Wang ◽  
Zhiyong Chen ◽  
Ru Xiao ◽  
Xiujuan Xie
Keyword(s):  
Complete Convergence ◽  
Law Of Large Numbers ◽  
Random Variables ◽  
Independent Random Variables ◽  
Weighted Sums ◽  
Complete Moment Convergence ◽  
Strong Law ◽  
Associated Random Variables ◽  
Moment Convergence ◽  
Large Numbers

In this paper, the complete moment convergence and the integrability of the supremum for weighted sums of negatively orthant dependent (NOD, in short) random variables are presented. As applications, the complete convergence and the Marcinkiewicz-Zygmund type strong law of large numbers for NODrandom variables are obtained. The results established in the paper generalize some corresponding ones for independent random variables and negatively associated random variables.

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