On Expected Figures and a Strong Law of Large Numbers for Random Elements in Quasi-Metric Spaces

1977 ◽  
pp. 591-602 ◽  
Author(s):  
Herbert Ziezold
Keyword(s):  
Law Of Large Numbers ◽  
Metric Spaces ◽  
Strong Law ◽  
Large Numbers ◽  
Random Elements

A REMARK ON THE STRONG LAW FOR B-VALUED ARRAYS OF RANDOM ELEMENTS

2010 ◽  
Vol 82 (1) ◽  
pp. 31-43 ◽  
Author(s):  
TIEN-CHUNG HU ◽  
PING YAN CHEN ◽  
N. C. WEBER
Keyword(s):  
Law Of Large Numbers ◽  
Moving Average ◽  
Average Process ◽  
Moving Average Process ◽  
Strong Law ◽  
Rate Law ◽  
Large Numbers ◽  
Random Elements

AbstractThe conditions in the strong law of large numbers given by Li et al. [‘A strong law for B-valued arrays’, Proc. Amer. Math. Soc.123 (1995), 3205–3212] for B-valued arrays are relaxed. Further, the compact logarithm rate law and the bounded logarithm rate law are discussed for the moving average process based on an array of random elements.

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.

Sample Behavior and Laws of Large Numbers for Gaussian Random Elements

2003 ◽  
Vol 10 (4) ◽  
pp. 637-676
Author(s):  
Z. Ergemlidze ◽  
A. Shangua ◽  
V. Tarieladze
Keyword(s):  
Banach Space ◽  
Law Of Large Numbers ◽  
Strong Law ◽  
Laws Of Large Numbers ◽  
Large Numbers ◽  
Random Elements ◽  
Vector Valued

Abstract Criteria for almost sure boundedness and convergence to zero almost surely of Banach space valued independent Gaussian random elements are found. The obtained statements can be viewed as vector-valued versions of the corresponding results due to N. Vakhania. Moreover, from the obtained statements a strong law of large numbers is derived in the form of Yu. V. Prokhorov.

scholarly journals Best Possible Sufficient Conditions for Strong Law of Large Numbers for Multi-Indexed Orthogonal Random Elements

2007 ◽  
Vol 2007 ◽  
pp. 1-15
Author(s):  
Kuo-Liang Su
Keyword(s):  
Banach Spaces ◽  
Law Of Large Numbers ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Strong Law ◽  
Large Numbers ◽  
Random Elements

It will be shown and induced that thed-dimensional indices in the Banach spaces version conditions∑n(E‖Xn‖p/|nα|p)<∞are sufficient to yieldlimmin1≤j≤d(nj)→∞(1/|nα|)∑k≤n∏j=1d(1−(kj−1)/nj)Xk=0a.s. for arrays of James-type orthogonal random elements. Particularly, it will be shown also that there are the best possible sufficient conditions for multi-indexed independent real-valued random variables.

scholarly journals Strong laws of large numbers for arrays of rowwise conditionally independent random elements

1993 ◽  
Vol 6 (1) ◽  
pp. 1-9 ◽  
Author(s):  
Ronald Frank Patterson ◽  
Abolghassem Bozorgnia ◽  
Robert Lee Taylor
Keyword(s):  
Banach Space ◽  
Separable Banach Space ◽  
Complete Convergence ◽  
Law Of Large Numbers ◽  
Strong Law ◽  
Moment Conditions ◽  
Laws Of Large Numbers ◽  
Large Numbers ◽  
Random Elements ◽  
Conditional Means

Let {Xnk} be an array of rowwise conditionally independent random elements in a separable Banach space of type p, 1≤p≤2. Complete convergence of n−1r∑k=1nXnk to 0, 0<r<p≤2 is obtained by using various conditions on the moments and conditional means. A Chung type strong law of large numbers is also obtained under suitable moment conditions on the conditional means.

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