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 On Chung's strong law of large numbers in general Banach spaces

1988 ◽  
Vol 37 (1) ◽  
pp. 93-100 ◽  
Author(s):  
Bong Dae Choi ◽  
Soo Hak Sung
Keyword(s):  
Banach Spaces ◽  
Law Of Large Numbers ◽  
Random Variables ◽  
Real Numbers ◽  
Strong Law ◽  
Laws Of Large Numbers ◽  
Strong Laws ◽  
Large Numbers

Let { Xn, n ≥ 1 } be a sequence of independent Banach valued random variables and { an, n, ≥ 1 } a sequence of real numbers such that 0 < an ↑ ∞. It is shown that, under the assumption with some restrictions on φ, Sn/an → 0 a.s. if and only if Sn/an → 0 in probability if and only if Sn/an → 0 in L1. From this result several known strong laws of large numbers in Banach spaces are easily derived.

scholarly journals Weighted Strong Law of Large Numbers for Random Variables Indexed by a Sector

2011 ◽  
Vol 2011 ◽  
pp. 1-16 ◽  
Author(s):  
Przemysław Matuła ◽  
Michał Seweryn
Keyword(s):  
Law Of Large Numbers ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Necessary And Sufficient Conditions ◽  
Independent Random Variables ◽  
Strong Law ◽  
Large Numbers ◽  
Multidimensional Indices ◽  
Necessary And Sufficient

We find necessary and sufficient conditions for the weighted strong law of large numbers for independent random variables with multidimensional indices belonging to some sector.

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 Strong laws of large numbers for the sequence of the maximum of partial sums of i.i.d. random variables

2019 ◽  
Vol 39 (1) ◽  
pp. 19-38
Author(s):  
Shuhua Chang ◽  
Deli Li ◽  
Andrew Rosalsky
Keyword(s):  
Law Of Large Numbers ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Random Variable ◽  
Partial Sums ◽  
Strong Law ◽  
Laws Of Large Numbers ◽  
Strong Laws ◽  
Large Numbers ◽  
Necessary And Sufficient

Let 0 < p ≤ 2, let {Xn; n ≥ 1} be a sequence of independent copies of a real-valued random variable X, and set Sn = X1 + . . . + Xn, n ≥ ­ 1. Motivated by a theorem of Mikosch 1984, this note is devoted to establishing a strong law of large numbers for the sequence {max1≤k≤n |Sk| ; n ≥ ­ 1}. More specifically, necessary and sufficient conditions are given forlimn→∞ max1≤k≤n |Sk|log n−1 = e1/p a.s.,where log x = loge max{e, x}, x ≥­ 0.

scholarly journals On Some Conditions for Strong Law of Large Numbers for Weighted Sums of END Random Variables under Sublinear Expectations

2019 ◽  
Vol 2019 ◽  
pp. 1-8
Author(s):  
Xiaochen Ma ◽  
Qunying Wu
Keyword(s):  
Probability Space ◽  
Law Of Large Numbers ◽  
Random Variables ◽  
Weighted Sums ◽  
Dependent Random Variables ◽  
Strong Law ◽  
Sublinear Expectation ◽  
Negatively Dependent Random Variables ◽  
Large Numbers ◽  
End Random Variables

In this article, we research some conditions for strong law of large numbers (SLLNs) for weighted sums of extended negatively dependent (END) random variables under sublinear expectation space. Our consequences contain the Kolmogorov strong law of large numbers and the Marcinkiewicz strong law of large numbers for weighted sums of extended negatively dependent random variables. Furthermore, our results extend strong law of large numbers for some sequences of random variables from the traditional probability space to the sublinear expectation space context.

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