Hyperfinite Law of Large Numbers

1996 ◽  
Vol 2 (2) ◽  
pp. 189-198 ◽  
Author(s):  
Yeneng Sun
Keyword(s):  
Law Of Large Numbers ◽  
Nonstandard Analysis ◽  
Random Variables ◽  
Triangular Array ◽  
Large Numbers ◽  
Loeb Space ◽  
Space Construction ◽  
Pairwise Independence ◽  
Necessary And Sufficient

AbstractThe Loeb space construction in nonstandard analysis is applied to the theory of processes to reveal basic phenomena which cannot be treated using classical methods. An asymptotic interpretation of results established here shows that for a triangular array (or a sequence) of random variables, asymptotic uncorrelatedness or asymptotic pairwise independence is necessary and sufficient for the validity of appropriate versions of the law of large numbers. Our intrinsic characterization of almost sure pairwise independence leads to the equivalence of various multiplicative properties of random variables.

scholarly journals The weak law of large numbers for nonnegative summands

2018 ◽  
Vol 50 (A) ◽  
pp. 241-252
Author(s):  
Eugene Seneta
Keyword(s):  
Branching Process ◽  
Law Of Large Numbers ◽  
Random Variables ◽  
Sufficient Condition ◽  
Specific Function ◽  
Slowly Varying Function ◽  
Necessary And Sufficient Condition ◽  
Large Numbers ◽  
Norming Sequence ◽  
Necessary And Sufficient

Abstract Khintchine's (necessary and sufficient) slowly varying function condition for the weak law of large numbers (WLLN) for the sum of n nonnegative, independent and identically distributed random variables is used as an overarching (sufficient) condition for the case that the number of summands is more generally [cn],cn→∞. Either the norming sequence {an},an→∞, or the number of summands sequence {cn}, can be chosen arbitrarily. This theorem generalizes results from a motivating branching process setting in which Khintchine's sufficient condition is automatically satisfied. A second theorem shows that Khintchine's condition is necessary for the generalized WLLN when it holds with cn→∞ and an→∞. Theorem 3, which is known, gives a necessary and sufficient condition for Khintchine's WLLN to hold with cn=n and an a specific function of n; it is extended to general cn subject to a growth restriction in Theorem 4. Section 6 returns to the branching process setting.

Remarks on the strong law of large numbers for a triangular array of associated random variables

Metrika ◽  
1997 ◽  
Vol 45 (1) ◽  
pp. 225-234 ◽  
Author(s):  
Isha Dewan ◽  
B. L. S. Prakasa Rao
Keyword(s):  
Law Of Large Numbers ◽  
Random Variables ◽  
Triangular Array ◽  
Strong Law ◽  
Associated Random Variables ◽  
Large Numbers

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 Two probability theorems and their application to some first passage problems

1964 ◽  
Vol 4 (2) ◽  
pp. 214-222 ◽  
Author(s):  
C. C. Heyde
Keyword(s):  
Rate Of Convergence ◽  
Law Of Large Numbers ◽  
Sufficient Conditions ◽  
Random Variables ◽  
First Passage ◽  
Sums Of Random Variables ◽  
Large Numbers ◽  
The Mean ◽  
Image Position ◽  
Necessary And Sufficient

Let Xi, i = 1, 2, 3,··· be a sequence of independent and identically distributed random variables and write Sn = X1+X2+…+Xn. If the mean of Xi is finite and positive, we have Pr(Sn ≦ x) → 0 as n → ∞ for all x1 – ∞ < x < ∞ using the weak law of large numbers. It is our purpose in this paper to study the rate of convergence of Pr(Sn ≦ x) to zero. Necessary and sufficient conditions are established for the convergence of the two series where k is a non-negative integer, and where r > 0. These conditions are applied to some first passage problems for sums of random variables. The former is also used in correcting a queueing Theorem of Finch [4].

Convergence rates in the weak law of large numbers for weighted sums of i.i.d. random variables and applications in errors-in-variables models

2019 ◽  
Vol 19 (06) ◽  
pp. 1950041 ◽  
Author(s):  
Mingyang Zhang ◽  
Pingyan Chen ◽  
Soo Hak Sung
Keyword(s):  
Regression Models ◽  
Law Of Large Numbers ◽  
Convergence Rates ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Weighted Sums ◽  
Errors In Variables ◽  
Error In Variables ◽  
Large Numbers ◽  
Necessary And Sufficient

We prove necessary and sufficient conditions for the convergence rates in the weak law of large numbers for weighted sums of independent and identically distributed (i.i.d.) random variables. This result is applied to simple linear error-in-variables regression models.

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.

Kolmogorov's strong law of large numbers for fuzzy random variables

2001 ◽  
Vol 120 (3) ◽  
pp. 499-503 ◽  
Author(s):  
Sang Yeol Joo ◽  
Yun Kyong Kim
Keyword(s):  
Law Of Large Numbers ◽  
Random Variables ◽  
Fuzzy Random Variables ◽  
Strong Law ◽  
Large Numbers ◽  
Fuzzy Random

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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