scholarly journals Law of Large Numbers under Choquet Expectations

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Jing Chen
Keyword(s):  
Law Of Large Numbers ◽  
Random Variables ◽  
Absolute Moment ◽  
Large Numbers ◽  
The Moment ◽  
Independence Of Random Variables ◽  
Independent And Identically Distributed

With a new notion of independence of random variables, we establish the nonadditive version of weak law of large numbers (LLN) for the independent and identically distributed (IID) random variables under Choquet expectations induced by 2-alternating capacities. Moreover, we weaken the moment assumptions to the first absolute moment and characterize the approximate distributions of random variables as well. Naturally, our theorem can be viewed as an extension of the classical LLN to the case where the probability is no longer additive.

scholarly journals On the rate of convergence in the strong law of large numbers for arrays

1992 ◽  
Vol 45 (3) ◽  
pp. 479-482 ◽  
Author(s):  
Tien-Chung Hu ◽  
N.C. Weber
Keyword(s):  
Rate Of Convergence ◽  
Law Of Large Numbers ◽  
Random Variables ◽  
Strong Law ◽  
Second Moment ◽  
Iterated Logarithm ◽  
Large Numbers ◽  
The Law ◽  
Independent And Identically Distributed

For sequences of independent and identically distributed random variables it is well known that the existence of the second moment implies the law of the iterated logarithm. We show that the law of the iterated logarithm does not extend to arrays of independent and identically distributed random variables and we develop an analogous rate result for such arrays under finite fourth moments.

A supplement to the strong law of large numbers

1975 ◽  
Vol 12 (1) ◽  
pp. 173-175 ◽  
Author(s):  
C. C. Heyde
Keyword(s):  
Law Of Large Numbers ◽  
Random Variables ◽  
Strong Law ◽  
Large Numbers ◽  
Independent And Identically Distributed

The strong law of large numbers for independent and identically distributed random variables Xi, i = 1,2,3, …, with finite mean µ can be stated as, for any ∊ > 0, the number of integers n such that |n−1 Σi=1nXi − μ| > ∊, N(∊), is finite a.s. It is known, furthermore, that EN(∊) < ∞ if and only if EX12 < ∞. Here it is shown that if EX12 < ∞ then ∊2EN(∊)→ var X1 as ∊ → 0.

A pair of complementary theorems on convergence rates in the law of large numbers

1967 ◽  
Vol 63 (1) ◽  
pp. 73-82 ◽  
Author(s):  
C. C. Heyde ◽  
V. K. Rohatgi
Keyword(s):  
Law Of Large Numbers ◽  
Convergence Rates ◽  
Random Variables ◽  
Strong Law ◽  
Large Numbers ◽  
The Law ◽  
Independent And Identically Distributed

Introduction. Let Xi (i= 1, 2, 3,…) be a sequence of independent and identically distributed random variables with law ℒ(X) and write The Kolmogorov-Marcinkiewicz strong law of large numbers (Loève(6), p. 243) has the following statement:If E|X|r < ∞, then with cr = 0 or EX according as r 1 or r ≥ 1.

A supplement to the strong law of large numbers

1975 ◽  
Vol 12 (01) ◽  
pp. 173-175 ◽  
Author(s):  
C. C. Heyde
Keyword(s):  
Law Of Large Numbers ◽  
Random Variables ◽  
Strong Law ◽  
Large Numbers ◽  
Independent And Identically Distributed

The strong law of large numbers for independent and identically distributed random variables Xi, i = 1,2,3, …, with finite mean µ can be stated as, for any ∊ &gt; 0, the number of integers n such that |n −1 Σ i=1 n X i − μ| &gt; ∊, N (∊), is finite a.s. It is known, furthermore, that EN (∊) &lt; ∞ if and only if EX 1 2 &lt; ∞. Here it is shown that if EX 1 2 &lt; ∞ then ∊ 2 EN (∊) → var X 1 as ∊ → 0.

scholarly journals Strong and Weak Convergence for Asymptotically Almost Negatively Associated Random Variables

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Aiting Shen ◽  
Ranchao Wu
Keyword(s):  
Weak Convergence ◽  
Law Of Large Numbers ◽  
Random Variables ◽  
Strong Law ◽  
Associated Random Variables ◽  
Negatively Associated ◽  
Strong And Weak Convergence ◽  
Large Numbers ◽  
Asymptotically Almost Negatively Associated ◽  
Independent And Identically Distributed

The strong law of large numbers for sequences of asymptotically almost negatively associated (AANA, in short) random variables is obtained, which generalizes and improves the corresponding one of Bai and Cheng (2000) for independent and identically distributed random variables to the case of AANA random variables. In addition, the Feller-type weak law of large number for sequences of AANA random variables is obtained, which generalizes the corresponding one of Feller (1946) for independent and identically distributed random variables.

scholarly journals Strong Limit Theorems for Dependent Random Variables

2021 ◽  
Vol 15 ◽  
pp. 15-17
Author(s):  
Libin Wu ◽  
Bainian Li
Keyword(s):  
Limit Theorems ◽  
Complete Convergence ◽  
Law Of Large Numbers ◽  
Random Variables ◽  
Moment Inequality ◽  
Strong Limit ◽  
Dependent Random Variables ◽  
Strong Law ◽  
Large Numbers ◽  
Independent And Identically Distributed

In this article We establish moment inequality of dependent random variables, furthermore some theorems of strong law of large numbers and complete convergence for sequences of dependent random variables. In particular, independent and identically distributed Marcinkiewicz Law of large numbers are generalized to the case of m₀ -dependent sequences.

scholarly journals On Complete Convergence for Weighted Sums ofρ*-Mixing Random Variables

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Aiting Shen ◽  
Xinghui Wang ◽  
Huayan Zhu
Keyword(s):  
Complete Convergence ◽  
Law Of Large Numbers ◽  
Random Variables ◽  
Independent Random Variables ◽  
Weighted Sums ◽  
Strong Law ◽  
Large Numbers ◽  
Mixing Random Variables ◽  
Independent And Identically Distributed

We prove the strong law of large numbers for weighted sums∑i=1n‍aniXi, which generalizes and improves the corresponding one for independent and identically distributed random variables andφ-mixing random variables. In addition, we present some results on complete convergence for weighted sums ofρ*-mixing random variables under some suitable conditions, which generalize the corresponding ones for independent random variables.

scholarly journals On the weak law of large numbers for normed weighted sums of I.I.D. random variables

1991 ◽  
Vol 14 (1) ◽  
pp. 191-202 ◽  
Author(s):  
André Adler ◽  
Andrew Rosalsky
Keyword(s):  
Law Of Large Numbers ◽  
Random Variables ◽  
Weighted Sums ◽  
Strong Law ◽  
Large Numbers ◽  
Independent And Identically Distributed

For weighted sums∑j=1najYjof independent and identically distributed random variables{Yn,n≥1}, a general weak law of large numbers of the form(∑j=1najYj−νn)/bn→P0is established where{νn,n≥1}and{bn,n≥1}are statable constants. The hypotheses involve both the behavior of the tail of the distribution of|Y1|and the growth behaviors of the constants{an,n≥1}and{bn,n≥1}. Moreover, a weak law is proved for weighted sums∑j=1najYjindexed by random variables{Tn,n≥1}. An example is presented wherein the weak law holds but the strong law fails thereby generalizing a classical example.

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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