scholarly journals Tauberian theorems for summability transforms

2005 ◽  
Vol 2005 (1) ◽  
pp. 55-66
Author(s):  
Rohitha Goonatilake
Keyword(s):  
Random Walk ◽  
Rate Of Convergence ◽  
Law Of Large Numbers ◽  
Summability Method ◽  
Tauberian Theorems ◽  
Strong Law ◽  
Random Walk Method ◽  
Large Numbers ◽  
The Law

The convolution summability method is introduced as a generalization of the random-walk method. In this paper, two well-known summability analogs concerning the strong law of large numbers (SLLN) and the law of the single logarithm (LSL), that gives the rate of convergence in SLLN for the random-walk method, are extended to this generalized method.

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.

scholarly journals Law of large numbers for the sequence of uncorrelated fuzzy random variables

2021 ◽  
pp. 1-8
Author(s):  
Li Guan ◽  
Jinping Zhang ◽  
Jieming Zhou
Keyword(s):  
Law Of Large Numbers ◽  
Hausdorff Metric ◽  
Random Variables ◽  
Fuzzy Random Variables ◽  
Strong Law ◽  
Fuzzy Variables ◽  
Large Numbers ◽  
The Law ◽  
Fuzzy Random

This work proposes the concept of uncorrelation for fuzzy random variables, which is weaker than independence. For the sequence of uncorrelated fuzzy variables, weak and strong law of large numbers are studied under the uniform Hausdorff metric d H ∞ . The results generalize the law of large numbers for independent fuzzy random variables.

scholarly journals Strong Law of Large Numbers for a Function of the Local Times of a Transient Random Walk in $${\mathbb {Z}}^d$$

2019 ◽  
Vol 33 (4) ◽  
pp. 2315-2336
Author(s):  
Inna M. Asymont ◽  
Dmitry Korshunov
Keyword(s):  
Random Walk ◽  
Law Of Large Numbers ◽  
Acta Math ◽  
Local Times ◽  
List Type ◽  
Strong Law ◽  
Statistics And Probability ◽  
Large Numbers ◽  
Berkeley Symposium ◽  
Transient Random Walk

Abstract For an arbitrary transient random walk $$(S_n)_{n\ge 0}$$ ( S n ) n ≥ 0 in $${\mathbb {Z}}^d$$ Z d , $$d\ge 1$$ d ≥ 1 , we prove a strong law of large numbers for the spatial sum $$\sum _{x\in {\mathbb {Z}}^d}f(l(n,x))$$ ∑ x ∈ Z d f ( l ( n , x ) ) of a function f of the local times $$l(n,x)=\sum _{i=0}^n{\mathbb {I}}\{S_i=x\}$$ l ( n , x ) = ∑ i = 0 n I { S i = x } . Particular cases are the number of visited sites [first considered by Dvoretzky and Erdős (Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, pp 353–367, 1951)], which corresponds to the function $$f(i)={\mathbb {I}}\{i\ge 1\}$$ f ( i ) = I { i ≥ 1 } ; $$\alpha $$ α -fold self-intersections of the random walk [studied by Becker and König (J Theor Probab 22:365–374, 2009)], which corresponds to $$f(i)=i^\alpha $$ f ( i ) = i α ; sites visited by the random walk exactly j times [considered by Erdős and Taylor (Acta Math Acad Sci Hung 11:137–162, 1960) and Pitt (Proc Am Math Soc 43:195–199, 1974)], where $$f(i)={\mathbb {I}}\{i=j\}$$ f ( i ) = I { i = j } .

COMPLETE CONVERGENCE FOR ARRAYS AND THE LAW OF THE SINGLE LOGARITHM

2017 ◽  
Vol 96 (2) ◽  
pp. 333-344
Author(s):  
ALLAN GUT ◽  
ULRICH STADTMÜLLER
Keyword(s):  
Limit Theorems ◽  
Complete Convergence ◽  
Law Of Large Numbers ◽  
Almost Sure Convergence ◽  
Strong Law ◽  
Moment Conditions ◽  
Large Numbers ◽  
Distributed Arrays ◽  
The Law ◽  
Independent And Identically Distributed

The present paper is devoted to complete convergence and the strong law of large numbers under moment conditions near those of the law of the single logarithm (LSL) for independent and identically distributed arrays. More precisely, we investigate limit theorems under moment conditions which are stronger than $2p$ for any $p<2$, in which case we know that there is almost sure convergence to 0, and weaker than $E\,X^{4}/(\log ^{+}|X|)^{2}<\infty$, in which case the LSL holds.

scholarly journals Excursions of a Random Walk Related to the Strong Law of Large Numbers

1998 ◽  
Vol 28 (2) ◽  
pp. 595-606
Author(s):  
Travis Lee ◽  
Max Minzner ◽  
Evan Fisher
Keyword(s):  
Random Walk ◽  
Law Of Large Numbers ◽  
Strong Law ◽  
Large Numbers
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