A central limit theorem for strongly multiplicative systems of functions

1969 ◽  
Vol 6 (4) ◽  
pp. 720-724
Author(s):  
V. F. Gaposhkin
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Multiplicative Systems

A Central Limit Theorem for Multiplicative Systems

1973 ◽  
Vol 16 (1) ◽  
pp. 67-73 ◽  
Author(s):  
J. Komlós
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Law Of Large Numbers ◽  
Law Of Iterated Logarithm ◽  
Independent Random Variables ◽  
Strong Law ◽  
Multiplicative Systems ◽  
Large Numbers ◽  
Strong Notion

The central limit theorem was originally proved for independent random variables. The independence is a very strong notion and hard to check. There are various efforts to prove different theorems on independent variables (e.g. strong law of large numbers, central limit theorem, the law of iterated logarithm, convergence theorem of Kolmogorov) under weaker conditions, like mixing, martingale-difference, orthogonality. Among these concepts the weakest one is orthogonality, but this ensures only the validity of law of large numbers.

A Mean Central Limit Theorem for Multiplicative Systems

1988 ◽  
Vol 139 (1) ◽  
pp. 87-94 ◽  
Author(s):  
Ludwig Paditz ◽  
Šaturgun Šarachmetov
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Multiplicative Systems

scholarly journals On the Markov Transition Kernels for First Passage Percolation on the Ladder

2011 ◽  
Vol 48 (02) ◽  
pp. 366-388 ◽  
Author(s):  
Eckhard Schlemm
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Asymptotic Variance ◽  
Almost Sure Convergence ◽  
First Passage Percolation ◽  
First Passage ◽  
Percolation Problem ◽  
Vertex Set ◽  
Step Transition

We consider the first passage percolation problem on the random graph with vertex set N x {0, 1}, edges joining vertices at a Euclidean distance equal to unity, and independent exponential edge weights. We provide a central limit theorem for the first passage times l n between the vertices (0, 0) and (n, 0), thus extending earlier results about the almost-sure convergence of l n / n as n → ∞. We use generating function techniques to compute the n-step transition kernels of a closely related Markov chain which can be used to explicitly calculate the asymptotic variance in the central limit theorem.

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