scholarly journals On the central limit theorem and the law of the iterated logarithm

2008 ◽  
Vol 78 (12) ◽  
pp. 1384-1387 ◽  
Author(s):  
Katusi Fukuyama ◽  
Yôhei Ueno
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
The Central Limit Theorem ◽  
Iterated Logarithm ◽  
The Law

scholarly journals Precise Asymptotics in the Law of the Iterated Logarithm under Sublinear Expectations

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Mingzhou Xu ◽  
Kun Cheng
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Uniform Convergence ◽  
Central Limit ◽  
Random Variables ◽  
Precise Asymptotics ◽  
Partial Sum ◽  
The Central Limit Theorem ◽  
Iterated Logarithm ◽  
The Law

By an inequality of partial sum and uniform convergence of the central limit theorem under sublinear expectations, we establish precise asymptotics in the law of the iterated logarithm for independent and identically distributed random variables under sublinear expectations.

scholarly journals On Feller's criterion for the law of the iterated logarithm

1994 ◽  
Vol 17 (2) ◽  
pp. 323-340 ◽  
Author(s):  
Deli Li ◽  
M. Bhaskara Rao ◽  
Xiangchen Wang
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Uniform Estimate ◽  
Random Variables ◽  
Independent Random Variables ◽  
Partial Sums ◽  
The Central Limit Theorem ◽  
Iterated Logarithm ◽  
The Law

Combining Feller's criterion with a non-uniform estimate result in the context of the Central Limit Theorem for partial sums of independent random variables, we obtain several results on the Law of the Iterated Logarithm. Two of these results refine corresponding results of Wittmann (1985) and Egorov (1971). In addition, these results are compared with the corresponding results of Teicher (1974), Tomkins (1983) and Tomkins (1990)

scholarly journals A law of the iterated logarithm for martingales

1991 ◽  
Vol 43 (2) ◽  
pp. 181-185 ◽  
Author(s):  
Michael Voit
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Rate Of Convergence ◽  
Central Limit ◽  
Upper Bound ◽  
The Central Limit Theorem ◽  
Iterated Logarithm ◽  
The Law

Using a slight generalisation of Brown's inequality, we show that for martingales the existence of a weak nonuniform bound on the rate of convergence in the central limit theorem yields the usual upper bound part of the law of the iterated logarithm.

A Central Limit Theorem and Law of the Iterated Logarithm for a Random Field with Exponential Decay of Correlations

2004 ◽  
Vol 56 (1) ◽  
pp. 209-224 ◽  
Author(s):  
Byron Schmuland ◽  
Wei Sun
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Random Field ◽  
Central Limit ◽  
Exponential Decay ◽  
Gibbs Measures ◽  
Strong Mixing ◽  
Iterated Logarithm ◽  
Exponential Decay Of Correlations ◽  
The Law

AbstractIn [6], Walter Philipp wrote that “… the law of the iterated logarithm holds for any process for which the Borel-Cantelli Lemma, the central limit theorem with a reasonably good remainder and a certain maximal inequality are valid.” Many authors [1], [2], [4], [5], [9] have followed this plan in proving the law of the iterated logarithm for sequences (or fields) of dependent random variables.We carry on this tradition by proving the law of the iterated logarithm for a random field whose correlations satisfy an exponential decay condition like the one obtained by Spohn [8] for certain Gibbs measures. These do not fall into the ϕ-mixing or strong mixing cases established in the literature, but are needed for our investigations [7] into diffusions on configuration space.The proofs are all obtained by patching together standard results from [5], [9] while keeping a careful eye on the correlations.

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