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The Other Law of the Iterated Logarithm
The Annals of Probability, 1975 Let $\{X_n\}$ be a sequence of independent, identically distributed random variables with $EX_1 = 0, EX_1^2 = 1$. Define $S_n = X_1 + \cdots + X_n$, and $A_n = \max_{1\leqq k\leqq n} |S_k|$. We prove that $\lim \inf A_n(n/\log \log n)^{-\frac{1}{2}} = \pi/8^{\frac{1}{2}}$ with probability one.
Jain, Naresh C., Pruitt, William E.
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On martingale tail sums in affine two-color urn models with multiple drawings [PDF]
, 2016 In two recent works, Kuba and Mahmoud (arXiv:1503.090691 and arXiv:1509.09053) introduced the family of two-color affine balanced Polya urn schemes with multiple drawings. We show that, in large-index urns (urn index between $1/2$ and $1$) and triangular
Kuba, Markus, Sulzbach, Henning
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Laws of the k-Iterated Logarithm of Weighted Sums in a Sub-Linear Expected Space
MathematicsThe law of the iterated logarithm precisely refines the law of large numbers and plays a fundamental role in probability limit theory. The framework of sub-linear expectation spaces substantially extends the classical concept of probability spaces.
Xiang Zeng
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The law of iterated logarithm for the estimations of diffusion-type processes
Advances in Difference Equations, 2020 This paper mainly discusses the asymptotic behaviours on the lasso-type estimators for diffusion-type processes with a small noise. By constructing the objective function on the estimation, in view of convexity argument, it is proved that the estimator ...
Mingzhi Mao, Gang Huang
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Metric discrepancy results for geometric progressions with small ratios
, 2018 The law of the iterated logarithm for discrepancies of geometric progressions with small ratios is ...
Fukuyama, K. +4 more
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The law of the iterated logarithm for exchangeable random variables
International Journal of Mathematics and Mathematical Sciences, 1995 In this note, necessary and sufficient conditions for laws of the iterated logarithm are developed for exchangeable random variables.
Hu-Ming Zhang, Robert L. Taylor
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A law of the iterated logarithm sublinear expectations
, 2013 In this paper, motivated by the notion of independent identically distributed (IID) random variables under sub-linear expectations initiated by Peng, we investigate a law of the iterated logarithm for capacities.
Khinchine A., Lévy P., Stout W. F.
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Law of the iterated logarithm for stationary processes [PDF]
, 2007 There has been recent interest in the conditional central limit question for (strictly) stationary, ergodic processes $...,X_{-1},X_0,X_1,...$ whose partial sums $S_n=X_1+...+X_n$ are of the form $S_n=M_n+R_n$, where $M_n$ is a square integrable ...
Woodroofe, Michael, Zhao, Ou
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On Feller's criterion for the law of the iterated logarithm
International Journal of Mathematics and Mathematical Sciences, 1994 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.
Deli Li, M. Bhaskara Rao, Xiangchen Wang
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Strong Approximation of Empirical Copula Processes by Gaussian Processes
, 2011 We provide the strong approximation of empirical copula processes by a Gaussian process. In addition we establish a strong approximation of the smoothed empirical copula processes and a law of iterated ...
Adler R. J. +21 more
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