scholarly journals Almost Sure Central Limit Theorem for Product of Partial Sums of Strongly Mixing Random Variables

2011 ◽  
Vol 2011 (1) ◽  
pp. 576301 ◽  
Author(s):  
Daxiang Ye ◽  
Qunying Wu
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Random Variables ◽  
Partial Sums ◽  
Strongly Mixing ◽  
Strongly Mixing Random Variables ◽  
Mixing Random Variables

scholarly journals An expansion related to the central limit theorem

1969 ◽  
Vol 10 (1-2) ◽  
pp. 219-230
Author(s):  
C. R. Heathcote
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Random Variables ◽  
Partial Sums ◽  
The Central Limit Theorem ◽  
Independent And Identically Distributed

Let X1, X2,…be independent and identically distributed non-lattice random variables with zero, varianceσ2<∞, and partial sums Sn = X1+X2+…+X.

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 Convergence of Weighted Linear Process forρ-Mixing Random Variables

2007 ◽  
Vol 2007 ◽  
pp. 1-7
Author(s):  
Guang-Hui Cai
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Random Variables ◽  
Linear Process ◽  
Functional Central Limit Theorem ◽  
Mixing Sequences ◽  
Functional Central Limit ◽  
Mixing Random Variables

A central limit theorem and a functional central limit theorem are obtained for weighted linear process ofρ-mixing sequences for theXt=∑i=0∞aiYt−i, where{Yi,0≤i<∞}is a sequence ofρ-mixing random variables withEYi=0,0<EYi2<∞,∑i=1∞ρ(2i)<∞. The results obtained generalize the results of Liang et al. (2004) toρ-mixing sequences.

scholarly journals Almost sure local central limit theorem for the product of some partial sums of negatively associated sequences

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Feng Xu ◽  
Binhui Wang ◽  
Yawen Hou
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
General Result ◽  
Random Variables ◽  
Partial Sums ◽  
Negatively Associated ◽  
Associated Sequences ◽  
Local Central Limit Theorem

AbstractThe almost sure local central limit theorem is a general result which contains the almost sure global central limit theorem. Let $\{X_{k},k\geq 1\}${Xk,k≥1} be a strictly stationary negatively associated sequence of positive random variables. Under the regular conditions, we discuss an almost sure local central limit theorem for the product of some partial sums $(\prod_{i=1}^{k} S_{k,i}/((k-1)^{k}\mu^{k}))^{\mu/(\sigma\sqrt{k})}$(∏i=1kSk,i/((k−1)kμk))μ/(σk), where $\mathbb{E}X_{1}=\mu$EX1=μ, $\sigma^{2}={\mathbb{E}(X_{1}-\mu)^{2}}+2\sum_{k=2}^{\infty}\mathbb{E}(X_{1}-\mu)(X_{k}-\mu)$σ2=E(X1−μ)2+2∑k=2∞E(X1−μ)(Xk−μ), $S_{k,i}=\sum_{j=1}^{k}X_{j}-X_{i}$Sk,i=∑j=1kXj−Xi.

scholarly journals Tail sums of convergent series of independent random variables

1974 ◽  
Vol 75 (3) ◽  
pp. 361-364 ◽  
Author(s):  
Andrew D. Barbour
Keyword(s):  
Distribution Function ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Random Variables ◽  
Convergent Series ◽  
Independent Random Variables ◽  
Partial Sums ◽  
Reversed Martingale ◽  
Tail Sums

Let X1, X2, … be a sequence of independent random variables such that, for each n ≥ 1, EXn = 0 and and assume that then converges almost surely as N → ∞. Let and let Fn(x) denote the distribution function of Xn. Loynes (2) observed that the sequence {Sn} is a reversed martingale, and applied his central limit theorem to it: however, stronger results are obtainable, in precise duality with the classical theory of partial sums of independent random variables. These results describe the fluctuations of the sequence {Sn}, and hence the way in which converges to its limit.

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