scholarly journals Precise Rates in Log Laws for NA Sequences

2007 ◽  
Vol 2007 ◽  
pp. 1-11
Author(s):  
Yuexu Zhao
Keyword(s):  
Random Variables ◽  
Random Variable ◽  
Stationary Sequence ◽  
Normal Random Variable ◽  
Negatively Associated ◽  
Standard Normal Random Variable ◽  
Na Random Variables ◽  
Standard Normal ◽  
Na Sequences

LetX1,X2,…be a strictly stationary sequence of negatively associated (NA) random variables withEX1=0, setSn=X1+⋯+Xn, suppose thatσ2=EX12+2∑n=2∞EX1Xn>0andEX12<∞,if−1<α≤1;EX12(log|X1|)α<∞, ifα>1. We provelimε↓0ε2α+2∑n=1∞((logn)α/n)P(|Sn|≥σ(ε+κn)2nlogn)=2−(α+1)(α+1)−1E|N|2α+2, whereκn=O(1/logn)and N is the standard normal random variable.

Asymptotic properties for the loglog laws under positive association

2012 ◽  
Vol 62 (5) ◽  
Author(s):  
Xiao-Rong Yang ◽  
Ke-Ang Fu
Keyword(s):  
Gamma Function ◽  
Asymptotic Properties ◽  
Random Variables ◽  
Positive Association ◽  
Random Variable ◽  
Stationary Sequence ◽  
Normal Random Variable ◽  
Finite Variance ◽  
Associated Random Variables ◽  
Standard Normal

AbstractLet {X n: n ≥ 1} be a strictly stationary sequence of positively associated random variables with mean zero and finite variance. Set $$S_n = \sum\limits_{k = 1}^n {X_k }$$, $$Mn = \mathop {\max }\limits_{k \leqslant n} \left| {S_k } \right|$$, n ≥ 1. Suppose that $$0 < \sigma ^2 = EX_1^2 + 2\sum\limits_{k = 2}^\infty {EX_1 X_k < \infty }$$. In this paper, we prove that if E|X 1|2+δ < for some δ ∈ (0, 1], and $$\sum\limits_{j = n + 1}^\infty {Cov\left( {X_1 ,X_j } \right) = O\left( {n^{ - \alpha } } \right)}$$ for some α > 1, then for any b > −1/2 $$\mathop {\lim }\limits_{\varepsilon \searrow 0} \varepsilon ^{2b + 1} \sum\limits_{n = 1}^\infty {\frac{{(\log \log n)^{b - 1/2} }} {{n^{3/2} \log n}}} E\left\{ {M_n - \sigma \varepsilon \sqrt {2n\log \log n} } \right\}_ + = \frac{{2^{ - 1/2 - b} E\left| N \right|^{2(b + 1)} }} {{(b + 1)(2b + 1)}}\sum\limits_{k = 0}^\infty {\frac{{( - 1)^k }} {{(2k + 1)^{2(b + 1)} }}}$$ and $$\mathop {\lim }\limits_{\varepsilon \nearrow \infty } \varepsilon ^{ - 2(b + 1)} \sum\limits_{n = 1}^\infty {\frac{{(\log \log n)^b }} {{n^{3/2} \log n}}E\left\{ {\sigma \varepsilon \sqrt {\frac{{\pi ^2 n}} {{8\log \log n}}} - M_n } \right\}} _ + = \frac{{\Gamma (b + 1/2)}} {{\sqrt 2 (b + 1)}}\sum\limits_{k = 0}^\infty {\frac{{( - 1)^k }} {{(2k + 1)^{2b + 2} }}} ,$$ where x + = max{x, 0}, N is a standard normal random variable, and Γ(·) is a Gamma function.

scholarly journals On the Bahadur representation of sample quantiles for widely orthant dependent sequences

Filomat ◽  
2014 ◽  
Vol 28 (7) ◽  
pp. 1333-1343 ◽  
Author(s):  
Wenzhi Yang ◽  
Tingting Liu ◽  
Xuejun Wang ◽  
Shuhe Hu
Keyword(s):  
Random Variables ◽  
Bahadur Representation ◽  
Negatively Associated ◽  
Statistics And Probability ◽  
Na Random Variables ◽  
End Random Variables ◽  
Widely Orthant Dependent ◽  
Na Sequences ◽  
Nod Sequence ◽  
Sample Quantiles

It can be found that widely orthant dependent (WOD) random variables are weaker than extended negatively orthant dependent (END) random variables, while END random variables are weaker than negatively orthant dependent (NOD) and negatively associated (NA) random variables. In this paper, we investigate the Bahadur representation of sample quantiles based on WOD sequences. Our results extend the corresponding ones of Ling [N.X. Ling, The Bahadur representation for sample quantiles under negatively associated sequence, Statistics and Probability Letters 78(16) (2008), 2660-2663], Xu et al. [S.F. Xu, L. Ge, Y. Miao, On the Bahadur representation of sample quantiles and order statistics for NA sequences, Journal of the Korean Statistical Society 42(1) (2013), 1-7] and Li et al. [X.Q. Li, W.Z. Yang, S.H. Hu, X.J. Wang, The Bahadur representation for sample quantile under NOD sequence, Journal of Nonparametric Statistics 23(1) (2011), 59-65] for the case of NA sequences or NOD sequences.

scholarly journals A Berry-Esseen bound for the lightbulb process

2011 ◽  
Vol 43 (03) ◽  
pp. 875-898 ◽  
Author(s):  
Larry Goldstein ◽  
Haimeng Zhang
Keyword(s):  
Conditional Expectation ◽  
Spectral Decomposition ◽  
Random Variable ◽  
Stein’S Method ◽  
Normal Random Variable ◽  
The Past ◽  
Terminal Time ◽  
Standard Normal Random Variable ◽  
Standard Normal ◽  
Size Bias

In the so-called lightbulb process, on daysr= 1,…,n, out ofnlightbulbs, all initially off, exactlyrbulbs, selected uniformly and independent of the past, have their status changed from off to on, or vice versa. WithXthe number of bulbs on at the terminal timen, an even integer, and μ =n/2, σ2= var(X), we have supz∈R| P((X- μ)/σ ≤z) - P(Z≤z) | ≤nΔ̅0/2σ2+ 1.64n/σ3+ 2/σ, whereZis a standard normal random variable and Δ̅0= 1/2√n+ 1/2n+ e−n/2/3 forn≥ 6, yielding a bound of orderO(n−1/2) asn→ ∞. A similar, though slightly larger bound, holds for oddn. The results are shown using a version of Stein's method for bounded, monotone size bias couplings. The argument for evenndepends on the construction of a variableXson the same space asXthat has theX-size bias distribution, that is, which satisfies E[Xg(X)] = μE[g(Xs)] for all bounded continuousg, and for which there exists aB≥ 0, in this caseB= 2, such thatX≤Xs≤X+Balmost surely. The argument for oddnis similar to that for evenn, but one first couplesXclosely toV, a symmetrized version ofX, for which a size bias coupling ofVtoVscan proceed as in the even case. In both the even and odd cases, the crucial calculation of the variance of a conditional expectation requires detailed information on the spectral decomposition of the lightbulb chain.

Asymptotic Approximation of an Integral Involving the Normal Distribution*

1986 ◽  
Vol 29 (2) ◽  
pp. 167-176 ◽  
Author(s):  
J. P. McClure ◽  
R. Wong
Keyword(s):  
Distribution Function ◽  
Normal Distribution ◽  
Positive Constant ◽  
Asymptotic Approximation ◽  
Cumulative Distribution Function ◽  
Random Variable ◽  
Normal Random Variable ◽  
Cumulative Distribution ◽  
Standard Normal Random Variable ◽  
Standard Normal

AbstractAn asymptotic approximation is obtained, as k → ∞, for the integralwhere Φ is the cumulative distribution function for a standard normal random variable, and L is a positive constant. The problem is motivated by a question in statistics, and an outline of'the application is given. Similar methods may be used to approximate other integrals involving the normal distribution.

scholarly journals A Berry-Esseen bound for the lightbulb process

2011 ◽  
Vol 43 (3) ◽  
pp. 875-898 ◽  
Author(s):  
Larry Goldstein ◽  
Haimeng Zhang
Keyword(s):  
Conditional Expectation ◽  
Spectral Decomposition ◽  
Random Variable ◽  
Stein’S Method ◽  
Normal Random Variable ◽  
The Past ◽  
Terminal Time ◽  
Standard Normal Random Variable ◽  
Standard Normal ◽  
Size Bias

In the so-called lightbulb process, on days r = 1,…,n, out of n lightbulbs, all initially off, exactly r bulbs, selected uniformly and independent of the past, have their status changed from off to on, or vice versa. With X the number of bulbs on at the terminal time n, an even integer, and μ = n/2, σ2 = var(X), we have supz ∈ R | P((X - μ)/σ ≤ z) - P(Z ≤ z) | ≤ nΔ̅0/2σ2 + 1.64n/σ3 + 2/σ, where Z is a standard normal random variable and Δ̅0 = 1/2√n + 1/2n + e−n/2/3 for n ≥ 6, yielding a bound of order O(n−1/2) as n → ∞. A similar, though slightly larger bound, holds for odd n. The results are shown using a version of Stein's method for bounded, monotone size bias couplings. The argument for even n depends on the construction of a variable Xs on the same space as X that has the X-size bias distribution, that is, which satisfies E[Xg(X)] = μE[g(Xs)] for all bounded continuous g, and for which there exists a B ≥ 0, in this case B = 2, such that X ≤ Xs ≤ X + B almost surely. The argument for odd n is similar to that for even n, but one first couples X closely to V, a symmetrized version of X, for which a size bias coupling of V to Vs can proceed as in the even case. In both the even and odd cases, the crucial calculation of the variance of a conditional expectation requires detailed information on the spectral decomposition of the lightbulb chain.

On an inequality for the normal distribution arising in bioequivalence studies

1999 ◽  
Vol 36 (1) ◽  
pp. 279-286
Author(s):  
Yi-Ching Yao ◽  
Hari Iyer
Keyword(s):  
Normal Distribution ◽  
Random Variable ◽  
Normal Random Variable ◽  
Individual Bioequivalence ◽  
Left Hand ◽  
Right Hand ◽  
Standard Normal Random Variable ◽  
Standard Normal ◽  
Image Position ◽  
The Right

For (μ,σ2) ≠ (0,1), and 0 < z < ∞, we prove that where φ and Φ are, respectively, the p.d.f. and the c.d.f. of a standard normal random variable. This inequality is sharp in the sense that the right-hand side cannot be replaced by a larger quantity which depends only on μ and σ. In other words, for any given (μ,σ) ≠ (0,1), the infimum, over 0 < z < ∞, of the left-hand side of the inequality is equal to the right-hand side. We also point out how this inequality arises in the context of defining individual bioequivalence.

On an inequality for the normal distribution arising in bioequivalence studies

1999 ◽  
Vol 36 (01) ◽  
pp. 279-286 ◽  
Author(s):  
Yi-Ching Yao ◽  
Hari Iyer
Keyword(s):  
Normal Distribution ◽  
Random Variable ◽  
Normal Random Variable ◽  
Individual Bioequivalence ◽  
Left Hand ◽  
Right Hand ◽  
Standard Normal Random Variable ◽  
Standard Normal ◽  
Image Position ◽  
The Right

For (μ,σ2) ≠ (0,1), and 0 &lt; z &lt; ∞, we prove that where φ and Φ are, respectively, the p.d.f. and the c.d.f. of a standard normal random variable. This inequality is sharp in the sense that the right-hand side cannot be replaced by a larger quantity which depends only on μ and σ. In other words, for any given (μ,σ) ≠ (0,1), the infimum, over 0 &lt; z &lt; ∞, of the left-hand side of the inequality is equal to the right-hand side. We also point out how this inequality arises in the context of defining individual bioequivalence.

scholarly journals A refinement of normal approximation to Poisson binomial

2005 ◽  
Vol 2005 (5) ◽  
pp. 717-728 ◽  
Author(s):  
K. Neammanee
Keyword(s):  
Normal Distribution ◽  
Normal Approximation ◽  
Random Variables ◽  
Random Variable ◽  
Standard Normal Distribution ◽  
Bernoulli Random Variables ◽  
Binomial Random Variable ◽  
Standard Normal ◽  
Correction Terms ◽  
Better Than

LetX1,X2,…,Xnbe independent Bernoulli random variables withP(Xj=1)=1−P(Xj=0)=pjand letSn:=X1+X2+⋯+Xn.Snis called a Poisson binomial random variable and it is well known that the distribution of a Poisson binomial random variable can be approximated by the standard normal distribution. In this paper, we use Taylor's formula to improve the approximation by adding some correction terms. Our result is better than before and is of order1/nin the casep1=p2=⋯=pn.

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