scholarly journals Normal approximation and fourth moment theorems for monochromatic triangles

2021 ◽  
Author(s):  
Bhaswar B. Bhattacharya ◽  
Xiao Fang ◽  
Han Yan
Keyword(s):  
Normal Approximation ◽  
Fourth Moment

scholarly journals Quantitative Fourth Moment Theorem of Functions on the Markov Triple and Orthogonal Polynomials

2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
Yoon Tae Kim ◽  
Hyun Suk Park
Keyword(s):  
Probability Measure ◽  
Orthogonal Polynomials ◽  
Normal Approximation ◽  
Random Variables ◽  
Random Variable ◽  
New Technique ◽  
Fourth Moment ◽  
Markov Diffusion ◽  
Carré Du Champ ◽  
A New Technique

In this paper, we consider a quantitative fourth moment theorem for functions (random variables) defined on the Markov triple E , μ , Γ , where μ is a probability measure and Γ is the carré du champ operator. A new technique is developed to derive the fourth moment bound in a normal approximation on the random variable of a general Markov diffusion generator, not necessarily belonging to a fixed eigenspace, while previous works deal with only random variables to belong to a fixed eigenspace. As this technique will be applied to the works studied by Bourguin et al. (2019), we obtain the new result in the case where the chaos grade of an eigenfunction of Markov diffusion generator is greater than two. Also, we introduce the chaos grade of a new notion, called the lower chaos grade, to find a better estimate than the previous one.

A Note on the Accuracy of Normal Approximation of Random Quantities

2021 ◽  
Vol 73 (1) ◽  
pp. 62-67
Author(s):  
Ibrahim A. Ahmad ◽  
A. R. Mugdadi
Keyword(s):  
Sample Size ◽  
Order Statistic ◽  
Normal Approximation ◽  
Order Approximation ◽  
Random Variables ◽  
Random Variable ◽  
Limiting Distribution ◽  
Exact Order ◽  
Fixed Sample ◽  
Independent Identically Distributed

For a sequence of independent, identically distributed random variable (iid rv's) [Formula: see text] and a sequence of integer-valued random variables [Formula: see text], define the random quantiles as [Formula: see text], where [Formula: see text] denote the largest integer less than or equal to [Formula: see text], and [Formula: see text] the [Formula: see text]th order statistic in a sample [Formula: see text] and [Formula: see text]. In this note, the limiting distribution and its exact order approximation are obtained for [Formula: see text]. The limiting distribution result we obtain extends the work of several including Wretman[Formula: see text]. The exact order of normal approximation generalizes the fixed sample size results of Reiss[Formula: see text]. AMS 2000 subject classification: 60F12; 60F05; 62G30.

scholarly journals The limits of normal approximation for adult height

2021 ◽  
Author(s):  
Sergei A. Slavskii ◽  
Ivan A. Kuznetsov ◽  
Tatiana I. Shashkova ◽  
Georgii A. Bazykin ◽  
Tatiana I. Axenovich ◽  
...  
Keyword(s):  
Complex Traits ◽  
Adult Height ◽  
Normal Approximation ◽  
Classical Model ◽  
Model Complexity ◽  
Test Case ◽  
Genetic Studies ◽  
Quantitative Genetic ◽  
Profound Influence ◽  
Log Normal

AbstractAdult height inspired the first biometrical and quantitative genetic studies and is a test-case trait for understanding heritability. The studies of height led to formulation of the classical polygenic model, that has a profound influence on the way we view and analyse complex traits. An essential part of the classical model is an assumption of additivity of effects and normality of the distribution of the residuals. However, it may be expected that the normal approximation will become insufficient in bigger studies. Here, we demonstrate that when the height of hundreds of thousands of individuals is analysed, the model complexity needs to be increased to include non-additive interactions between sex, environment and genes. Alternatively, the use of log-normal approximation allowed us to still use the additive effects model. These findings are important for future genetic and methodologic studies that make use of adult height as an exemplar trait.

The fourth moment of Ramanujan ?-function

1983 ◽  
Vol 266 (2) ◽  
pp. 233-239 ◽  
Author(s):  
Carlos J. Moreno ◽  
Freydoon Shahidi
Keyword(s):  
Fourth Moment
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