Super Generalized Central Limit Theorem —Limit Distributions for Sums of Non-identical Random Variables with Power Laws—

Shintani, Masaru; Umeno, Ken

doi:10.7566/jpsj.87.043003

Cited by 13 publications

(8 citation statements)

References 19 publications

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“…It is also known that as another universal law, there is a statistical law, such as the super generalized central limit theorem based on the power law [16,17],…”

Section: Discussionmentioning

confidence: 99%

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The Cross-Industry Universal Laws of Exponential Booking Curves

Shintani

Umeno

2021

Preprint

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In recent years, the e-commerce market has grown with the spread of the internet worldwide every year. Accordingly, in service industries, purchasing products with reservations has become common. With the spread of online reservations, the booking curve, which is the concept of the time series in the cumulative number of reservations and has been used for sales optimization in the airline ticket and hotel industries, has been used in various industries. Booking curves in specific industries have been studied, but a universally applicable model across various industries has not been developed. In this study, we show that booking curves can be modeled universally by the exponential decay function, and we also show that the model is valid by using real data from some industries before and after the COVID-19 pandemic, that is, under completely different market conditions. The cross-industry exponential laws of booking curves constitute an important discovery in regard to mathematical laws in the social sciences and can be applied to give leading microeconomic indicators.

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“…It is also known that as another universal law, there is a statistical law, such as the super generalized central limit theorem based on the power law [16,17],…”

Section: Discussionmentioning

confidence: 99%

“…as the super generalized central limit theorem based on the power law [16,17], which is a ubiquitous characteristic found in such systems. In the present case, considering the generality based on Eqs.…”

Section: Discussionmentioning

confidence: 99%

The Cross-Industry Universal Laws of Exponential Booking Curves

Shintani

Umeno

2021

Preprint

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“…Below we first show that when n is sufficiently large, we can apply generalized central limit theorem (GCLT) from Shintani and Umeno (2018) to approximate the null distribution of T CA tr below.…”

Section: Large Negative Penalty Issue In Cauchy and A Truncated Cauch...mentioning

confidence: 99%

Heavy-tailed distribution for combining dependent $p$-values with asymptotic robustness

Fang¹,

Tseng²,

Chang³

2021

Preprint

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The issue of combining individual p-values to aggregate multiple small effects is prevalent in many scientific investigations and is a long-standing statistical topic. Many classical methods are designed for combining independent and frequent signals in a traditional meta-analysis sense using the sum of transformed p-values with the transformation of light-tailed distributions, in which Fisher's method and Stouffer's method are the most well-known. Since the early 2000, advances in big data promoted methods to aggregate independent, sparse and weak signals, such as the renowned higher criticism and Berk-Jones tests. Recently, Liu and Xie (2020) and Wilson (2019) independently proposed Cauchy and harmonic mean combination tests to robustly combine p-values under "arbitrary" dependency structure, where a notable application is to combine p-values from a set of often correlated SNPs in genome-wide association studies. The proposed tests are the transformation of heavy-tailed distributions for improved power with the sparse signal. It calls for a natural question to investigate heavy-tailed distribution transformation, to understand the connection among existing methods, and to explore the conditions for a method to possess robustness to dependency. In this paper, we investigate the regularly varying distribution, which is a rich family of heavy-tailed distribution and includes Pareto distribution as a special case. We show that only an equivalent class of Cauchy and harmonic mean tests have sufficient robustness to dependency in a practical sense. We also show an issue caused by large negative penalty in the Cauchy method and propose a simple, yet practical modification. Finally, we present simulations and apply to a neuroticism GWAS application to verify the discovered theoretical insights and provide practical guidance.

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“…The statistical properties of the wave function envelope in the long-chain limit (n → ∞) can be determined by appealing to the central limit theorem (CLT) and its generalized version [33,34]. The CLT specifies that the form of the limiting distribution for sums of random variables is a Gaussian N (µ, σ 2 ), with some mean µ and standard deviation σ, so long as each of the random variables has finite variance.…”

Section: The Zero Energy Statementioning

confidence: 99%

Beyond the universal Dyson singularity for 1-D chains with hopping disorder

Krishna,

Bhatt

2021

Preprint

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We study a simple non-interacting nearest neighbor tight-binding model in one dimension with disorder, where the hopping terms are chosen randomly. This model exhibits a well-known singularity at the band center both in the density of states and localization length. If the probability distribution of the hopping terms is well-behaved, then the singularities exhibit universal behavior, the functional form of which was first discovered by Freeman Dyson in the context of a chain of classical harmonic oscillators. We show here that this universal form can be violated in a tunable manner if the hopping elements are chosen from a divergent probability distribution. We also demonstrate a connection between a breakdown of universality in this quantum problem and an analogous scenario in the classical domain -that of random walks and diffusion with anomalous exponents.

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Super Generalized Central Limit Theorem —Limit Distributions for Sums of Non-identical Random Variables with Power Laws—

Cited by 13 publications

References 19 publications

The Cross-Industry Universal Laws of Exponential Booking Curves

The Cross-Industry Universal Laws of Exponential Booking Curves

Heavy-tailed distribution for combining dependent $p$-values with asymptotic robustness

Beyond the universal Dyson singularity for 1-D chains with hopping disorder

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