From the central limit theorem to heavy-tailed distributions

2003 ◽  
Vol 40 (3) ◽  
pp. 803-806 ◽  
Author(s):  
Jinwen Chen
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Power Law ◽  
Power Law Distribution ◽  
The Central Limit Theorem ◽  
Heavy Tailed Distributions ◽  
Heavy Tailed ◽  
Random Indices ◽  
Power Law Distributions

It has been observed that in many practical situations randomly stopped products of random variables have power law distributions. In this note we show that, in order for such a product to have a power law distribution, the only random indices are the exponentially distributed ones. We also consider a more general problem, which is closely related to problems concerning transformation from the central limit theorem to heavy-tailed distributions.

From the central limit theorem to heavy-tailed distributions

2003 ◽  
Vol 40 (03) ◽  
pp. 803-806
Author(s):  
Jinwen Chen
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Power Law ◽  
Power Law Distribution ◽  
The Central Limit Theorem ◽  
Heavy Tailed Distributions ◽  
Heavy Tailed ◽  
Random Indices ◽  
Power Law Distributions

It has been observed that in many practical situations randomly stopped products of random variables have power law distributions. In this note we show that, in order for such a product to have a power law distribution, the only random indices are the exponentially distributed ones. We also consider a more general problem, which is closely related to problems concerning transformation from the central limit theorem to heavy-tailed distributions.

An analysis of power law distributions and tipping points during the global financial crisis

2018 ◽  
Vol 13 (1) ◽  
pp. 80-91 ◽  
Author(s):  
Yifei Li ◽  
Lei Shi ◽  
Neil Allan ◽  
John Evans
Keyword(s):  
Financial Crisis ◽  
Power Law ◽  
Global Financial Crisis ◽  
Operational Risk ◽  
Tipping Point ◽  
Power Law Distribution ◽  
Heavy Tailed Distributions ◽  
Heavy Tailed ◽  
The Global Financial Crisis ◽  
Power Law Distributions

AbstractHeavy-tailed distributions have been observed for various financial risks and papers have observed that these heavy-tailed distributions are power law distributions. The breakdown of a power law distribution is also seen as an indicator of a tipping point being reached and a system then moves from stability through instability to a new equilibrium. In this paper, we analyse the distribution of operational risk losses in US banks, credit defaults in US corporates and market risk events in the US during the global financial crisis (GFC). We conclude that market risk and credit risk do not follow a power law distribution, and even though operational risk follows a power law distribution, there is a better distribution fit for operational risk. We also conclude that whilst there is evidence that credit defaults and market risks did reach a tipping point, operational risk losses did not. We conclude that the government intervention in the banking system during the GFC was a possible cause of banks avoiding a tipping point.

scholarly journals Corrections to the Central Limit Theorem for Heavy-tailed Probability Densities

2011 ◽  
Vol 24 (4) ◽  
pp. 895-927 ◽  
Author(s):  
Henry Lam ◽  
Jose Blanchet ◽  
Damian Burch ◽  
Martin Z. Bazant
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
The Central Limit Theorem ◽  
Probability Densities ◽  
Heavy Tailed

scholarly journals Central limit theorem for linear processes generated by IID random variables under the sub-linear expectation

2021 ◽  
Vol 36 (2) ◽  
pp. 243-255
Author(s):  
Wei Liu ◽  
Yong Zhang
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Invariance Principle ◽  
Random Variables ◽  
Probability Measures ◽  
Linear Processes ◽  
The Central Limit Theorem

AbstractIn this paper, we investigate the central limit theorem and the invariance principle for linear processes generated by a new notion of independently and identically distributed (IID) random variables for sub-linear expectations initiated by Peng [19]. It turns out that these theorems are natural and fairly neat extensions of the classical Kolmogorov’s central limit theorem and invariance principle to the case where probability measures are no longer additive.

scholarly journals Does a central limit theorem hold for the k-skeleton of Poisson hyperplanes in hyperbolic space?

2021 ◽  
Author(s):  
Felix Herold ◽  
Daniel Hug ◽  
Christoph Thäle
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Hyperbolic Space ◽  
Central Limit ◽  
Hausdorff Measure ◽  
Rates Of Convergence ◽  
Limit Problem ◽  
Dimensional Hausdorff Measure ◽  
The Central Limit Theorem ◽  
Order Quantities

AbstractPoisson processes in the space of $$(d-1)$$ ( d - 1 ) -dimensional totally geodesic subspaces (hyperplanes) in a d-dimensional hyperbolic space of constant curvature $$-1$$ - 1 are studied. The k-dimensional Hausdorff measure of their k-skeleton is considered. Explicit formulas for first- and second-order quantities restricted to bounded observation windows are obtained. The central limit problem for the k-dimensional Hausdorff measure of the k-skeleton is approached in two different set-ups: (i) for a fixed window and growing intensities, and (ii) for fixed intensity and growing spherical windows. While in case (i) the central limit theorem is valid for all $$d\ge 2$$ d ≥ 2 , it is shown that in case (ii) the central limit theorem holds for $$d\in \{2,3\}$$ d ∈ { 2 , 3 } and fails if $$d\ge 4$$ d ≥ 4 and $$k=d-1$$ k = d - 1 or if $$d\ge 7$$ d ≥ 7 and for general k. Also rates of convergence are studied and multivariate central limit theorems are obtained. Moreover, the situation in which the intensity and the spherical window are growing simultaneously is discussed. In the background are the Malliavin–Stein method for normal approximation and the combinatorial moment structure of Poisson U-statistics as well as tools from hyperbolic integral geometry.

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