An I-Dimensional Limiting Distribution Function of Largest Values and Its Relevance to the Statistical Theory of Extremes

1981 ◽  
pp. 389-410 ◽  
Author(s):  
M. Ivette Gomes
Keyword(s):  
Distribution Function ◽  
Statistical Theory ◽  
Limiting Distribution ◽  
Statistical Theory Of Extremes

The exponential rate of convergence of the distribution of the maximum of a random walk

1975 ◽  
Vol 12 (02) ◽  
pp. 279-288 ◽  
Author(s):  
N. Veraverbeke ◽  
J. L. Teugels
Keyword(s):  
Distribution Function ◽  
Random Walk ◽  
Rate Of Convergence ◽  
Exponential Convergence ◽  
Random Variables ◽  
Limiting Distribution ◽  
Partial Sums ◽  
Exponential Rate ◽  
Distribution Of The Maximum ◽  
Exponential Rate Of Convergence

Let Gn (x) be the distribution function of the maximum of the successive partial sums of independent and identically distributed random variables and G(x) its limiting distribution function. Under conditions, typical for complete exponential convergence, the decay of Gn (x) — G(x) is asymptotically equal to c.H(x)n −3/2 γn as n → ∞ where c and γ are known constants and H(x) is a function solely depending on x.

Probability distribution function of complex systems

2018 ◽  
Vol 32 (03) ◽  
pp. 1850022
Author(s):  
Jincan Chen ◽  
Tie Liu ◽  
Zhifu Huang ◽  
Guozhen Su
Keyword(s):  
Complex System ◽  
Distribution Function ◽  
Statistical Theory ◽  
Probability Distribution ◽  
Complex Systems ◽  
Probability Distribution Function ◽  
Single Complex

Based on the probability distribution observed in some complex systems and an assumption that the entropies of complex systems satisfy a pseudo additivity, it is expounded that a new probability distribution function is suitable for not only a single complex system but also a coupling complex system, and consequently, a statistical theory of complex systems can be established in the extensive-like framework.

Asymptotic approximations for coupon collectors

2009 ◽  
Vol 46 (1) ◽  
pp. 61-96
Author(s):  
Anna Pósfai ◽  
Sándor Csörgő
Keyword(s):  
Distribution Function ◽  
Asymptotic Expansion ◽  
Rate Of Convergence ◽  
Limit Theorems ◽  
Random Number ◽  
Limiting Distribution ◽  
Asymptotic Approximations ◽  
The One ◽  
First Time

A collector samples with replacement a set of n ≧ 2 distinct coupons until he has n − m , 0 ≦ m < n , distinct coupons for the first time. We refine the limit theorems concerning the standardized random number of necessary draws if n → ∞ and m is fixed: we give a one-term asymptotic expansion of the distribution function in question, providing a better approximation of it, than the one given by the limiting distribution function, and proving in particular that the rate of convergence in these limiting theorems is of order (log n )/ n .

Non-Equilibrium Statistical Theory of Void Microstructure Evolution in Irradiated Metals

2013 ◽  
Vol 364 ◽  
pp. 568-572
Author(s):  
Xiang Hui Guo ◽  
Hai Yun Hu
Keyword(s):  
Distribution Function ◽  
Statistical Theory ◽  
Density Distribution ◽  
Probability Density ◽  
Microstructure Evolution ◽  
Void Growth ◽  
Probability Density Distribution ◽  
Density Distribution Function ◽  
Void Evolution ◽  
Non Equilibrium

The non-equilibrium statistical theory was used as a theoretical approach to modeling and predicting void evolution in metal materials. Fokker-Plank equation was introduced as the kinetic equation for the void evolution, from which the probability density distribution function of voids could be obtained. From the micro-mechanism of metal's irradiation damage, void growth rate equation was obtained using spherical Weilv model and control diffusion theory, and then was simplified based on appropriate assumptions. According to the probability density distribution function of void, a series of macro-mechanical characteristics caused by void growth can be calculated, such as: the critical radius of the void nucleation, the average radius of void. Thus the correlation between the void microstructure evolution and the macroscopic properties of metals can be achieved.

The exponential rate of convergence of the distribution of the maximum of a random walk

1975 ◽  
Vol 12 (2) ◽  
pp. 279-288 ◽  
Author(s):  
N. Veraverbeke ◽  
J. L. Teugels
Keyword(s):  
Distribution Function ◽  
Random Walk ◽  
Rate Of Convergence ◽  
Exponential Convergence ◽  
Random Variables ◽  
Limiting Distribution ◽  
Partial Sums ◽  
Exponential Rate ◽  
Distribution Of The Maximum ◽  
Exponential Rate Of Convergence

Let Gn(x) be the distribution function of the maximum of the successive partial sums of independent and identically distributed random variables and G(x) its limiting distribution function. Under conditions, typical for complete exponential convergence, the decay of Gn(x) — G(x) is asymptotically equal to c.H(x)n−3/2γn as n → ∞ where c and γ are known constants and H(x) is a function solely depending on x.

On the limiting distribution of a supercritical branching process in a random environment

1992 ◽  
Vol 29 (03) ◽  
pp. 499-518 ◽  
Author(s):  
Ben Hambly
Keyword(s):  
Distribution Function ◽  
Laplace Transform ◽  
Family Size ◽  
Random Environment ◽  
Branching Process ◽  
Random Variable ◽  
Limiting Distribution ◽  
The Laplace Transform

We consider an increasing supercritical branching process in a random environment and obtain bounds on the Laplace transform and distribution function of the limiting random variable. There are two possibilities that can be distinguished depending on the nature of the component distributions of the environment. If the minimum family size of each is 1, the growth will be as a power depending on a parameter α. If the minimum family sizes of some are greater than 1, it will be exponential, depending on a parameter γ. We obtain bounds on the distribution function analogous to those found for the simple Galton-Watson case. It is not possible to obtain exact estimates and we are only able to obtain bounds to within ε of the parameters.

Statistical Theory for Predicting the Failure of Brittle Materials

1991 ◽  
Vol 58 (1) ◽  
pp. 43-49 ◽  
Author(s):  
S. She ◽  
J. D. Landes ◽  
J. A. M. Boulet ◽  
J. E. Stoneking
Keyword(s):  
Distribution Function ◽  
Statistical Theory ◽  
Failure Probability ◽  
Statistical Models ◽  
Brittle Materials ◽  
Boundary Carbide ◽  
Physical Consideration ◽  
Different Temperatures ◽  
Statistical Distribution Function ◽  
Carbide Particles

Statistical models for predicting failure probability of brittle materials are investigated. A formula is derived from a physical consideration for the fracture of microcracks in materials based on the general forms of a fracture criterion and a statistical distribution function incorporating the weakest link principle. The relationships of this model and other statistical models in the literature are discussed; they were found to be equivalent for isotropic materials in which microcracks are randomly distributed in all directions. The statistical model is also used in a failure analysis of the round-notch four-point bending specimen made of an AISI 1008 steel. The grain boundary carbide particles are considered to be microcracks in the plastic zone near the notch tip. The distribution function in the statistical theory is derived from the density and size distribution of carbide particles in the steel. The statistical theory for a triaxial stress state is used to predict the failure probability for any given load on the specimen. The failure loads (loads corresponding to 50 percent of failure probability) are calculated for the specimen at different temperatures. The results are compared with experimental data; good agreement is obtained.

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