Limit Theorems for Identically Distributed Summands Asuming the Convergence of the Distribution Functions on a Half Axis

1980 ◽  
Vol 24 (4) ◽  
pp. 693-711 ◽  
Author(s):  
H.-J. Rossberg
Keyword(s):  
Limit Theorems ◽  
Distribution Functions ◽  
Half Axis

scholarly journals On the time spent above a level by Brownian motion with negative drift

1986 ◽  
Vol 18 (04) ◽  
pp. 1017-1018 ◽  
Author(s):  
J.-P. Imhof
Keyword(s):  
Brownian Motion ◽  
Limit Theorems ◽  
Distribution Functions ◽  
Probability Densities ◽  
Negative Drift

Limit theorems of Berman involve the total time spent by Brownian motion with negative drift above a fixed or exponentially distributed negative level. We give explicitly the probability densities and distribution functions, obtained via an equivalence of laws.

Regenerative processes in the infinite mean cycle case

2001 ◽  
Vol 38 (01) ◽  
pp. 165-179 ◽  
Author(s):  
K. V. Mitov ◽  
N. M. Yanev
Keyword(s):  
Renewal Process ◽  
Limit Theorems ◽  
Domain Of Attraction ◽  
Regenerative Process ◽  
Distribution Functions ◽  
Regenerative Processes ◽  
Alternating Renewal Process ◽  
Different Types ◽  
Mean Cycle ◽  
Regenerative Cycle

A class of non-negative alternating regenerative processes is considered, where the process stays at zero random time (waiting period), then it jumps to a random positive level and hits zero after some random period (life period), depending on the evolution of the process. It is assumed that the waiting time and the lifetime belong to the domain of attraction of stable laws with parameters in the interval (½,1]. An integral representation for the distribution functions of the regenerative process is obtained, using the spent time distributions of the corresponding alternating renewal process. Given the asymptotic behaviour of the process in the regenerative cycle, different types of limiting distributions are proved, applying some new results for the corresponding renewal process and two limit theorems for the convergence in distribution.

The order of the approximation to a Wiener process by its Fourier series

1982 ◽  
Vol 92 (3) ◽  
pp. 547-562 ◽  
Author(s):  
Peter Hall
Keyword(s):  
Distribution Function ◽  
Fourier Series ◽  
Wiener Process ◽  
Limit Theorems ◽  
Brownian Bridge ◽  
Distribution Functions ◽  
Practical Significance ◽  
Unusual Property ◽  
Theoretical Interest ◽  
Statistical Procedures

AbstractLet F be a distribution function with support confined to ( − π, π), W a Wiener process on (0, 1), and W0 a Brownian bridge. We examine the order of the approximation to W(F) and W0(F) by their Fourier series. The results we obtain are of theoretical interest since the Wiener process possesses the unusual property of satisfying a Lipschitz condition of order ∝ for each 0 < ∝ < ½, but not one of order ∝. = ½. The classical results on the rate of convergence of Fourier series are not sufficiently specific to provide an exact rate in the case of the Wiener process. The work also has practical significance because many statistical procedures can be closely approximated by functions of a Wiener process, and several procedures rely heavily on Fourier series methods. Thus, some of our results may be applied to obtain limit theorems for quantities such as trigonometric series estimates of density and distribution functions.

Regenerative processes in the infinite mean cycle case

2001 ◽  
Vol 38 (1) ◽  
pp. 165-179 ◽  
Author(s):  
K. V. Mitov ◽  
N. M. Yanev
Keyword(s):  
Renewal Process ◽  
Limit Theorems ◽  
Domain Of Attraction ◽  
Regenerative Process ◽  
Distribution Functions ◽  
Regenerative Processes ◽  
Alternating Renewal Process ◽  
Different Types ◽  
Mean Cycle ◽  
Regenerative Cycle

A class of non-negative alternating regenerative processes is considered, where the process stays at zero random time (waiting period), then it jumps to a random positive level and hits zero after some random period (life period), depending on the evolution of the process. It is assumed that the waiting time and the lifetime belong to the domain of attraction of stable laws with parameters in the interval (½,1]. An integral representation for the distribution functions of the regenerative process is obtained, using the spent time distributions of the corresponding alternating renewal process. Given the asymptotic behaviour of the process in the regenerative cycle, different types of limiting distributions are proved, applying some new results for the corresponding renewal process and two limit theorems for the convergence in distribution.

scholarly journals A Note on the Central Limit Theorems for Dependent Random Variables

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Yilun Shang
Keyword(s):  
Central Limit ◽  
Limit Theorems ◽  
Random Variables ◽  
Distribution Functions ◽  
Central Limit Theorems ◽  
Cumulative Distribution ◽  
Characteristic Functions ◽  
Mixing Conditions ◽  
Cumulative Distribution Functions ◽  
Probability And Statistics

Classical central limit theorem is considered the heart of probability and statistics theory. Our interest in this paper is central limit theorems for functions of random variables under mixing conditions. We impose mixing conditions on the differences between the joint cumulative distribution functions and the product of the marginal cumulative distribution functions. By using characteristic functions, we obtain several limit theorems extending previous results.

scholarly journals A generalization of the global limit theorems of R. P. Agnew

1988 ◽  
Vol 11 (2) ◽  
pp. 365-374
Author(s):  
Andrew Rosalsky
Keyword(s):  
Weak Convergence ◽  
Limit Theorems ◽  
Nondecreasing Function ◽  
Distribution Functions ◽  
Independent Random Variables ◽  
Limiting Behavior ◽  
The Central Limit Theorem ◽  
Almost Everywhere ◽  
The Relationship ◽  
Special Case

For distribution functions{Fn,n≥0}, the relationship between the weak convergence ofFntoF0and the convergence of∫Rϕ(|Fn−F0|)dxto0is studied whereϕis a nonnegative, nondecreasing function. Sufficient and, separately, necessary conditions are given for the latter convergence thereby generalizing the so-called global limit theorems of Agnew whereinϕ(t)=|t|r. The sufficiency results are shown to be sharp and, as a special case, yield a global version of the central limit theorem for independent random variables obeying the Liapounov condition. Moreover, weak convergence of distribution functions is characterized in terms of their almost everywhere limiting behavior with respect to Lebesgue measure on the line.

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