scholarly journals On the Asymptotic Distribution of the Sequences of Random Variables with Random Indices

1971 ◽  
Vol 42 (6) ◽  
pp. 2018-2028 ◽  
Author(s):  
Silviu Guiasu
Keyword(s):  
Asymptotic Distribution ◽  
Random Variables ◽  
Random Indices

Limit distributions for compounded sums of extreme order statistics

1992 ◽  
Vol 29 (03) ◽  
pp. 557-574 ◽  
Author(s):  
Jan Beirlant ◽  
Jozef L. Teugels
Keyword(s):  
Order Statistics ◽  
Asymptotic Distribution ◽  
Counting Process ◽  
Extreme Values ◽  
Random Variables ◽  
Limit Distributions ◽  
Extreme Order Statistics ◽  
Renewal Counting Process ◽  
Independent Identically Distributed

Let X (1) ≦ X (2) ≦ ·· ·≦ X (N(t)) be the order statistics of the first N(t) elements from a sequence of independent identically distributed random variables, where {N(t); t ≧ 0} is a renewal counting process independent of the sequence of X's. We give a complete description of the asymptotic distribution of sums made from the top kt extreme values, for any sequence kt such that kt → ∞, kt /t → 0 as t → ∞. We discuss applications to reinsurance policies based on large claims.

scholarly journals Asymptotic distribution of sum and maximum for Gaussian processes

1999 ◽  
Vol 36 (4) ◽  
pp. 1031-1044 ◽  
Author(s):  
Hwai-Chung Ho ◽  
William P. McCormick
Keyword(s):  
Asymptotic Distribution ◽  
Gaussian Processes ◽  
Continuous Time ◽  
Joint Distribution ◽  
Covariance Function ◽  
Random Variables ◽  
Regularity Conditions ◽  
Gaussian Sequence ◽  
Stationary Gaussian Sequence ◽  
Standard Normal

Let {Xn, n ≥ 0} be a stationary Gaussian sequence of standard normal random variables with covariance function r(n) = EX0Xn. Let Under some mild regularity conditions on r(n) and the condition that r(n)lnn = o(1) or (r(n)lnn)−1 = O(1), the asymptotic distribution of is obtained. Continuous-time results are also presented as well as a tube formula tail area approximation to the joint distribution of the sum and maximum.

Time spent below a random threshold by a Poisson driven sequence of observations

2003 ◽  
Vol 40 (03) ◽  
pp. 807-814 ◽  
Author(s):  
S. N. U. A. Kirmani ◽  
Jacek Wesołowski
Keyword(s):  
Poisson Process ◽  
Asymptotic Distribution ◽  
Random Variables ◽  
Nonhomogeneous Poisson Process ◽  
System Level ◽  
Homogeneous Poisson Process ◽  
Random Threshold ◽  
The Mean ◽  
Current Value ◽  
Independent And Identically Distributed

The mean and the variance of the time S(t) spent by a system below a random threshold until t are obtained when the system level is modelled by the current value of a sequence of independent and identically distributed random variables appearing at the epochs of a nonhomogeneous Poisson process. In the case of the homogeneous Poisson process, the asymptotic distribution of S(t)/t as t → ∞ is derived.

On road traffic with free overtaking

1969 ◽  
Vol 6 (3) ◽  
pp. 524-549 ◽  
Author(s):  
Torbjörn Thedéen
Keyword(s):  
Asymptotic Distribution ◽  
Road Traffic ◽  
Random Variables ◽  
Time Axis ◽  
Independent Identically Distributed

The cars are considered as points on an infinite road with no intersections. They can overtake each other without any delay and they travel at constant speeds. These are independent identically distributed random variables also independent of the initial positions of the cars. The main purpose of the paper is the study of the asymptotic distribution for the number of overtakings (and/or meetings) in increasing rectangles in the time-road plane. Under the assumption of (weighted) Poisson distributed cars along the time-axis we deduce the asymptotic distribution of the standardized number of overtakings (and/or meetings) for large rectangles in the time-road plane. Lastly we shall indicate an application of the results.

Extreme values of phase-type and mixed random variables with parallel-processing examples

1999 ◽  
Vol 36 (01) ◽  
pp. 194-210 ◽  
Author(s):  
Sungyeol Kang ◽  
Richard F. Serfozo
Keyword(s):  
Parallel Processing ◽  
Asymptotic Distribution ◽  
Extreme Values ◽  
Random Variables ◽  
Extreme Value Distribution ◽  
Rates Of Convergence ◽  
Value Theory ◽  
Extreme Value ◽  
Independent Random Variables ◽  
Phase Type

A basic issue in extreme value theory is the characterization of the asymptotic distribution of the maximum of a number of random variables as the number tends to infinity. We address this issue in several settings. For independent identically distributed random variables where the distribution is a mixture, we show that the convergence of their maxima is determined by one of the distributions in the mixture that has a dominant tail. We use this result to characterize the asymptotic distribution of maxima associated with mixtures of convolutions of Erlang distributions and of normal distributions. Normalizing constants and bounds on the rates of convergence are also established. The next result is that the distribution of the maxima of independent random variables with phase type distributions converges to the Gumbel extreme-value distribution. These results are applied to describe completion times for jobs consisting of the parallel-processing of tasks represented by Markovian PERT networks or task-graphs. In these contexts, which arise in manufacturing and computer systems, the job completion time is the maximum of the task times and the number of tasks is fairly large. We also consider maxima of dependent random variables for which distributions are selected by an ergodic random environment process that may depend on the variables. We show under certain conditions that their distributions may converge to one of the three classical extreme-value distributions. This applies to parallel-processing where the subtasks are selected by a Markov chain.

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