A stochastic ordering of partial sums of independent random variables and of some random processes

1992 ◽  
Vol 29 (03) ◽  
pp. 645-654 ◽  
Author(s):  
Philip J. Boland ◽  
Frank Proschan ◽  
Y. L. Tong
Keyword(s):  
Random Variables ◽  
Stochastic Ordering ◽  
Random Processes ◽  
Reliability Theory ◽  
Independent Random Variables ◽  
Partial Sums ◽  
Similar Argument ◽  
First Passage ◽  
Wiener Processes ◽  
Likelihood Ratio Ordering

In this paper we first prove an arrangement-decreasing property of partial sums of independent random variables when they are partially ordered through the likelihood ratio ordering. We then apply a similar argument to obtain a stochastic ordering of random processes via a comparison of their parameter functions, with special applications to Poisson and Wiener processes. Finally, in Section 4 we present some applications in reliability theory, queueing, and first-passage problems.

A stochastic ordering of partial sums of independent random variables and of some random processes

1992 ◽  
Vol 29 (3) ◽  
pp. 645-654 ◽  
Author(s):  
Philip J. Boland ◽  
Frank Proschan ◽  
Y. L. Tong
Keyword(s):  
Random Variables ◽  
Stochastic Ordering ◽  
Random Processes ◽  
Reliability Theory ◽  
Independent Random Variables ◽  
Partial Sums ◽  
Similar Argument ◽  
First Passage ◽  
Wiener Processes ◽  
Likelihood Ratio Ordering

In this paper we first prove an arrangement-decreasing property of partial sums of independent random variables when they are partially ordered through the likelihood ratio ordering. We then apply a similar argument to obtain a stochastic ordering of random processes via a comparison of their parameter functions, with special applications to Poisson and Wiener processes. Finally, in Section 4 we present some applications in reliability theory, queueing, and first-passage problems.

Characterization of Some First Passage Times Using Log-Concavity and Log-Convexity as Aging Notions

1987 ◽  
Vol 1 (3) ◽  
pp. 279-291 ◽  
Author(s):  
Moshe Shaked ◽  
J. George Shanthikumar
Keyword(s):  
Branching Processes ◽  
Stochastic Ordering ◽  
Reliability Theory ◽  
First Passage Times ◽  
First Passage ◽  
Likelihood Ratio Ordering ◽  
New Better Than Used ◽  
Better Than ◽  
Passage Times

An interpretation of log-concavity and log-convexity as aging notions is given in this paper. It imitates a stochastic ordering characterization of the NBU (new better than used) and the NWU (new worse than used) notions but stochastic ordering is now replaced by the likelihood ratio ordering. The new characterization of log-concavity and log-convexity sheds new light on these properties and enables one to obtain intuitively simple proofs of the log-convexity and log-concavity of some first passage times of interest in branching processes and in reliability theory.

A characterization of the negative exponential distribution with application to reliability theory

1981 ◽  
Vol 18 (3) ◽  
pp. 652-659 ◽  
Author(s):  
M. J. Phillips
Keyword(s):  
Exponential Distribution ◽  
Random Variables ◽  
Reliability Theory ◽  
The Other ◽  
Independent Random Variables ◽  
System Availability ◽  
Failure Times ◽  
Negative Exponential Distribution ◽  
Negative Exponential

The negative exponential distribution is characterized in terms of two independent random variables. Only one of the random variables has a negative exponential distribution whilst the other can belong to a wide class of distributions. This result is then applied to two models for the reliability of a system of two modules subject to revealed and unrevealed faults to show when the models are equivalent. It is also shown, under certain conditions, that the system availability is only independent of the distribution of revealed failure times in one module when unrevealed failure times in the other module have a negative exponential distribution.

A characterization of the negative exponential distribution with application to reliability theory

1981 ◽  
Vol 18 (03) ◽  
pp. 652-659 ◽  
Author(s):  
M. J. Phillips
Keyword(s):  
Exponential Distribution ◽  
Random Variables ◽  
Reliability Theory ◽  
The Other ◽  
Independent Random Variables ◽  
System Availability ◽  
Failure Times ◽  
Negative Exponential Distribution ◽  
Negative Exponential

The negative exponential distribution is characterized in terms of two independent random variables. Only one of the random variables has a negative exponential distribution whilst the other can belong to a wide class of distributions. This result is then applied to two models for the reliability of a system of two modules subject to revealed and unrevealed faults to show when the models are equivalent. It is also shown, under certain conditions, that the system availability is only independent of the distribution of revealed failure times in one module when unrevealed failure times in the other module have a negative exponential distribution.

The Distribution of the Maximum of Partial Sums of Independent Random Variables

1950 ◽  
Vol 2 ◽  
pp. 375-384 ◽  
Author(s):  
Mark Kac ◽  
Harry Pollard
Keyword(s):  
Random Variables ◽  
Independent Random Variables ◽  
Partial Sums ◽  
Distribution Of The Maximum

1. The problem. It has been shown [1] that if Xi, + X2, … are independent random variables each of density then(1.1)

Comparing sums of exchangeable Bernoulli random variables

1996 ◽  
Vol 33 (02) ◽  
pp. 285-310 ◽  
Author(s):  
Claude Lefèvre ◽  
Sergey Utev
Keyword(s):  
Iterative Procedure ◽  
First Passage Time ◽  
Random Variables ◽  
Passage Time ◽  
Partial Sums ◽  
First Passage ◽  
Sampling Schemes ◽  
Bernoulli Random Variables ◽  
Discrete Random Variables ◽  
Comparison Results

The paper is first concerned with a comparison of the partial sums associated with two sequences of n exchangeable Bernoulli random variables. It then considers a situation where such partial sums are obtained through an iterative procedure of branching type stopped at the first-passage time in a linearly decreasing upper barrier. These comparison results are illustrated with applications to certain urn models, sampling schemes and epidemic processes. A key tool is a non-standard hierarchical class of stochastic orderings between discrete random variables valued in {0, 1,· ··, n}.

scholarly journals On Feller's criterion for the law of the iterated logarithm

1994 ◽  
Vol 17 (2) ◽  
pp. 323-340 ◽  
Author(s):  
Deli Li ◽  
M. Bhaskara Rao ◽  
Xiangchen Wang
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Uniform Estimate ◽  
Random Variables ◽  
Independent Random Variables ◽  
Partial Sums ◽  
The Central Limit Theorem ◽  
Iterated Logarithm ◽  
The Law

Combining Feller's criterion with a non-uniform estimate result in the context of the Central Limit Theorem for partial sums of independent random variables, we obtain several results on the Law of the Iterated Logarithm. Two of these results refine corresponding results of Wittmann (1985) and Egorov (1971). In addition, these results are compared with the corresponding results of Teicher (1974), Tomkins (1983) and Tomkins (1990)

scholarly journals Maxima of Sums of Heavy-Tailed Random Variables

2002 ◽  
Vol 32 (1) ◽  
pp. 43-55 ◽  
Author(s):  
K.W. Ng ◽  
Q.H. Tang ◽  
H. Yang
Keyword(s):  
Asymptotic Properties ◽  
Random Variables ◽  
Independent Random Variables ◽  
Partial Sums ◽  
Tail Probabilities ◽  
Large Classes ◽  
Heavy Tailed Distributions ◽  
Heavy Tailed ◽  
The Individual ◽  
Risk Processes

AbstractIn this paper, we investigate asymptotic properties of the tail probabilities of the maxima of partial sums of independent random variables. For some large classes of heavy-tailed distributions, we show that the tail probabilities of the maxima of the partial sums asymptotically equal to the sum of the tail probabilities of the individual random variables. Then we partially extend the result to the case of random sums. Applications to some commonly used risk processes are proposed. All heavy-tailed distributions involved in this paper are supposed on the whole real line.

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