Topic
Renewal theory
About: Renewal theory is a research topic. Over the lifetime, 2381 publications have been published within this topic receiving 54908 citations.
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TL;DR: In this paper, it was shown that in many cases it is more suitable to expand functions of that type not into power series but into infinite series of appropriate Poissonian functions of tf.
Abstract: SUMMARY It has been shown by Smith, Leadbetter and White how various functions describing the Weibull Renewal Process can be evaluated numerically by being expanded into power series of t/l, where,8 is the Weibull 'shape' parameter. It seems that in many cases it is more suitable to expand functions of that type not into power series but into infinite series of appropriate Poissonian functions of tf.
47 citations
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TL;DR: In this paper, sufficient and necessary conditions for the existence of moments of the first passage time of a random walk are given under the condition that the random walk remains above the level x on K consecutive occasions, which has applications in option pricing.
Abstract: Necessary and sufficient conditions for the existence of moments of the first passage time of a random walk S
n
into [x, ∞) for fixed x≧ 0, and the last exit time of the walk from (−∞, x], are given under the condition that S
n
→∞ a.s. The methods, which are quite different from those applied in the previously studied case of a positive mean for the increments of S
n
, are further developed to obtain the “order of magnitude” as x→∞ of the moments of the first passage and last exit times, when these are finite. A number of other conditions of interest in renewal theory are also discussed, and some results for the first time for which the random walk remains above the level x on K consecutive occasions, which has applications in option pricing, are given.
47 citations
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47 citations
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TL;DR: The process of holding times of items at each of k stations is defined, a generalized vector renewal process that is represented by an activity network and relationships among random variables of interest are simplified through the use of equivalent networks.
47 citations
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TL;DR: In this paper, a statistical distribution of the first-occurrence and first-recurrence times of the crossing of a given level in a continuous random process is presented. But the distribution is restricted to the case when the crossings of the level under consideration are statistically rare events.
Abstract: This paper deals with the statistical distribution of the first‐occurrence and first‐recurrence times of the crossing of a given level in a continuous random process. Approximate forms of the first‐occurrence and first‐recurrence time densities are found by considering the successive crossings to form a renewal process. A relatively simple exponential distribution is found to give an appropriate representation of the limiting case when the crossings of the level under consideration are statistically rare events. Numerical examples are worked out for some stationary Gaussian processes. The method is of use in evaluating survival probabilities for randomly excited mechanical systems subject to failure upon occurrence of a sufficiently high load.
46 citations