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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, the authors investigate three kinds of strong laws of large numbers for capacities with a new notion of independently and identically distributed (IID) random variables for sublinear expectations initiated by Peng.
Abstract: We investigate three kinds of strong laws of large numbers for capacities with a new notion of independently and identically distributed (IID) random variables for sub-linear expectations initiated by Peng. It turns out that these theorems are natural and fairly neat extensions of the classical Kolmogorov’s strong law of large numbers to the case where probability measures are no longer additive. An important feature of these strong laws of large numbers is to provide a frequentist perspective on capacities.
37 citations
01 Jan 2013
TL;DR: The topic of Markov processes is huge and a number of volumes can be written on this topic, but what is given in this chapter is the minimum required in order to follow what is presented afterwards.
Abstract: The topic of Markov processes is huge. A number of volumes can be, and in fact were, written on this topic. We have no intentions to be complete in this area. What is given in this chapter is the minimum required in order to follow what is presented afterwards. In particular, we will refer at times to this chapter when results we present here are called for. For more comprehensive coverage of the topic of Markov chains and stochastic matrices, see [9, 19, 41] or [42].
36 citations
TL;DR: In this paper, the renewal risk process with stochastic interest is considered and exact expressions and integral equations for the Gerber-Shiu expected discounted penalty function and the ultimate ruin probability are derived.
36 citations
TL;DR: The transient behavior of a system with operational and repair times distributed following phase-type distributions is studied, and an algorithmic approach is performed to determine the transition probabilities for the Markov process which governs the system.
Abstract: The transient behavior of a system with operational and repair times distributed following phase-type distributions is studied. These times are alternated in the evolution of the system, and they form 2 separate geometric processes. The stationary study of this system when the repair times form a renewal process has been made . This paper also considers that operational times are partitioned into two well-distinguished classes successively occupied: good, and preventive. An algorithmic approach is performed to determine the transition probabilities for the Markov process which governs the system, and other performance measures beyond those in are calculated in a well-structured form. The results are applied to a numerical example, and the transient quantities are compared with the ones obtained in the stationary case. The computational implementation of the mathematical expressions formulated are performed using the Matlab program.
36 citations
TL;DR: This paper shows that in the classical age replacement setting, with known failure time distribution with increasing hazard rate, the one-cycle criterion leads to earlier replacement than the renewal reward criterion, and presents adaptive age replacement with a one- cycle criterion within the nonparametric predictive inferential framework.
36 citations