Abstract
The paper studies the behavior of an (l + 3)th‐dimensional, delayed renewal process with dependent components, the first three (called active) of which are to cross one of their respective thresholds. More specifically, the crossing takes place when at least one of the active components reaches or exceeds its assigned level. The values of the other two active components, as well as the rest of the components (passive), are to be registered. The analysis yields the joint functional of the “crossing level” and other characteristics (some of which can be interpreted as the first passage time) in a closed form, refining earlier results of the author. A brief, informal discussion of various applications to stochastic models is presented.
Highlights
The analysis yields the joint functional of the "crossing level" and other characteristics in a closed form, refining earlier results of the author
The behavior of sums of independent random variables about a fixed level is a classical theme in the theory of fluctuations
In various applied probability models, the information about the level crossing and first passage time of the sum of random variables is of significant importance
Summary
The paper studies the behavior of an (1 + 3)th-dimensional, delayed renewal process with dependent components, the first three (called active) of which are to cross one of their respective thresholds. The crossing takes place when at least one of the active components reaches or exceeds its assigned level. The values of the other two active components, as well as the rest of the components (passive), are to be registered. The analysis yields the joint functional of the "crossing level" and other characteristics (some of which can be interpreted as the first passage time) in a closed form, refining earlier results of the author. A brief, informal discussion of various applications to stochastic models is presented AMS subject classifications: 60K05, 60K10, 60K25, 60G25, 68M20, 90A09, 90A60, 90B22
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