Crossings Problems in Random Processes Theory and Their Applications in Aviation

The behavior of any real system is a process to a greater or lesser degree probabilistic. As a rule, it is impossible to specify exactly which external influences and internal mechanisms of interaction of the system components will be decisive in the future. As a consequence, we cannot accurately predict the behavior of the system. We can only talk about the probability that, in the future, the system will come to a particular state. While the problem of the probabilistic description of all possible future states of the system is very difficult to answer, it is often enough, for research purposes, to obtain answers to questions that are more simple, for example, “For how long will, on average, the system operate in a given mode?” or “What is the probability that the process of functioning of the system will come out of given mode to a specific point in time?” Problems of this type concern the crossings of a level by a random process.

This book states some of the most important fundamental results related to crossings problems. The known problem of first reaching boundaries by a random process is discussed in detail. It explores the solution of this problem for arbitrary continuous processes, and also considers the application of obtained mathematical results to the investigation of the safety of an airplane landing.

The book will appeal to engineers and scientists who are interested in the applications of random processes theory and its methods. Its results will also be of interest for mathematicians who study crossings problems.