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The crossings of boundaries by vector Gaussian processes with applications to problems in reliability

Illsley, R.G. (1992). The crossings of boundaries by vector Gaussian processes with applications to problems in reliability. (Unpublished Doctoral thesis, City, University of London)

Abstract

The problem of 'nuisance disconnects' in high integrity redundant systems is shown to be mathematically equivalent to the problem of the crossings of the boundary of a region by a vector stochastic process. A number of other engineering situations are similarly modelled by such multivariate crossing processes.

Let X(t) be a vector valued stationary Gaussian process, having continuous sample paths a.s., and let U be the number of exits, in the interval (0,1], from a region T having a boundary ՁГ consisting of a finite number of regular elements. We prove the formula of Belyaev for the factorial moments of U under conditions on the process similar to those of Ylvisaker(1966). We further show that these conditions are sufficient to guarantee E(U) < ∞ .

The validity of the formula of Belyaev does not imply the existence of the moments of all orders. We show that, for a two- dimensional Gaussian process, the variance of U is finite if

[Formula] (Please see inside thesis)

for some δ > 0, where ϴ”(t) = R "(t) - R "(0) and R”(t) is the second derivative of the covariance matrix of the process.

It is well known that the duration of an exceedence above a high level, by a stationary Gaussian process, has an asymptotic Rayleigh distribution. In chapter 4, we show that, for the two dimensional processes of the present study, the Rayleigh distribution is but one of three asymptotic distributions possible for the duration of an exceedence above a large boundary.

In the final chapters we comment on the problem of 'nuisance disconnects' in the light of the theoretical developments of the previous chapters. A discussion of the relation of our work to that of earlier authors and of possible avenues for future research is also included.

Publication Type: Thesis (Doctoral)
Subjects: Q Science > QA Mathematics
Departments: School of Science & Technology > Mathematics
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