The moments of the number of exits from a simply connected region

Advances in Applied Probability - Tập 30 Số 1 - Trang 167-180 - 1998
Robert Illsley1
1London Guildhall University ,

Tóm tắt

We generalise the work of Cramér and Leadbetter, Ylvisaker and Ito on the level crossings of a stationary Gaussian process to multivariate processes. Necessary and sufficient conditions for the existence of the expected number of crossingsE(C) of the boundary of a region of ℝpby a stationary vector stochastic process are obtained, along with an explicit formula forE(C) in the Gaussian case. A rigorous proof of Belyaev's integral formula for the factorial moments of the number of exits from a region of ℝpis given for a class of processes which includes Gaussian processes having a finite second order spectral moment matrix. Applications to χ2processes are briefly considered.

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