On Gaussian noise envelopes

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Abstract
The first-passage time problem for a continuous one-dimensional Markov process is reviewed, and upper bounds are obtained for both the probability of failure (or passage and the moments of the time to failure, in terms of the mean time to failure. In addition, stationary Gaussian variables arising from systems withNdegrees of freedom that have autocorrelation functions of the form \begin{equation} R(r) = e^{-b \mid \tau \mid} \sum_{k=1}^{N} d_k^2 \cos \omega_k \tau \end{equation} are shown to be derivable from a2N-dimensional (or2N- 1, if one of the\omega_kis zero) Markov process that possesses a "pseudoenvelope," which is itself the result of a one-dimensional Markov process. This pseudo-envelope can be used as a bound on the magnitude of the Gaussian variable, and its first-passage time problem can be solved explicitly or utilized to obtain convenient bounds for the probability of failure of the Gaussian process.

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