Define the first passage time of y(t) from within the interval (−ymin, ymax) as where "inf" is the infimum.
The mean of τ(y0) is the residence time,[1][2] For a Gaussian process and a boundary far from the mean, the residence time equals the inverse of the frequency of exceedance of the smaller critical value,[2] where the frequency of exceedance N is σy2 is the variance of the Gaussian distribution, and Φy(f) is the power spectral density of the Gaussian distribution over a frequency f. Suppose that instead of being scalar, y(t) has dimension p, or y(t) ∈ ℝp.
Define a domain Ψ ⊂ ℝp that contains yavg and has a smooth boundary ∂Ψ.
Noting the exponential in Equation (1), the logarithmic residence time of a Gaussian process is defined as[5][6] This is closely related to another dimensionless descriptor of this system, the number of standard deviations between the boundary and the mean, min(ymin, ymax)/σy.
In general, the normalization factor N0 can be difficult or impossible to compute, so the dimensionless quantities can be more useful in applications.