On a Class of Gaussian Processes for Which the Mean Rate of Crossings is Infinite
J. A. McFadden · Journal of the Royal Statistical Society Series B (Statistical Methodology) · 1967
Summary A stationary Gaussian process X(t) is defined as locally Markov if the slope of its autocorrelation function has a finite jump at the origin. For such a process the mean rate of crossings at an arbitrary level a is infinite. For small (t2 − t1), the probability that X(t1) − a and X(t2) − a have opposite signs is calculated, both with and without the auxiliary condition, X(t3)∈ (x3, x3+dx3], …, X(tn+2)∈(xn+2, xn+2+dxn+2], where the times t3,…, tn+2 are all outside [t1,t2]. By inversion, it is shown that the conditional probability of this multiple event in X(t3), …, X(tn+2), given the sign change in (t1,t2), is asymptotically equal to the conditional probability of the same multiple event, given that X(t2) ϵ (a, a + da]. Again for small (t2 − t1). the probability that X(t) - a has one or more zeros is calculated, both marginally and conditionally. This event includes the case of a positive even number of crossings, which is not negligible. The above asymptotic equivalence between conditional probabilities can be extended also to this condition of one or more crossings. As an application, the probability of crossings in two small intervals (at different levels) is considered.