A general Rice formula, Palm measures, and horizontal-window conditioning for random fields
A classical problem in the theory of random processes is to compute the mean number (or another suitable measure) of (up-)crossings of a fixed level by a random process or field. In the present paper the level sets of m-dimensional random vector fields with values in \(R^ n\) are measured by the (m-n)-dimensional Hausdorff measure. Based on the powerful concept of using the Hausdorff area and Federer's co-area theorem, the intensity measure of this (m-n)-dimensional ''volume'' of the level sets is computed. The sample paths are assumed to be approximately differentiable. A conditional in the horizontal-window sense process, crossing a fixed level y at the origin, is the limit of the distribution of the process under the condition that in \([0,\epsilon]\) an y-level crossing occurs, \(\epsilon \downarrow 0\). In the terminology of point processes this is the Palm distribution of the process relative to the crossing point process. (For the point process case, this connection is well-known; for the case of surface measures see the author, Local interpretation of Palm distributions of surface measures, Math. Nachr. 119 (1984).) In the paper such horizontal-window conditional (Palm) distributions are computed for (not necessary stationary) random vector fields. As an example the Gaussian case is considered [see also the author, On level crossings by Gaussian fields. Math. Operationsforsch. Stat., Ser. Stat. to appear].
scientific article; zbMATH DE number 3860106
scientific article