Envelopes of Vector Random Processes and Their Crossing Rates

Abstract

Vector-valued random processes, $\mathbf{X}(t)$, can be "enveloped" by set-valued random processes, $\mathscr{S}(t)$, to which they belong with probability 1 during any finite length of time. When applied to scalar processes, the set-definition of envelope includes and is richer than the familiar point-definitions. Several random set-envelope processes in $n$-dimensional space, $R_n$, are defined and the mean rates at which they "cross" given regions of $R_n$ are calculated. Comparison is made with the mean crossing rates of associated enveloped Gaussian processes, $\mathbf{X}(t)$.