On Streams of Events and Mixtures of Streams

Abstract

Distribution functions for streams of events (and, in particular, for the length of the interval 'between successive events') are defined within the framework of the theory of Khintchine (1955). Relationships such as that between the distributions for the length of an 'arbitrary interval' between successive events, the interval containing a fixed point and that of the time to the $n$th event after $t=0$ are discussed briefly in this context. The definitions and results are extended to deal with mixtures of two streams, in which the events are alternately of opposite type. An important application concerns upcrossings and downcrossings of a level by a stationary stochastic process