On a zero-one process and some of its applications

We have many occasions to analyse the pattern of occurrences of serial random events, for example, automobile flows, dropping ends of cocoon filaments. Many works have been done for the case when the lengthes of gaps independently follow one and the same negative exponential distribution. In this case, as is well known, the number of occurrences in one time interval is entirely independent of that in another time interval which does not overlap the former, and follows the Poisson distribution. Nevertheless, we sometimes observe random events concerning which the numbers of occurrences in some disjoint time intervals are not entirely independent, that is, they show wave-like movement. In some cases, this is due to the fact that the distribution of gaps are not necessarily of negative exponential type. In this paper we shall treat a discrete parameter processes for the purpose to analyse practically the processes of the type just stated.