Recurrent events in a Bernoulli sequence

The "point of regeneration" method is used to obtain simple sequential equations for determining the complete probability density function for multiple occurrences of events in a Bernoulli sequence. Both the independent and overlapping classes of recurrent events are included in the general framework of these equations. The equations also lead to the generating function for the probability distribution. This is used to obtain the expected recurrence times for the different classes of recurrent events. A distinction is made between the probability distributions for the occurrence of the th event at the th trial and the occurrence of events in trials. The latter case is of primary concern. The methods employed and the results obtained have extensive applications in problems in automatic control, communications, and information processing.