Let be the number of events which will be observed in the time interval and define as the average number of events per time unit if this limit exists. In the case of i.i.d. waiting-times between the events, is the renewal function and it follows from well-known results of renewal theory that A exists and is equal to , if is the expectation of the waiting-times. This holds true also when . A may be estimated by or where is the mean of the first n waiting-times . Both estimators converge with probability 1 to if the are i.i.d.; but the expectation of may be infinite for all n and also if it is finite, is in general a positively biased estimator of A. For a stationary renewal process, is unbiased for each t; if the X i are i.i.d. with density , then has this property only if is of the exponential type and only for this type the numbers of events in consecutive time intervals are i.i.d. random variables for arbitrary .
