The average number of events per time unit and its estimation

Abstract

Let ζ _t be the number of events which will be observed in the time interval [0;t] and define$A = \mathop {\lim }\limits_{t \to \infty } E$ 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,E[ζ _t ] is the renewal function and it follows from well-known results of renewal theory thatA exists and is equal to 1/τ, if τ>0 is the expectation of the waiting-times.

This holds true also when τ = ∞.A may be estimate by ζ _t /t or$1/\bar X$ where$\bar X$ is the mean of the firstn waiting-timesX ₁,X ₂, ...,X _n . Both estimators converage with probability 1 to 1/τ if theX _i are i.i.d.; but the expectation of$1/\bar X$ may be infinite for alln and also if it is finite,$1/\bar X$ is in general a positively biased estimator ofA. For a stationary renewal process, ζ _t /t is unbiased for eacht; if theX _i are i.i.d. with densityf(x), then ζ _t /t has this property only iff(x) is of the exponential type and only for this type the numbers of events in consecutive time intervals [0,t], [t, 2t], ... are i.i.d. random variables for arbitraryt > 0.