4. Counting Processes and Poisson Processes
Emanuel Parzen · Society for Industrial and Applied Mathematics eBooks · 1999
By a counting process we mean an integer-valued process {N(t), t≥0} which counts the number of points occurring in an interval, these points having been distributed by some stochastic mechanism. In a typical case, the points represent the times τ1,τ2,⋯ at which events of a specified character have occurred, where 0<τ1<τ2<⋯ . The random variablesT1=τ1,T2=τ2−τ1,⋯,Tn=τn−τn−1,⋯are called the successive inter-arrival times. If for t≥0 , N(t) represents the number of points lying in the interval (0, t], then {N(t), t≥0} is called the counting process of the series of points.