Mathematical Statistics and Stochastic Processes

Chapter 11

Poisson Processes — A Probabilistic andStatistical Study

11.1. Introduction

A Poisson process models a sequence of events that occur at random instants.

Let 0 < T1 < T2 < … < Tn < … be the arrival time of a sequence of events such that limn↑ ∞ Tn = + ∞. For reasons that will be apparent later, we set T0 = 0.

Supposing the Tn to be random variables defined on the same probability space, is said to be a point process.

Equivalently, the process may be defined, where Nt denotes the number of events that occur in the interval [0, t]. We say that (Nt) is the counting process associated with (Tn).

(Nt) and (Tn) are connected by the following relations:

– N0 =T0 = 0;

–

– {Nt = n} = {Tn ≤ t < Tn + 1};

– {Nt ≥ n} = {Tn ≤ t};

– {s < Tn ≤ t} = {Ns < n ≤ Nt}.

Consequently, (Nt) is entirely determined by (Tn), and vice versa.

11.2. The axioms of Poisson processes

To specify the distribution of the process, we make the following hypotheses:

  A1: (Nt, t ≥ 0) has independent increments, i.e. N_t2 − N_t1 ,…, N_tk − N_tk−1 are independent for all k > 2, 0 ≤ t1 < t2 < … < tk.

  A2: (Nt, t ≥ 0) has stationary increments, i.e. P(Nt + h − Ns + h) = P(Nt − Ns) for 0 ≤ s < t, h > 0.

THEOREM 11.1.– ...