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Poisson Process: Special Case of Many Things
It is useful to be aware that a Poisson process is a special case of several important stochastic processes. That leads to different equivalent definitions of a Poisson process, as in Definitions 5.2 and 5.3 of the Ross text. It also leads to different ways to analyze a Poisson process.
- A Point Process and a Counting Process A point process on the positive half line, i.e., on the interval [0, ∞), is a random distribu- tion of points on the positive half line. We may specify the distribution in three ways: (i) by specifying the distribution of the locations of the points, (ii) by specifying the distribution of the intervals between successive points and (iii) by specifying the distribution of the associated counting process. Let Sn be the location of the nth^ point, where S 0 ≡ 0 (without there being a 0 th^ point). Let Xn ≡ Sn − Sn− 1 be the interval between the (n − 1)st^ point and the nth^ point. Let the associated counting process be defined by
N (t) ≡ max {k ≥ 0 : Sk ≤ t}, t ≥ 0.
In other words, a point process may be specified in three ways, via the stochastic processes: (i) {Sn : n ≥ 0 }, (ii) {Xn : n ≥ 1 } and (iii) {N (t) : t ≥ 0 }. The first representation {Sn : n ≥ 0 } is the typical form for a point process. The last representation {N (t) : t ≥ 0 } is the typical form for a counting process. A picture makes this clear; see Figure 1.
Figure 1: A Sample Path of a Counting Process.
For any point process or counting process, there is an important inverse relation, men- tioned in Section 5.3.3 after Proposition 5.1 and discussed at greater length in (7.2) of Chapter
- For any nonnegative n and t,
Sn ≤ t if and only if N (t) ≥ n.
This is for any possible realization; i.e., it is valid with probability 1. Again, a picture makes this clear: In Figure 1, the counting process view looks at the horizontal x axis as the domain and the vertical y axis as the range (mapping time t into the number N (t)), while the point process view looks at the vertical y axis as the domain and the horizontal x axis as the range (mapping the nonnegative integers n into the nth^ point Sn).
- Alternative Definitions of a Poisson Process (a) Standard Definition The standard definition of a Poisson process is a counting process {N (t) : t ≥ 0 } such that (i) N (t) has a Poisson distribution with mean λt for each t > 0, where λ > 0 is some parameter, and (ii) the process has independent increments.
Property (i) means that P (N (t) = k) =
e−λt(λt)k k!
for any nonnegative integer k (where x^0 = 1 and 0! = 1). As a consequence,
E[N (t)] = Variance(N (t)) = λt.
The stochastic process N ≡ {N (t) : t ≥ 0 } has independent increments if the number of point in any number of disjoint intervals are independent random variables. The number of points in the interval (a, b], closed on the right and open on the left, is N (b) − N (a). (The probability of a point at any specific location will be 0, because the distance between successive points has a density.) An increment of the stochastic process N ≡ {N (t) : t ≥ 0 } is N (b) − N (a). Suppose that (t 1 , t 2 ], (t 3 , t 4 ]... (t 2 k− 1 , t 2 k] are k disjoint intervals, i.e., with
0 ≤ t 1 < t 2 ≤ t 3 < t 4 ≤ · · · ≤ t 2 k− 1 < t 2 k.
Then the k random variables N (t 2 ) − N (t 1 ), N (t 4 ) − N (t 3 ),... N (t 2 k) − N (t 2 k− 1 ) are mutually independent random variables. As a consequence, if 0 ≤ t 1 < t 2 ≤ t 3 < t 4 , then
P (N (t 2 ) − N (t 1 ) = j, N (t 4 ) − N (t 3 ) = k) = P (N (t 2 ) − N (t 1 ) = j)P (N (t 4 ) − N (t 3 ) = k)
= e−λ(t^2 −t^1 )(λ(t 2 − t 1 ))j j!
e−λ(t^4 −t^3 )(λ(t 4 − t 3 ))k k!
(b) renewal process A point process (or counting process) is a renewal process if the intervals between points, i.e., the random variables Xn defined above, are independent and identically distributed (i.i.d.). In renewal theory (Chapter 7), much attention is given to the associated renewal counting process {N (t) : t ≥ 0 }.
A Poisson process (Chapter 5) is a special case of a renewal process (Chapter 7) in which the times between renewals have an exponential distribution. This corresponds to Proposition 5.1 in Section 5.3.
