Page 203 - Elements of Distribution Theory

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052184472Xc06 CUNY148/Severini May 24, 2005 2:41

6.5 Counting Processes 189

If the number of possible arrivals is bounded by M, then
M

µ(t) = F n (t).
n=1

Proof. Fix t. Let A n denote the indicator of the event that the nth arrival occurs before t,
that is, that S n ≤ t. Then F n (t) = E(A n ) and
∞

N(t) = A n .
n=1
Hence,
∞ ∞

µ(t) = E[N(t)] = E(A n ) = F n (t),
n=1 n=1
proving the first result.
If there are at most M arrivals, then the indicator A n is identically equal to 0 for n =
M + 1, M + 2,.... The second result follows.

Poisson processes
The most important counting process is the Poisson process.A counting process {N(t):
t ≥ 0} is said to be a Poisson process if the following conditions are satisfied:
(PP1) N(0) = 0
(PP2) {N(t): t ≥ 0} has independent increments: if
0 ≤ t 0 ≤ t 1 ≤· · · ≤ t m ,
then the random variables
N(t 1 ) − N(t 0 ), N(t 2 ) − N(t 1 ),..., N(t m ) − N(t m−1 )
are independent.
(PP3) There exists a constant λ> 0 such that, for any nonnegative s, t, N(t + s) − N(s)
has a Poisson distribution with mean λt.

The condition that differences of the form N(t + s) − N(s) follow a Poisson distribution,
condition (PP3), may be replaced by a condition on the behavior of N(t) for small t, provided
that the distribution of N(t + s) − N(s) does not depend on s. Consider the following
conditions.

(PP4)For any t > 0, the distribution of N(t + s) − N(s)is the same for all s ≥ 0.
(PP5) lim t→0 Pr[N(t) ≥ 2]/t = 0 and for some positive constant λ,
Pr[N(t) = 1]
lim = λ.
t→0 t
The equivalence of condition (PP3) and conditions (PP4) and (PP5) is established in the
following theorem.

Theorem 6.11. Suppose that a given counting process {N(t): t ≥ 0} satisfies conditions
(PP1) and (PP2). The process satisfies condition (PP3) if and only if it satisfies conditions
(PP4) and (PP5).