Page 206 - Elements of Distribution Theory

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192 Stochastic Processes

Hence, T 1 has an exponential distribution with rate parameter λ. More generally, the interar-
rival times T 1 , T 2 ,... are independent, identically distributed exponential random variables.
Conversely, if T 1 , T 2 ,... are known to be independent exponential random variables then
the counting process is a Poisson process. Thus, a counting process is a Poisson process if
and only if the interarrival times are independent exponential random variables.
A formal proof of this result is surprisingly difficult and will not be attempted here; see,
for example, Kingman (1993, Section 4.1). It is easy, however, to give an informal argument
showing why we expect the result to hold.
We have seen that T 1 has an exponential distribution. Now consider
Pr(T 2 > t|T 1 = t 1 ) = Pr[N(t 1 + t) − N(t 1 ) = 0|T 1 = t 1 ].

Since a Poisson process has independent increments, we expect that
Pr[N(t 1 + t) − N(t 1 ) = 0|T 1 = t 1 ] = Pr[N(t 1 + t) − N(t 1 ) = 0] = exp(−λt). (6.8)

Hence, T 1 and T 2 are independent and the marginal distribution of T 2 is an exponential
distribution with parameter λ. This approach may be carried out indefinitely:

Pr(T m > t|T 1 = t 1 ,..., T m−1 = t m−1 )
= Pr[N(t 1 +· · · + t m−1 + t) − N(t 1 + ··· + t m−1 ) = 0]
= exp{−λt}.
The difficulty in carrying out this argument rigorously is in showing that (6.8) actually
follows from (PP2). For instance, the event that T 1 = t 1 is the event that N(t) jumps from
0to1at t = t 1 and, hence, it is not a straightforward function of differences of the form
N(t j ) − N(t j−1 ) for some set of t j .

6.6 Wiener Processes

A Wiener process or Brownian motion is a continuous time process {W t : t ≥ 0} with the
following properties:

(W1) Pr(W 0 = 0) = 1
(W2) The process has independent increments:if

0 ≤ t 0 ≤ t 1 ≤· · · ≤ t m ,
then the random variables
W t 1 − W t 0 , W t 2 − W t 1 ,..., W t m − W t m−1
are independent.
has a normal distribution with mean 0 and variance
(W3)For t 2 > t 1 ≥ 0, W t 2 − W t 1
t 2 − t 1 .
Processes satisfying (W1)–(W3) can always be defined in such a way so as to be con-
tinuous; hence, we assume that (W4) is satisfied as well:
(W4)For every realization of the process, W t is a continuous function of t.

Two basic properties of Wiener processes are given in the following theorem.