# What is the Poisson process

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### Homogeneous Poisson process

In this section we consider (renewal) counting processes With

 and (22)

in which is a sequence of independent and identically distributed random variables. We also assume that the inter-arrival times are exponentially distributed, i.e.
 (23)

definition
If (23) holds, then the counting process becomes a homogeneous Poisson process with the intensity called.
Notice
• A fundamental property of homogeneous Poisson processes is that they form a class of stochastic processes with independent and stationary increments.
• We will analyze this in more detail in the following theorem, which contains various equivalent definitions of the homogeneous Poisson process, where we omit the adjective "homogeneous" for brevity.
Theorem 2.9 Be any counting process. Then the following statements are equivalent:
(a)
is a Poisson process with intensity .
(b)
For any and is the random variable poisson distributed with parameter i.e. , and on the condition that , has the random vector the same distribution as the order statistics of independent, in uniformly distributed random variables.
(c)
has independent gains with , and for any and has the random vector under the condition that , the same distribution as the order statistics of independent, in uniformly distributed random variables.
(d)
has stationary and independent gains, and for applies
 (24)

(e)
has stationary and independent gains, and for each is a -distributed random variable.
proof
We carry out a cyclic proof, i.e. we show that (a) (b) (c) (d) (e) (a) applies.

(a) (b)

• From (a) it follows that a sum of independent and exp-distributed random variables, i.e., is Erl -distributed, with Erl the Erlang distribution with the parameters and designated.
• It follows that and

for each , i.e., is poi -distributed.
• Because the common density of is given by
for any and otherwise, applies to the common conditional density of under the condition that , the formula

For and otherwise.
• That is the density of the order statistics of independent, in uniformly distributed random variables.

(b) (c)

(c) (d)

• We assume that the random vector under the condition that , has the same distribution as the order statistics of independent, in uniformly distributed random variables.
• Then applies to any , and that
• With the help of the formula of the total probability it now follows that the counting process has stationary gains.
• In addition, it follows from the conditional uniform distribution property, which is assumed in (c), that for

• Thus

• From the inequality that for any and holds, it follows that the functions in the last sum the common bound have.
• This limit can be integrated because it applies
• By interchanging the summation and the limit value, it now results that
and thus the first part of (24).
• In the same way we get that
which is equivalent to the second part of (24).

(d) (e)

• Be , and . Then applies to
 (25)

and for
 (26)

• It follows that the function steadily in and right steady in the point is.
• Because , it follows from (25) and (26) that for any
• The function is thus differentiable, and it satisfies the differential equation
 (27)

For .
• Because of the (uniquely determined) solution of (27) is given by
 (28)

• To show that
 (29)

for any holds, one can proceed in exactly the same way as in the proof of (28).
• For this it suffices to note that (24) is the validity of For implies.
• With the help of (28) we now get by complete induction after that (29) holds.

(e) (a)

Next page:Compound Poisson processes Upwards:Poisson-type counting processes Previous page:Poisson-type counting processes & nbsp content Ursa Pantle 2005-07-13