# 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

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