0

The textbook wrote like the following:

A natural generalization is to consider a counting process for which the interarrival times are independent and identically distributed with an arbitrary distribution. Such a counting process is called a renewal process.

I understand from the above sentences that Renewal Process is a subset(specific type) of Counting Process. enter image description here

However, the textbook wrote the definition like the following:

Definition 3.1.1

The counting process $\{N(t), t\ge0\}$ is called a renewal process.

I understand from the definition 3.1.1 the Counting Process is equal to the Renewal Process.

I want to know the relationship of the counting and renewal process.

Which one is subset of the other? Or Equal?

Thank you for reading my question.

Community
  • 1
Danny_Kim
  • 3,423
  • I suppose, since definition 3.1.1 says "THE counting..." that it refers to especifiacally process $N(t)$ (which i conjecture is the process that measures the arrivals) – sinbadh Jan 17 '16 at 04:56
  • Aha, so you mean my picture is correct? (renewal process is a subset of counting process) – Danny_Kim Jan 17 '16 at 04:57
  • Yes , of course. Forgetting the name "renewal ", this processes have the property of "memory loss" . That is , by independence , once that happened an arrival , you can forget all history and think in time $ t = 0 $ for measure for the next arrival – sinbadh Jan 17 '16 at 04:58
  • @sinbadh Thank you very much – Danny_Kim Jan 17 '16 at 04:58
  • 1
    @sinbadh Most misleading comment. As should be well known, loss of memory holds only for renewal processes whose increments are i.i.d. and exponentially distributed. – Did Jan 18 '16 at 07:50

0 Answers0