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A continuous-time process $\{N(t) : t\in[0,\infty)\}$ is a counting process if it satisfies

  1. $N(t)\geqslant0$ a.s. (nonnegative)
  2. $\mathbb P(N(t)\in\mathbb N\cup\{0\}) = 1$ (integer-valued)
  3. If $s\leqslant t$ then $N(s)\leqslant N(t)$ a.s. (non-decreasing),

and a counting process $N(t)$ is a renewal process if $$N(t) = \sum_{n=0}^\infty\mathsf 1_{(0,t]}(W_n) $$ where $\{W_n\}$ is an i.i.d. sequence of nonnegative random variables.

My questions are as follows:

  1. In the answer to this question, there is a family of counting processes that is not a renewal process. Are there any other interesting such families?
  2. Is there another characterization of renewal processes, i.e. necessary and sufficient conditions for a counting process $N(t)$ to be a renewal process?
  3. It is well-known that a renewal process with independent and stationary increments is a Poisson process. A nonhomogenous Poisson process has the first property but not the second; and a delayed renewal process whose asymptotic excess life distribution is the same as the distribution of the first renewal. Are there any other interesting examples?
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    $\begingroup$ For question 1 just drop one of the two conditions (independence or identical distribution) for the jump time intervals. For example, the sum of two Poisson counting processes with different frequencies has nonindependent time intervals between jumps. And if you consider the counting process where the intervals alternate between a low-frequency Poisson and a high-frequency Poisson then the intervals are obviously not identically distributed. $\endgroup$ – Justpassingby Dec 18 '15 at 9:50

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