Strong Renewal Assumption?

Question

Just started a Stochastic Processes course and I am a bit confused over the Strong Renewal Assumption we make for Renewal (and Poisson?) processes. The assumption in my text goes as follows: "At each fixed time and at each arrival time, the process starts over independently of the past".

From my understanding, renewal processes have independent interarrival times with the Poisson process adding the memorylessness property (Assumption also holds for fixed times).

Shouldn't the Renewal Property only mention the arrival times in order to hold for all renewal processes? The text I am reading is confusing me cause it sounds like the Renewal property implies memorylessness - which doesn't apply to all Renewal processes.

Answer 1

To shun the impending confusion, OP needs to comprehend regenerative process.

If for a stochastic process $\{X(t): t\geq 0\}$ with state space $0, 1,2,\ldots, $ there exists instances from the point of which the process probabilistically restarts itself i.e. with probability unity, there exists $T_1$ such that the process post $T_1$ is the probabilistic replica of the process starting at $0, $ then it is regenerative (from $\rm [I], $ section $7.5, $ p. $442$).

In a renewal process $\{N(t): t\geq 0\}, $ the interarrival times are iid, so the process probabilistically starts over at each renewal. That is, specifically, a renewal process is regenerative where $T_1$ is the first renewal.

The quoted sentence in OP could have been framed better. But this is all about renewal processes being regenerative.

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Reference:

$\rm [I]$ Introduction to Probability Models, Sheldon M. Ross, Elsevier Inc., $2007.$

$\rm [II]$ Stochastic Processes, Sheldon M. Ross, John Wiley & Sons, $1996, $ section $3.1, $ p. $99.$