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Interarrival times are essential in understanding processes that involve events occurring randomly over time. In the case of renewal processes, these are the periods between consecutive events happening within the system. For instance, in a queuing scenario, interarrival times might represent the time between customers arriving at a checkout counter.

Now, when analyzing the interarrival times of two independent renewal processes like in our exercise example, we need to determine whether these times for a combined process remain independent. If the two original processes are genuinely independent, their respective interarrival times are not influenced by each other and are solely determined by their own process characteristics. However, once we start observing the combined process, any particular interarrival time for the new process is affected by the events of both original processes, making the interarrival times of the combined process dependent on each other. This shows that for the combined process, the time until the next event cannot be predicted based purely on the previous interarrival time, hence they're not independent.

When discussing renewal processes, a process is labeled 'independent' if the interarrival times between events are not influenced by previous interarrival times. In essence, each event occurs as if 'resetting the clock', not pertaining to when the last event took place.

With independent renewal processes the scenario is such that the behavior or occurrence of one event provides no information about when the next event will happen. This is similar to flipping a fair coin; no matter how many times it comes up heads, the probability of it landing heads on the next flip remains the same. In our textbook problem, we examine two independent renewal processes. While each of these processes, separately, upholds this independence characteristic, their combination results in a new set of interarrival times that lose this independence, as the timing of events from both processes starts intermingling.

The term 'identically distributed' pertains to a scenario where random variables have the same probability distribution. For renewal processes, this means that each interarrival time is drawn from the same statistical distribution, ensuring consistency in the random behavior from one event to the next.

In individual independent renewal processes, the consistency of the random nature of events is preserved; each interarrival time follows the same pattern with no variations in distribution. When dealing with interarrival times of a combined process from two independent processes, such uniformity cannot be guaranteed unless the original processes have interarrival times following precisely the same distribution. The exercise highlights that once we combine two renewal processes, the resultant process’s interarrival times are a convolution of individual distributions, and hence, they are not identically distributed unless the initial processes were characterized by identical distributions.