One of the exercices in Sheldon M. Ross's book, "Introduction to Probability Models", is the following:
Let {N1(t), t ≥ 0} and {N2(t), t ≥ 0} be independent renewal processes. Let N (t) = N1(t) + N2(t).
(a) Are the interarrival times of {N (t), t ≥ 0} independent?
(b) Are they identically distributed?
(c) Is {N (t), t ≥ 0} a renewal process?
It looks a good exercise to get a deep understanding of the renewal processes theory. I can suspect the answer, but I don't know how to prove it. Any help?
Consider the following example. Assume $N_1(t)$ is a "deterministic" process with arrival times at $\{1,2,3,4,5,...\}$. The interarrival of the second process $N_2(t)$ is $\mathrm{Unif}[0,1]$. For instance, for item (a), consider the first arrival of $N(t)=N_1(t)+N_2(t)$, which is denoted by $X_1$. we have $Pr(X_1<=a)=a$ for $a\in[0,1]$. Now consider the second arrival. Let say the first arrival happens at time $0.9$, i.e., $X_1=0.9$. Then, since $N_1(t)$ is based on a deterministic arrival we have $Pr(X_2<=0.1|X_1=0.9)=1$ which clearly shows the interarrivals are not independent and identically distributed.
You can use this example to answer other parts