Q: What does it mean mathematically when we say that a regenerative cycle can be split into i.i.d. cycles? Asmussen's Applied Probability and Queues says:

The classical definition of a stochastic process $\{X(t)\}$ to be regenerative means in intuitive terms that the process can be split into i.i.d. cycles.

What does this mean mathematically? That for $\{X(t)\}$ to be regenerative means  there exists a sequence $\{S_n: n=0,1,2,\ldots\}$ of random variables with $S_n=0$ and $S_{n+1} = \inf\{t>S_n : t\geqslant 0\}$ such that $$\{X(t): t\geqslant 0\} = \bigcup_{n=0}^\infty \{X(S_n + t) : 0\leqslant t\leqslant S_n\}$$ where $\{X(S_n+t):t\geqslant 0\}\stackrel{\mathrm d}= \{X(t):t\geqslant 0\}$ and $\{X(S_n+t):t\geqslant 0\}$ is independent of $\{X(t):0\leqslant t \leqslant S_n\}$? The cycles would then be e.g. $C_n = \{X(S_n+t):0\leqslant t\leqslant S_n\}$ for nonnegative integer $n$?
