ThmDex – An index of mathematical definitions, results, and conjectures.
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Definition D3745
Renewal process

Let $X_1, X_2, X_3, \ldots \in \text{Random}[0, \infty)$ each be a D5452: Random unsigned real number such that
 (i) $X_1, X_2, X_3, \ldots$ is an D3358: I.I.D. random collection (ii) $$\mathbb{E} |X_1| < \infty$$
A D6140: Random natural number process $N : [0, \infty) \to \text{Random}(\mathbb{N})$ is a revewal process with respect to $X_1, X_2, X_3, \ldots$ if and only if $$\forall \, t \in [0, \infty) : N_t \overset{d}{=} \max \left\{ M \in \mathbb{N} : \sum_{m = 1}^M X_m \leq t \right\}$$
Subdefinitions
 ▶ Real poisson process
Children
 ▶ Real poisson process