Let $(N_t)_{t \geq 0}$ be a ordinary renewal process, i.e. the interarrival times of the processes jumps say $(T_k-T_{k-1})_{k\in\mathbb{N}}$ is an i.i.d sequence.
In an example in my book on the subject they consider a function $Z(t)=E(T_{N_t+1}-t)$. Then it says "Consider $E(T_{N_t+1}-t|T_1=s)$. If $s \leq t$ then $E(T_{N_t+1}-t|T_1=s)=Z(t-s)$. [...]."
I have tried to prove the last equality for some hours now, but i cant seem to formalize it. Would a kind soul, show how to prove this?
Conditionally on $T_1=s$, the process $(N_{t+s})_{t\geqslant0}$ is distributed as $(1+N_t)_{t\geqslant0}$ hence $T_{N_{t+s}}$ is distributed, conditionally on $T_1=s$, as $T_{1+N_t}+s$, unconditionally.
In particular $E(T_{N_{t+s}}\mid T_1=s)=E(T_{1+N_t}+s)$ hence $E(T_{N_{t+s}}-(t+s)\mid T_1=s)=E(T_{1+N_t}-t)=Z(t)$ for every $s\geqslant0$, as desired.