age process is a Markov chain If $T_n=\sum_{k=1}^nX_k$ is a renewal process where $X_k$ are identical and independent with values in the positive integers, and $N(t)=\max\{j:T_j\leq t\}$ is the associated counting process, define $C_t=t-T_{N(t)}$ to be the "age" at time $t\in \mathbb{N}$. I'm wondering how to show rigorously  that $C_t$ is a Markov chain. 
Heuristically, I can see that $C_{t+1}$ is $C_t+1$ if there is no renewal at $t+1$ and $0$ if there is a renewal at $t+1$, and that the probability of a renewal at $t+1$ should depend on the location of the last renewal and the renewals before that should be irrelevant. But I don't know how to make this "rigorous" or calculate the transition probabilities. So my questions are: what is a rigorous proof that $\{C_t\}$ satisfies the Markov property, and how do you calculate the transition probabilities
\begin{align*}
\Pr(C_{t+1}=i+1|C_t=i),\\
\Pr(C_{t+1}=0|C_t=i).
\end{align*}
The texts I'm reading treat this as easy enough to leave out the details, but I haven't grasped it yet.
 A: This just summarizes my comments to formally provide an answer. Let $t$ be a positive integer.  For a given trajectory $i_0, i_1, ..., i_{t-1}$ of the $C_t$ process, define the history event $$H = \{C_j = i_j \: \forall j \in \{0, 1, ..., t-1\}\}$$  
Easy way:
Let $i \in \{0, 1, 2, ...\}$.  Then 
$$ P[C_{t+1}=0|C_t=i, H] = P[X_1=i+1|X_1>i] $$
and so this is conditionally independent of the history $H$, given that $C_t=i$. 

More formal:
You can condition on $N_t=k$ and $(X_1, ..., X_k)=x$ for all $k \in \{0, 1, 2, ...\}$ and $x \in \mathbb{Z}^k$.  
\begin{align*}
&P[C_{t+1}=0|C_t=i, H] \\
&= \sum_{k=0}^{\infty} \sum_{x \in \mathbb{Z}^k} \underbrace{P[C_{t+1}=0|C_t=i, H, N_t=k, (X_1, ..., X_k)=x]}_{P[X_{k+1}=i+1|X_{k+1}>i]}P[N_t=k, (X_1, ..., X_k)=x|C_t=i, H]\\
&= P[X_1=i+1|X_1>i]\underbrace{\sum_{k=0}^{\infty} \sum_{x \in \mathbb{Z}^k}P[N_t=k, (X_1, ..., X_k)=x|C_t=i, H]}_{1}
\end{align*}
where (even more formally) we can restrict the double sum to consider only those $k$ and $x$ values that satisfy: 
$$P[N_t=k, (X_1, ..., X_k)=k|C_t=i, H]>0$$
and note that for such cases, the following two events are the same: 
\begin{align}
&\{C_t=i, H, N_t=k, (X_1, ..., X_k) = x\} \\
&\{X_{k+1}>i, (X_1, ..., X_k)=x\}
\end{align}
