= Solution
A <renewal process> counts arrivals $N(t)=\max\{j:A_j\leq t\}$, where $A_0=0,A_j=S_1+\cdots+S_j$ and the holding times are <independent>, identically distributed positive random variables. Their sum tends to infinity almost surely, so the count is finite on bounded time intervals. Use $N(s)=X_s$ in the question's notation. The index $J=N(s)+1=\min\{j:A_j>s\}$ is a stopping index for the holding-time sequence. Conditional on $J=j$ and the first $j$ holding times, every later $S_{j+k}$ retains its original <probability distribution>. Summing over $j$ proves
$$
\boxed{P(T_n>t)=P(S_1>t)\quad(n\geq2).}
$$
The straddling time $T_1=S_J$ is different. Let $\overline F(t)=P(S_1>t)$ and let $U$ be the <renewal measure> $U(B)=\sum_{j\geq0}P(A_j\in B)$. <Independence> of the next <holding time> from its start gives
$$
P(T_1>t)=\int_{[0,s]}\overline F(\max(t,s-u))\,U(du),
\qquad1=\int_{[0,s]}\overline F(s-u)\,U(du).
$$
For two numbers in $[0,1]$, their minimum is at least their product. Since $\overline F(\max(t,a))=\min(\overline F(t),\overline F(a))$, subtracting $\overline F(t)$ times the second integral proves
$$
\boxed{P(T_1>t)\geq P(S_1>t).}
$$
These formulae handle holding-time atoms with the stated renewal convention $A_j\leq s<A_{j+1}$; sums may be interchanged by nonnegativity.
For rate-$\lambda$ exponentials, the <renewal process> is Poisson. The elapsed age $B_s=s-A_{N(s)}$ has <probability distribution> $\min(E,s)$ with $E\sim\operatorname{Exp}(\lambda)$: for $0\leq b<s$, $P(B_s>b)$ is the <probability> of no arrivals in $(s-b,s]$, namely $e^{-\lambda b}$, with an atom $e^{-\lambda s}$ at $s$. The forward residual $R_s$ is an <independent> rate-$\lambda$ exponential by <independent> future increments. Thus $T_1=B_s+R_s$. Conditioning on the age gives the exact survival law
$$
\boxed{P(T_1>t)=e^{-\lambda t}\big[1+\lambda\min(s,t)\big],\qquad t\geq0.}
$$
For $s,t>0$ the extra term is strictly positive, proving strict inequality. For fixed $t$ and $s\to\infty$, the limit is $e^{-\lambda t}(1+\lambda t)$, which equals the survival <probability> of the sum of two <independent> rate-$\lambda$ exponentials. This is the finite-time form of the <stochastic length bias of a renewal interval>.
Back to article page
