Stochastic length bias of a renewal interval
= Stochastic length bias of a renewal interval
At every deterministic $t>0$, the containing interval stochastically dominates an unselected interarrival time:
$$
\mathbb P(L(t)\geq x)\geq\mathbb P(\xi_1\geq x),\qquad x>0.
$$
Conditioning on the first interarrival gives a renewal equation; subtracting the constant tail probability leaves a renewal equation with a nonnegative forcing term.