Near-optimal independent inventory windows at both horizons

Consider the mission's lost-sales inventory model with integer lead time $L\ge1$, holding cost $h>0$, lost-sales penalty $p>0$, and i.i.d. nonnegative demands of finite mean $\mu>0$. Let $\pi$ be any admissible measurable history-dependent randomized policy, started from zero inventory and an empty pipeline, with finite limsup expected average cost $C(\pi)$.

For every $\varepsilon>0$, there exist an integrable nonnegative input vector $(I_0,A_0,\ldots,A_L)$, independent of a fresh demand sequence, and real numbers $r,z$, such that

and, for the recursion $I_{k+1}=(I_k+A_k-D_k)^+$,

This auxiliary existence theorem supplies the independent inventory window needed for both horizon constraints in Proposition 1. It is a new approximate formulation of the manuscript's stationary-policy step: it does not assert that the original optimum is attained. Its scope includes the full policy class and both horizons, so it is an additional obligation beyond the cited 2016 theorem's $L$-window conclusion.

Formalization Note Policy cost and the final cost comparison take values in the extended nonnegative reals; finiteness of the given policy cost is explicit. Window coordinates have finite real first moments, and independence from demands is represented by a product measure.