Proposition 2 — Finite-cap policy cost bound

For every probability measure $P$ on the nonnegative real numbers with integrable identity function and strictly positive real mean $\mu=\int d\,P(\mathrm{d}d)$, every integer $L\ge1$, every pair of real numbers $h>0$ and $p>0$, and all real numbers $r,z$ satisfying the following feasibility conditions, the stated cost bound holds. For every nonnegative real demand sequence $d=(d_i)_{i\ge0}$ and integer $m\ge0$, write $F_m(d,r)=\max_{0\le k\le m}\sum_{i=0}^{k-1}(r-d_i)$, where $k=0$ contributes the empty sum $0$, and $E_m(d,r,z)=(F_m(d,r)+\sum_{i=0}^{m-1}(d_i-r)-z)^+$, with $y^+=\max(y,0)$. Feasibility means exactly $0\le r\le\mu$, $z\ge0$, and $\int E_m(d,r,z)\,P^{\otimes\mathbb N}(\mathrm dd)\le m(\mu-r)$ for both $m=L$ and $m=L+1$; these are nonnegative Lebesgue integrals and inequalities in the extended nonnegative reals $[0,\infty]$, with the positive parts and the finite right sides embedded into that space. Set $S=((L+1)r+z)^+$ and $a=r^+$, both finite nonnegative real numbers; the feasibility conditions make these equal to $(L+1)r+z$ and $r$. Let $(D_t)_{t\ge0}$ have the infinite independent product law $P^{\otimes\mathbb N}$, and independently let $U$ have law equal to Lebesgue measure restricted to $[0,1]$. Starting with $I_0=0$ and $x_{0,j}=0$ for every $0\le j<L$, use $q_t=\min\{(S-I_t-\sum_{j=0}^{L-1}x_{t,j})^+,a\}$ and the recursion $I_{t+1}=(I_t+x_{t,0}-D_t)^+$, $x_{t+1,j}=x_{t,j+1}$ for $0\le j<L-1$, and $x_{t+1,L-1}=q_t$ for every integer $t\ge0$. This recursion does not use $U$, although its cost is integrated over the stated joint law. Define $C_t=hI_{t+1}+p(D_t-I_t-x_{t,0})^+$ as an extended nonnegative real and $J_{S,a}=\limsup_{T\to\infty}\frac{1}{T+1}\sum_{t=0}^{T}\int C_t\,\mathrm d(P^{\otimes\mathbb N}\otimes\mathrm{Leb}|_{[0,1]})$, where $T$ ranges over nonnegative integers and each integral is a nonnegative Lebesgue integral. The assertion is $J_{S,a}\le\left[(h+\frac{p}{L+1})z+(2-\frac{1}{L+1})p(\mu-r)\right]^+$, with the finite right side embedded in $[0,\infty]$; it is already nonnegative under the hypotheses. All subtractions shown inside positive parts are truncated at zero, whereas the sums in $F_m$ and the expression inside $E_m$ use ordinary real subtraction. The average cost uses extended nonnegative arithmetic and can in its definition equal $+\infty$; its denominator is the positive finite number $T+1$. This is a bound for the specified pair $(S,a)$, with no infimum or assertion of optimality or attainment. The quantifiers include $L=1$, zero demand values, $r=0$, $r=\mu$, and $z=0$ when feasible, and $L\ge1$ ensures that $L+1$ is nonzero.