The Expiring Coupon Collector: Sliding-Window Surjection Flux and Rare-Entry Laws

Abstract

We study the coupon collector with deterministic expiration: one coupon is drawn at each time, and each coupon remains active for exactly $M$ draws. Completion occurs when all $n$ coupon types are simultaneously active. Equivalently, the current length-$M$ sliding window of draws must contain all $n$ types. The central object is not the one-time probability that a random window is onto, but the stationary flux of new entries into the onto-window set. We compute this flux exactly: \[ \mu_{n,M} =\Pbb(W_{t-1}\text{ is not onto},\ W_t\text{ is onto}) =\frac{(n-1)(n-1)!S(M-1,n-1)}{n^M}, \] where $S(\cdot,\cdot)$ denotes a Stirling number of the second kind. Under a quantitative subcritical separation condition, satisfied in particular by every fixed integer scale $M=\floor{\alpha n\log n}$, $0<\alpha<1$, we prove local declumping and obtain \[ \mu_{n,M}T_{n,M}\Rightarrow \Exp(1). \] For the fixed subcritical scale $M=\floor{\alpha n\log n}$, $0<\alpha<1$, this gives the logarithmic scale \[ \log T_{n,M}=n^{1-\alpha}+o_{\mathbb P}(n^{1-\alpha}), \qquad \log \Ebb T_{n,M}=n^{1-\alpha}+o(n^{1-\alpha}), \] and, when $\alpha>1/2$, the sharper normalization \[ n^{-\alpha}e^{-n^{1-\alpha}}T_{n,M}\Rightarrow \Exp(1), \qquad \Ebb T_{n,M}\sim n^\alpha e^{n^{1-\alpha}}. \] Thus the leading scale proposed in the Math StackExchange discussion is made rigorous; the exact finite-$n$ flux gives the canonical normalization throughout the subcritical range. The result is a sliding-window companion to rare-void entry-flux methods for nonmonotone coupon collectors.