The Asymptotic Behavior of the Reward Sequence in the Optimal Stopping of I.I.D. Random Variables

Abstract

Let $X_1, X_2,\ldots$ be integrable, i.i.d. r.v.'s with common distribution function $F$ and let $\{v_n\}_{n \geq 1}$ be the sequence of optimal rewards or values in the associated optimal stopping problem, i.e., $v_n = \sup\{E(X_T): T \text{is a stopping time for} \{X_m\}_{m\geq 1} \text{and} T \leq n\}$ for $n \geq 1$. For distribution functions $F$ in the domain of attraction of one of the three classical extreme-value laws $G_I, G^\alpha_{II}$ or $G^\alpha_{III}$, it is shown that $\lim_n n(1 - F(v_n)) = 1, 1 - \alpha^{-1}$, or $1 + \alpha^{-1}$ if $F \in \mathscr{D}(G_1), F \in \mathscr{D}(G^\alpha_{II})$ and $\alpha > 1$, or $F \in \mathscr{D}(G^\alpha_{III})$ and $\alpha > 0$, respectively. From this result, the growth rate of $\{v_n\}_{n\geq 1}$ is obtained and compared to the growth rate of the expected maximum sequence. Also, the limit distribution of the optimal reward r.v.'s $\{X_{T^\ast_n}\}_{n\geq 1}$ is derived, where $\{T^\ast_n\}_{n\geq 1}$ are the optimal stopping times defined by $T^\ast_n \equiv 1$ if $n = 1$ and, for $n = 2,3,\ldots$, by $T^\ast_n = \min\{1 \leq k < n: X_k > v_{n-k}\}$ if this set is not equal to $\varnothing$ and equal to $n$ otherwise. This tail-distribution growth rate is shown to be sufficient for any threshold sequence to be asymptotically optimal.