Abstract
Let $\{Z_i\}$ be i.i.d., let $\{\varepsilon_i\}$ be i.i.d. Bernoulli, independent of $\{Z_i\}$, let $T_0 = z$ and $T_n = \varepsilon_n(T_{n-1} + Z_n)$ for $n \geqq 1$. Under a moment condition, optimal stopping rules are found for stopping $T_n - nc$ where $c > 0$ (the cost model), and for stopping $\beta^nT_n$ where $0 < \beta < 1$ (the discount model). Special cases are treated in detail. The cost model generalizes results of N. Starr, and the discount model generalizes results of Dubins and Teicher.
Citation
Thomas S. Ferguson. "Stopping a Sum During a Success Run." Ann. Statist. 4 (2) 252 - 264, March, 1976. https://doi.org/10.1214/aos/1176343405
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