A Curious Connection Between Optimal Stopping and Branching Processes

A curious connection exists between the theory of branching processes and optimal stopping for independent random variables. For the branching process Zn with offspring distribution Y , there exists a random variable X such that the probability P (Zn = 0) of extinction of the n generation in the branching process equals the value obtained by optimally stopping the independent sequence X1, . . . , Xn with distribution X. The correspondence can be generalized to the inhomogeneous and infinite horizon cases, and sometimes furnishes simple ‘stopping rule’ methods for computing various characteristics of branching processes, such as rates of convergence of the n generation’s extinction probability to the eventual extinction probability. It also provides a method for computing bounds on branching extinction probabilities by applying prophet inequalities and suboptimal rules in the corresponding stopping problem. This curious correspondence may also be used in the other direction, to inform the theory of optimal stopping using results from branching. We have no probabilistic explanation for the correspondence. Two Choice Stopping: How much is an extra chance worth? David Assaf, Larry Goldstein and Ester Samuel-Cahn Abstract A statistician sequentially observes the values in the independent, identically distributed sequence Xn, . . . , X1 with known distribution F , and is given two chances to choose as small a value as possible using stopping rules. Let V 2 n equal the expectation of the smaller of the two values chosen when proceeding optimally. Then for a large class of F ’s belonging to the domain of attraction (for the minimum) of D(G), where G(x) = [1− exp(−x)]I(x ≥ 0),A statistician sequentially observes the values in the independent, identically distributed sequence Xn, . . . , X1 with known distribution F , and is given two chances to choose as small a value as possible using stopping rules. Let V 2 n equal the expectation of the smaller of the two values chosen when proceeding optimally. Then for a large class of F ’s belonging to the domain of attraction (for the minimum) of D(G), where G(x) = [1− exp(−x)]I(x ≥ 0), lim n→∞ nF (V 2 n ) = h (dα) where dα > 0 is the unique solution d to ∫ d 0 h(y)dy + (1/α− d)h(d) = 0, and h(y) is the function h(y) = ( y 1 + αy/(α+ 1) )1/α for y ≥ 0. For ‘most’ α, having two choices is a substantial improvment over having one as measured by asymptotic distance to the “prophet” sequence E(min{Xn, . . . , X1}). Berry Esseen Bounds for Local Extremes and Combinatorial Central Limit Theorems, using Size and Zero Biasing