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

Kennedy, Douglas P.; Kertz, Robert P.

doi:10.1214/aop/1176990547

Cited by 38 publications

(35 citation statements)

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“…(c) We shall draw on the results of Kennedy and Kertz (1991), who show that the asymptotic behavior of the value sequence V n for optimal stopping of i.i.d. random variables depends on to which extremal distribution domain X belongs.…”

Section: Examplesmentioning

confidence: 99%

An unexpected connection between branching processes and optimal stopping

Assaf

Goldstein

Samuel‐Cahn

2000

Journal of Applied Probability

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A curious connection exists between the theory of optimal stopping for independent random variables, and branching processes. In particular, for the branching process Z n with offspring distribution Y , there exists a random variable X such that the probability P (Z n = 0) of extinction of the nth generation in the branching process equals the value obtained by optimally stopping the sequence X 1 , . . . , X n , where these variables are i.i.d distributed as X. Generalizations to the inhomogeneous and infinite horizon cases are also considered. This correspondence furnishes a simple 'stopping rule' method for computing various characteristics of branching processes, including rates of convergence of the n th generation's extinction probability to the eventual extinction probability, for the supercritical, critical and subcritical Galton-Watson process. Examples, bounds, further generalizations and a connection to classical prophet inequalities are presented. Throughout, the aim is to show how this unexpected connection can be used to translate methods from one area of applied probability to another, rather than to provide the most general results.

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Section: Examplesmentioning

confidence: 99%

An unexpected connection between branching processes and optimal stopping

Assaf

Goldstein

Samuel‐Cahn

2000

Journal of Applied Probability

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“…It is well known that there are three types of asymptotic distributions for the maximum (see Leadbetter et al, 1983, p. 4), corresponding to three domains of attraction. The asymptotic behavior of V 1 n for the one-choice situation has been studied by Kennedy and Kertz (1991), who show that the limiting behavior of V 1 n depends upon the domain of attraction to which F belongs. This will therefore clearly also be the case for the two-choice value sequence, V 2 n .…”

Section: Introductionmentioning

confidence: 99%

“…where is Euler's constant (see, e.g., Leadbetter et al, 1983), and, as obtained by Kennedy and Kertz (1991),…”

Section: Introductionmentioning

confidence: 99%

Optimal Two-Choice Stopping on an Exponential Sequence

Goldstein

Samuel‐Cahn

2006

Sequential Analysis

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Let X 1 X 2 X n be independent and identically distributed with distribution function F . A statistician may choose two X values from the sequence by means of two stopping rules t 1 t 2 , with the goal of maximizing E X t 1 ∨ X t 2 . We describe the optimal stopping rules and the asymptotic behavior of the optimal expected stopping values, V 2 n , as n → , when F is the exponential distribution. Specifically, we show that lim n→ n 1 − F V 2 n = 1 − e −1 , and conjecture that this same limit obtains for any F in the (Type I) domain of attraction of exp −e −x .

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“…sequences with discount and observation costs. In the case m = 1 this problem has been considered in various degree of generality in [8], [9], [11], [4]. m-stopping in the max case has been considered in [6].…”

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confidence: 99%