Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-08T19:27:32.063Z Has data issue: false hasContentIssue false
  • English
  • Français

Moments of Random Sums and Robbins' Problem of Optimal Stopping

Part of: Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Alexander Gnedin  and
Alexander Iksanov
Alexander Gnedin*
Affiliation:
Utrecht University
Alexander Iksanov*
Affiliation:
National Taras Shevchenko University of Kiev
*
∗ Postal address: Department of Mathematics, Utrecht University, Postbus 80010, 3508 TA Utrecht, The Netherlands. Email address: [email protected]
∗∗ Postal address: Faculty of Cybernetics, National Taras Shevchenko University of Kiev, Kiev 01033, Ukraine. Email address: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Robbins' problem of optimal stopping is that of minimising the expected rank of an observation chosen by some nonanticipating stopping rule. We settle a conjecture regarding the value of the stopped variable under the rule that yields the minimal expected rank, by embedding the problem in a much more general context of selection problems with the nonanticipation constraint lifted, and with the payoff growing like a power function of the rank.

Keywords

Stopping timeRobbins' problem of minimising the expected rankPoisson embeddingrandom sum

MSC classification

Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory 60G50: Sums of independent random variables; random walks
Type
Short Communications
Information
Journal of Applied Probability , Volume 48 , Issue 4 , December 2011 , pp. 1197 - 1199
Copyright
Copyright © Applied Probability Trust 2011 

References

[1] Alsmeyer, G. and Iksanov, A. (2009). A log-type moment result for perpetuities and its application to martingales in supercritical branching random walks. Electron. J. Prob. 14, 289–312.Google Scholar
[2] Assaf, D. and Samuel-Cahn, E. (1996). The secretary problem: minimizing the expected rank with i.i.d. random variables. Adv. Appl. Prob. 28, 828–852.Google Scholar
[3] Bruss, F. T. (2005). What is known about Robbinsö problem? J. Appl. Prob. 42, 108–120.Google Scholar
[4] Bruss, F. T. and Swan, Y. C. (2009). A continuous-time approach to Robbins' problem of minimizing the expected rank. J. Appl. Prob. 46, 1–18.Google Scholar
[5] Gianini, J. and Samuels, S. M. (1976). The infinite secretary problem. Ann. Prob. 3, 418–432.Google Scholar
[6] Gnedin, A. V. (2007). Optimal stopping with rank-dependent loss. J. Appl. Prob. 44, 996–1011.Google Scholar