A note on last-success-problem

Author:
J. M. Grau Ribas

Journal:
Theor. Probability and Math. Statist. **103** (2020), 155-165

MSC (2020):
Primary 60G40, 62L15, 91A60

DOI:
https://doi.org/10.1090/tpms/1139

Published electronically:
June 16, 2021

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Additional Information

Abstract: We consider the Last-Success-Problem with $n$ independent Bernoulli random variables with parameters $p_i>0$. We improve the lower bound provided by F.T. Bruss for the probability of winning and provide an alternative proof to the one given in [**3**] for the lower bound ($1/e$) when $R≔\sum _{i=1}^n (p_i/(1-p_i))\geq 1$. We also consider a modification of the game which consists in not considering it a failure when all the random variables take the value of 0 and the game is repeated as many times as necessary until a “$1$” appears. We prove that the probability of winning in this game when $R\leq 1$ is lower-bounded by $0.5819\ldots =\frac {1}{e-1}$. Finally, we consider the variant in which the player can choose between participating in the game in its standard version or predict that all the random variables will take the value 0.

References
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References
- P. Allaart and J. A. Islas
*A sharp lower bound for choosing the maximum of an independent sequence*, J. Appl. Prob. **53** (2016), no. 4, 1041–1051. MR **3581240**
- F. T. Bruss,
*Sum the odds to one and stop*, Ann. Probab. **28** (2000), no. 3, 1384–1391. MR **1797879**
- F. T. Bruss,
*A note on bounds for the odds theorem of optimal stopping*, Ann. Probab. **31** (2003), no. 4, 1859–1861. MR **2016602**
- F. T. Bruss,
*Odds-theorem and monotonicity*, Math. Applicanda **47** (2019), no. 1, 25–43. MR **3988930**
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*The sum-the-odds theorem with application to a stopping game of Sakaguchi*, Math. Appl. **44** (2016), no. 1, 45–61. MR **3557090**
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*Optimal stopping and applications*, Electronic Text at http://www.math.ucla.edu/{$\thicksim $}tom/Stopping/Contents.html (2006).
- J. M. Grau Ribas,
*An extension of the Last-Success-Problem*, Stat. Probab. Lett. **156** (2020), Article 108591. MR **4007552**
- J. M. Grau Ribas,
*A turn-based game related to the Last-Success-Problem*, Dyn. Games Appl. **10** (2019), no. 4, 836–844. MR **4181815**
- T. P. Hill and U. Krengel,
*A prophet inequality related to the secretary problem*, Contemp. Math. **125** (1992), 209–215. MR **1160621**
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Additional Information

**J. M. Grau Ribas**

Affiliation:
Departamento de Matemáticas, Universidad de Oviedo, Avda. Calvo Sotelo s/n, 33007 Oviedo, Spain

Email:
grau@uniovi.es

Keywords:
Last-Success-Problem,
lower bounds,
odds-theorem,
optimal stopping,
optimal threshold

Received by editor(s):
June 7, 2020

Published electronically:
June 16, 2021

Article copyright:
© Copyright 2020
Taras Shevchenko National University of Kyiv