Maximizing the Probability of Stopping on Any of the Last $m$ Successes in Independent Bernoulli Trials with Random Horizona. Tamaki, M. Advances in Applied Probability, 43(3):760–781, September, 2011.
Maximizing the Probability of Stopping on Any of the Last $m$ Successes in Independent Bernoulli Trials with Random Horizona [link]Paper  doi  bibtex   
@Article{Tamaki:2011:MPS,
  author =       "Mitsushi Tamaki",
  title =        "Maximizing the Probability of Stopping on Any of the
                 Last $m$ Successes in Independent {Bernoulli} Trials
                 with Random Horizona",
  journal =      j-ADV-APPL-PROB,
  volume =       "43",
  number =       "3",
  pages =        "760--781",
  month =        sep,
  year =         "2011",
  CODEN =        "AAPBBD",
  DOI =          "https://doi.org/10.2307/23024574",
  ISSN =         "0001-8678 (print), 1475-6064 (electronic)",
  ISSN-L =       "0001-8678",
  bibdate =      "Sat May 3 07:43:13 MDT 2014",
  bibsource =    "http://www.jstor.org/stable/i23024564;
                 http://www.math.utah.edu/pub/tex/bib/advapplprob.bib",
  URL =          "http://www.jstor.org/stable/23024574",
  acknowledgement = ack-nhfb,
  ajournal =     "Adv. Appl. Probab.",
  fjournal =     "Advances in Applied Probability",
  journal-URL =  "http://www.jstor.org/journals/00018678.html",
}

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