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Degenerate and poisson convergence criteria for success runs

Author

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  • Godbole, Anant P.

Abstract

Let N(k)n be the number of success runs of length k > 1 in n Bernoulli trials, each with success probability pn. We show that N(k)n converges weakly to the distribution degenerate at zero as n --> [infinity], nf(pn) --> [lambda] (0 [infinity]). This answers, in the negative, a question posed by Philippou and Makri (1986) who suspected that a Poisson distribution of order k might be the target limit (if [is proportional to](pn) = pn). If, instead, npkn --> [lambda], we prove that N(k)n tends in law to a Poisson([lambda]) random variable. This improves a classical result of von Mises (1921) which required, in addition, that k --> [infinity]. Rates of convergence are provided for the above results.

Suggested Citation

  • Godbole, Anant P., 1990. "Degenerate and poisson convergence criteria for success runs," Statistics & Probability Letters, Elsevier, vol. 10(3), pages 247-255, August.
    • Handle: RePEc:eee:stapro:v:10:y:1990:i:3:p:247-255
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