Degenerate and poisson convergence criteria for success runs

Anant P. Godbole

Statistics & Probability Letters, 1990, vol. 10, issue 3, 247-255

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.

Keywords: Bernoulli; trials; success; runs; rare; events; discrete; distributions; of; order; k; Chernoff--Okamoto; inequalities (search for similar items in EconPapers)
Date: 1990
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