Degenerate and poisson convergence criteria for success runs

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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 → ∞, nf(pn) → λ (0 < λ < ∞) for any ∝ satisfying pkn = o(∝(pn)) (n → ∞). 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 ∝(pn) = pn). If, instead, npkn → λ, we prove that N(k)n tends in law to a Poisson(λ) random variable. This improves a classical result of von Mises (1921) which required, in addition, that k → ∞. Rates of convergence are provided for the above results. © 1990.

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Statistics and Probability Letters