Degenerate and poisson convergence criteria for success runs
Document Type
Article
Publication Date
8-1990
Department
Department of Mathematical Sciences
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 → ∞, 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.
Publication Title
Statistics and Probability Letters
Recommended Citation
Godbole, A.
(1990).
Degenerate and poisson convergence criteria for success runs.
Statistics and Probability Letters,
10(3), 247-255.
http://doi.org/10.1016/0167-7152(90)90082-I
Retrieved from: https://digitalcommons.mtu.edu/michigantech-p/5604