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Published in:
2012 | OriginalPaper | Chapter
3. Clubbed Binomial Approximation for the Lightbulb Process
Authors : Larry Goldstein, Aihua Xia
Published in: Probability Approximations and Beyond
Publisher: Springer New York
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Abstract
In the so called lightbulb process, on days \(r=1,\ldots,n,\) out of n lightbulbs, all initially off, exactly r bulbs selected uniformly and independent of the past have their status changed from off to on, or vice versa. With \(W_n\) the number of bulbs on at the terminal time n and \(C_n\) a suitable clubbed binomial distribution,
The result is shown using Stein’s method.
$$ d_{{{\rm TV}}}(W_n,C_n) \leqslant 2.7314 \sqrt{n} e^{-(n+1)/3} \quad \hbox{for all}\,n \geqslant 1. $$