The Maximum of Independent Geometric Random Variables as the Time for Genomic Evolution |
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| Authors: | Aristides V. Doumas Vassilis G. Papanicolaou |
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| Affiliation: | 1. Department of Mathematics , National Technical University of Athens , Athens , Greece;2. Department of Mathematics , National Technical University of Athens , Athens , Greece;3. Boeing Center for Technology, Information, and Manufacturing , Olin School of Business, Washington University in St. Louis , St. Louis , Missouri |
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| Abstract: | □ We calculate the asymptotics of the moments as well as the limiting distribution (after the appropriate normalization) of the maximum of independent, not identically distributed, geometric random variables. In many cases, the limit distribution turns out to be the standard Gumbel. The motivation comes from a variant of the genomic evolutionary model proposed by Wilf and Ewens[ 15 Wilf , H.S. , Ewens , W.J . There's plenty of time for evolution . Proc. Nat. Acad. Sci. 2010 , 107 ( 52 ), 22454 – 22456 , doi: 10.1073/pnas.1016207107. [Crossref], [PubMed] , [Google Scholar] ] as an answer to the criticism of the Darwinian theory of evolution stating that the time required for the appropriate mutations is huge. A byproduct of our analysis is the asymptotics of the moments as well as the limiting distribution (after the appropriate normalization) of the maximum of independent, not identically distributed, exponential random variables. |
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| Keywords: | Coupon collector's problem (CCP) Genomic word Gumbel distribution Independent geometric random variables Limit distribution Maximum Moment-asymptotics Mutations |
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