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Random subsequences of {αn} with asymptotically independent successive terms

Abstract : Fix an irrational number α and let Y n be the number of attempts needed to get the nth success in a non-stationary sequence of independent Bernoulli trials. It is known that the law of the fractional part of αY n converges weakly to the uniform distribution on [0, 1) , as n → +∞, when the probabilities of success decrease to 0 and sum to +∞. We provide sufficient conditions on the probabilities of success ensuring that the fractional parts of αY n and αY n+1 are asymptotically independent. We extend our results to any number of successive terms, compute upper bounds of the convergence rates depending on a measure of irrationality of α and on the probabilities of success and apply our results to discuss the mantissae of 2 Yn and 2 Y n+1 .
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https://hal.archives-ouvertes.fr/hal-03502439
Contributor : Bruno Massé Connect in order to contact the contributor
Submitted on : Saturday, December 25, 2021 - 11:18:33 AM
Last modification on : Friday, January 7, 2022 - 3:39:26 AM
Long-term archiving on: : Saturday, March 26, 2022 - 6:07:55 PM

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  • HAL Id : hal-03502439, version 1

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Bruno Massé. Random subsequences of {αn} with asymptotically independent successive terms. [Research Report] Université du Littoral - Côte d'Opale. 2021. ⟨hal-03502439⟩

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