Averaging along Uniform Random Integers

Abstract : Motivated by giving a meaning to ''The probability that a random integer has initial digit d'', we define a URI-set as a random set E of natural integers such that each n>0 belongs to E with probability 1/n, independently of other integers. This enables us to introduce two notions of densities on natural numbers: The URI-density, obtained by averaging along the elements of E, and the local URI-density, which we get by considering the k-th element of E and letting k go to infinity. We prove that the elements of E satisfy Benford's law, both in the sense of URI-density and in the sense of local URI-density. Moreover, if b_1 and b_2 are two multiplicatively independent integers, then the mantissae of a natural number in base b_1 and in base b_2 are independent. Connections of URI-density and local URI-density with other well-known notions of densities are established: Both are stronger than the natural density, and URI-density is equivalent to log-density. We also give a stochastic interpretation, in terms of URI-set, of the H_\infty-density.
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Submitted on : Monday, September 5, 2011 - 2:02:05 PM
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  • HAL Id : hal-00574623, version 2
  • ARXIV : 1103.1719

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Élise Janvresse, Thierry de La Rue. Averaging along Uniform Random Integers. Uniform Distribution Theory, Mathematical Institute of the Slovak Academy of Sciences, 2012, 7 (2), pp.35-60. ⟨hal-00574623v2⟩

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