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### Janvresse, É and de la Rue, T (2012)

#### Averaging along Uniform Random Integers

Uniform Distribution Theory 7(2), pp. 35–60.

ISSN/ISBN: Not available at this time. DOI: Not available at this time.

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.

Bibtex:
@article{, title={Averaging along uniform random integers}, author={Janvresse, {\'E}lise and De La Rue, Thierry}, journal={Uniform Distribution Theory}, volume={7}, issue={2}, year={2012}, pages={35--60}, }

Reference Type: Journal Article

Subject Area(s): Analysis