<abstract xmlns="http://www.w3.org/1999/xhtml">

<p>The aim of this paper is to study sequences of numbers as random variables. The asymptotic density will play the role of the probability.</p>
<p>In the first part of this paper, the notion of natural metric on the set of natural numbers is defined. It is a metric so that the completion of ℕ is a compact metric space on which a probability Borel measure exists so that the sequence {<italic>n</italic>} is uniformly distributed. This condition connects the asymptotic density and the mentioned measure. A necessary and sufficient condition is derived so that a given metric is natural. Later, we study the properties of sequences uniformly continuous with respect to the given natural metric. Inter alia, the continuity ofdistribution function is characterized.</p>
</abstract>

The aim of this paper is to study sequences of numbers as random variables. The asymptotic density will play the role of the probability.
In the first part of this paper, the notion of natural metric on the set of natural numbers is defined. It is a metric so that the completion of ℕ is a compact metric space on which a probability Borel measure exists so that the sequence {n} is uniformly distributed. This condition connects the asymptotic density and the mentioned measure. A necessary and sufficient condition is derived so that a given metric is natural. Later, we study the properties of sequences uniformly continuous with respect to the given natural metric. Inter alia, the continuity ofdistribution function is characterized.

Metrics on ℕ and the Distribution of Sequences

Department of Mathematics and Informatics, Faculty of Education

University of Trnava, Trnava and Institute of Mathematics, Slovak Academy of Sciences

Tatra Mountains Mathematical Publications

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

{"article-title":"Metrics on ℕ and the Distribution of Sequences"}

The aim of this paper is to study sequences of numbers as random variables. The asymptotic density will play the role of the probability.
In the first part of...

Metrics on ℕ and the Distribution of Sequences

Published Online: Feb 15, 2023

Page range: 29 - 52

Received: Jun 06, 2022

DOI: https://doi.org/10.2478/tmmp-2022-0017

Keywords
uniform distribution, distribution function, asymptotic density, Weyl criterion, natural metric, natural measure

© 2022 Milan Paštéka, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Metrics on ℕ and the Distribution of Sequences

Published Online: Feb 15, 2023

Page range: 29 - 52

Received: Jun 06, 2022

DOI: https://doi.org/10.2478/tmmp-2022-0017

Keywordsuniform distribution, distribution function, asymptotic density, Weyl criterion, natural metric, natural measure

© 2022 Milan Paštéka, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Keywords
uniform distribution, distribution function, asymptotic density, Weyl criterion, natural metric, natural measure