# Harmonic statistics

## Abstract

The exponential, the normal, and the Poisson statistical laws are of major importance due to their universality. Harmonic statistics are as universal as the three aforementioned laws, but yet they fall short in their ‘public relations’ for the following reason: the full scope of harmonic statistics cannot be described in terms of a statistical law. In this paper we describe harmonic statistics, in their full scope, via an object termed harmonic Poisson process: a Poisson process, over the positive half-line, with a harmonic intensity. The paper reviews the harmonic Poisson process, investigates its properties, and presents the connections of this object to an assortment of topics: uniform statistics, scale invariance, random multiplicative perturbations, Pareto and inverse-Pareto statistics, exponential growth and exponential decay, power-law renormalization, convergence and domains of attraction, the Langevin equation, diffusions, Benford’s law, and 1/f noise. - Highlights: • Harmonic statistics are described and reviewed in detail. • Connections to various statistical laws are established. • Connections to perturbation, renormalization and dynamics are established.

- Authors:

- Publication Date:

- OSTI Identifier:
- 22617503

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Annals of Physics; Journal Volume: 380; Other Information: Copyright (c) 2017 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; HARMONICS; LANGEVIN EQUATION; RENORMALIZATION; STATISTICS

### Citation Formats

```
Eliazar, Iddo, E-mail: eliazar@post.tau.ac.il.
```*Harmonic statistics*. United States: N. p., 2017.
Web. doi:10.1016/J.AOP.2017.03.016.

```
Eliazar, Iddo, E-mail: eliazar@post.tau.ac.il.
```*Harmonic statistics*. United States. doi:10.1016/J.AOP.2017.03.016.

```
Eliazar, Iddo, E-mail: eliazar@post.tau.ac.il. Mon .
"Harmonic statistics". United States.
doi:10.1016/J.AOP.2017.03.016.
```

```
@article{osti_22617503,
```

title = {Harmonic statistics},

author = {Eliazar, Iddo, E-mail: eliazar@post.tau.ac.il},

abstractNote = {The exponential, the normal, and the Poisson statistical laws are of major importance due to their universality. Harmonic statistics are as universal as the three aforementioned laws, but yet they fall short in their ‘public relations’ for the following reason: the full scope of harmonic statistics cannot be described in terms of a statistical law. In this paper we describe harmonic statistics, in their full scope, via an object termed harmonic Poisson process: a Poisson process, over the positive half-line, with a harmonic intensity. The paper reviews the harmonic Poisson process, investigates its properties, and presents the connections of this object to an assortment of topics: uniform statistics, scale invariance, random multiplicative perturbations, Pareto and inverse-Pareto statistics, exponential growth and exponential decay, power-law renormalization, convergence and domains of attraction, the Langevin equation, diffusions, Benford’s law, and 1/f noise. - Highlights: • Harmonic statistics are described and reviewed in detail. • Connections to various statistical laws are established. • Connections to perturbation, renormalization and dynamics are established.},

doi = {10.1016/J.AOP.2017.03.016},

journal = {Annals of Physics},

number = ,

volume = 380,

place = {United States},

year = {Mon May 15 00:00:00 EDT 2017},

month = {Mon May 15 00:00:00 EDT 2017}

}