## Electronic Journal of Probability

- Electron. J. Probab.
- Volume 22 (2017), paper no. 17, 38 pp.

### Functional central limit theorem for subgraph counting processes

#### Abstract

The objective of this study is to investigate the limiting behavior of a subgraph counting process built over random points from an inhomogeneous Poisson point process on $\mathbb R^d$. The subgraph counting process we consider counts the number of subgraphs having a specific shape that exist outside an expanding ball as the sample size increases. As underlying laws, we consider distributions with either a regularly varying tail or an exponentially decaying tail. In both cases, the nature of the resulting functional central limit theorem differs according to the speed at which the ball expands. More specifically, the normalizations in the central limit theorems and the properties of the limiting Gaussian processes are all determined by whether or not an expanding ball covers a region - called a weak core - in which the random points are highly densely scattered and form a giant geometric graph.

#### Article information

**Source**

Electron. J. Probab., Volume 22 (2017), paper no. 17, 38 pp.

**Dates**

Received: 11 February 2016

Accepted: 26 January 2017

First available in Project Euclid: 15 February 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.ejp/1487127645

**Digital Object Identifier**

doi:10.1214/17-EJP30

**Mathematical Reviews number (MathSciNet)**

MR3622887

**Zentralblatt MATH identifier**

1357.60056

**Subjects**

Primary: 60G70: Extreme value theory; extremal processes 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]

Secondary: 60G15: Gaussian processes 60G18: Self-similar processes

**Keywords**

extreme value theory functional central limit theorem geometric graph regular variation von-Mises function

**Rights**

Creative Commons Attribution 4.0 International License.

#### Citation

Owada, Takashi. Functional central limit theorem for subgraph counting processes. Electron. J. Probab. 22 (2017), paper no. 17, 38 pp. doi:10.1214/17-EJP30. https://projecteuclid.org/euclid.ejp/1487127645