## The Annals of Probability

- Ann. Probab.
- Volume 16, Number 1 (1988), 407-413.

### Moment and Geometric Probability Inequalities Arising from Arrangement Increasing Functions

Philip J. Boland, Frank Proschan, and Y. L. Tong

#### Abstract

A real-valued function $g$ of two vector arguments $\mathbf{x}$ and $\mathbf{y} \in R^n$ is said to be arrangement increasing if it increases in value as the arrangement of components in $\mathbf{x}$ becomes increasingly similar to the arrangement of components in $\mathbf{y}$. Hollander, Proschan and Sethuraman (1977) show that the convolution of arrangement increasing functions is arrangement increasing. This result is used to generate some interesting probability inequalities of a geometric nature for exchangeable random vectors. Other geometric inequalities for families of arrangement increasing multivariate densities are also given, and some moment inequalities are obtained.

#### Article information

**Source**

Ann. Probab., Volume 16, Number 1 (1988), 407-413.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176991911

**Digital Object Identifier**

doi:10.1214/aop/1176991911

**Mathematical Reviews number (MathSciNet)**

MR920281

**Zentralblatt MATH identifier**

0638.60013

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]

Secondary: 62H10: Distribution of statistics

**Keywords**

Arrangement increasing decreasing in transposition exchangeable random vector family of arrangement increasing densities inequalities moments Laplace transforms permutation

#### Citation

Boland, Philip J.; Proschan, Frank; Tong, Y. L. Moment and Geometric Probability Inequalities Arising from Arrangement Increasing Functions. Ann. Probab. 16 (1988), no. 1, 407--413. doi:10.1214/aop/1176991911. https://projecteuclid.org/euclid.aop/1176991911