## Brazilian Journal of Probability and Statistics

- Braz. J. Probab. Stat.
- Volume 29, Number 4 (2015), 878-896.

### Generalizations of some probability inequalities and $L^{p}$ convergence of random variables for any monotone measure

Hamzeh Agahi, Adel Mohammadpour, and Radko Mesiar

#### Abstract

This paper has three specific aims. First, some probability inequalities, including Hölder’s inequality, Lyapunov’s inequality, Minkowski’s inequality, concentration inequalities and Fatou’s lemma for Choquet-like expectation based on a monotone measure are shown, extending previous work of many researchers. Second, we generalize some theorems about the convergence of sequences of random variables on monotone measure spaces for Choquet-like expectation. Third, we extend the concept of uniform integrability for Choquet-like expectation. These results are useful for the solution of various problems in machine learning and made it possible to derive new efficient algorithms in any monotone system. Corresponding results are valid for capacities, the usefulness of which has been demonstrated by the rapidly expanding literature on generalized probability theory.

#### Article information

**Source**

Braz. J. Probab. Stat., Volume 29, Number 4 (2015), 878-896.

**Dates**

Received: July 2012

Accepted: May 2014

First available in Project Euclid: 17 September 2015

**Permanent link to this document**

https://projecteuclid.org/euclid.bjps/1442513451

**Digital Object Identifier**

doi:10.1214/14-BJPS251

**Mathematical Reviews number (MathSciNet)**

MR3397398

**Zentralblatt MATH identifier**

1328.60051

**Keywords**

Capacities probability inequalities Choquet-like expectation

#### Citation

Agahi, Hamzeh; Mohammadpour, Adel; Mesiar, Radko. Generalizations of some probability inequalities and $L^{p}$ convergence of random variables for any monotone measure. Braz. J. Probab. Stat. 29 (2015), no. 4, 878--896. doi:10.1214/14-BJPS251. https://projecteuclid.org/euclid.bjps/1442513451