## Journal of Applied Probability

- J. Appl. Probab.
- Volume 50, Number 1 (2013), 42-53.

### Sharp bounds for sums of dependent risks

Giovanni Puccetti and Ludger Rüschendorf

#### Abstract

Sharp tail bounds for the sum of *d* random variables with given
marginal distributions and arbitrary dependence structure have been
known since Makarov (1981) and Rüschendorf (1982) for *d*=2 and, in
some examples, for *d*≥3. Based on a duality result, dual bounds have
been introduced in Embrechts and Puccetti (2006b). In the homogeneous
case,\break $*F*_{1}=···=*F*_{n}, with monotone density, sharp tail bounds
were recently found in Wang and Wang (2011). In this paper we establish
the sharpness of the dual bounds in the homogeneous case under general
conditions which include, in particular, the case of monotone densities
and concave densities. We derive the corresponding optimal couplings
and also give an effective method to calculate the sharp bounds.

#### Article information

**Source**

J. Appl. Probab., Volume 50, Number 1 (2013), 42-53.

**Dates**

First available in Project Euclid: 20 March 2013

**Permanent link to this document**

https://projecteuclid.org/euclid.jap/1363784423

**Digital Object Identifier**

doi:10.1239/jap/1363784423

**Mathematical Reviews number (MathSciNet)**

MR3076771

**Zentralblatt MATH identifier**

1282.60017

**Subjects**

Primary: 60E05: Distributions: general theory 91B30: Risk theory, insurance

**Keywords**

Bounds for dependent risks Fréchet bound mass transportation theory

#### Citation

Puccetti, Giovanni; Rüschendorf, Ludger. Sharp bounds for sums of dependent risks. J. Appl. Probab. 50 (2013), no. 1, 42--53. doi:10.1239/jap/1363784423. https://projecteuclid.org/euclid.jap/1363784423