## Brazilian Journal of Probability and Statistics

- Braz. J. Probab. Stat.
- Volume 33, Number 1 (2019), 39-56.

### Inventory model of type $(s,S)$ under heavy tailed demand with infinite variance

Aslı Bektaş Kamışlık, Tülay Kesemen, and Tahir Khaniyev

#### Abstract

In this study, a stochastic process $X(t)$, which describes an inventory model of type $(s,S)$ is considered in the presence of heavy tailed demands with infinite variance. The aim of this study is observing the impact of regularly varying demand distributions with infinite variance on the stochastic process $X(t)$. The main motivation of this work is, the publication by Geluk [*Proceedings of the American Mathematical Society* **125** (1997) 3407–3413] where he provided a special asymptotic expansion for renewal function generated by regularly varying random variables. Two term asymptotic expansion for the ergodic distribution function of the process $X(t)$ is obtained based on the main results proposed by Geluk [*Proceedings of the American Mathematical Society* **125** (1997) 3407–3413]. Finally, weak convergence theorem for the ergodic distribution of this process is proved by using Karamata theory.

#### Article information

**Source**

Braz. J. Probab. Stat., Volume 33, Number 1 (2019), 39-56.

**Dates**

Received: September 2016

Accepted: September 2017

First available in Project Euclid: 14 January 2019

**Permanent link to this document**

https://projecteuclid.org/euclid.bjps/1547456486

**Digital Object Identifier**

doi:10.1214/17-BJPS376

**Mathematical Reviews number (MathSciNet)**

MR3898721

**Zentralblatt MATH identifier**

07031063

**Keywords**

Semi-Markovian inventory model of type $(s,S)$ heavy tailed distributions with infinite variance regular variation renewal reward process asymptotic expansion Karamata theorem

#### Citation

Bektaş Kamışlık, Aslı; Kesemen, Tülay; Khaniyev, Tahir. Inventory model of type $(s,S)$ under heavy tailed demand with infinite variance. Braz. J. Probab. Stat. 33 (2019), no. 1, 39--56. doi:10.1214/17-BJPS376. https://projecteuclid.org/euclid.bjps/1547456486