Mittag-Leffler L\'evy Processes

TL;DR

This paper introduces the Mittag-Leffler Le9vy process, providing novel representations, analyzing its properties including fractional moments, density, and heavy-tailed behavior, thus extending the understanding of Mittag-Leffler distributions.

Contribution

It presents new representations of the Mittag-Leffler Le9vy process and analyzes its properties, including fractional moments and self-similarity, expanding existing knowledge.

Findings

Fractional moments are derived due to infinite integer moments.

Density function and Le9vy measure are obtained.

Heavy-tailed behavior and self-similarity are demonstrated.

Abstract

In this article, we introduce Mittag-Leffler L\'evy process and provide two alternative representations of this process. First, in terms of Laplace transform of the marginal densities and next as a subordinated stochastic process. Both these representations are useful in analyzing the properties of the process. Since integer order moments for this process are not finite, we obtain fractional order moments. Its density function and corresponding L\'evy measure density is also obtained. Further, heavy tailed behavior of densities and stochastic self-similarity of the process is demonstrated. Our results generalize and complement the results available on Mittag-Leffler distribution in several directions.