A note on the Cauchy-type mixture distributions

ABSTRACT

An interesting class of continuous distributions, called Cauchy-type mixture, with potential applications in modelling erratic phenomena is introduced by Soltani and Tafakori [A class of continuous kernels and Cauchy type heavy tail distributions. Statist Probab Lett. 2013;83:1018–1027]. In this work, we provide more insights into the Cauchy-type mixture distributions, involving certain characterizations, connections with the generalized Linnik distributions and the class of discrete distributions induced by stable laws. We also prove that the Laplace transform of Cauchy-type mixture distributions when normalized by constant terms become as a density functions in terms of distributional conjugate property.

KEYWORDS:

• One-sided Cauchy distributions
• Linnik distribution
• stable distributions
• Cauchy-type mixture distributions

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 60E05
• 60E07
• 60E10

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

L. Tafakori  http://orcid.org/0000-0002-0815-8094

Additional information

Funding

The first author would like to thank the Australian Research Council for supporting this work through Laureate Fellowship FL130100039.