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Product-Convolution of Heavy-Tailed and Related Distributions

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Closure Properties for Heavy-Tailed and Related Distributions

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Abstract

Products of random variables and related distribution problems appear in physics, engineering, number theory, and many probability and statistical problems, such as multivariate statistical modelling, asymptotic analysis of randomly weighted sums, etc. In financial time series, the multiplicative structures occur in modelling conditional heteroskedasticity as in GARCH or stochastic volatility models. In this chapter, we mainly are interested in the following questions: (1) when a given class of distributions is closed with respect product-convolution?; (2) for which distributions G the product-convolution \(F \otimes G\) remains in the same class as a primary distribution F? (3) for given classes of distributions F and G, which class of distributions the product-convolution \(F \otimes G\) belongs to. Also, we discuss the phenomena of producing heavy tails from the light-tailed multipliers.

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Authors and Affiliations

  1. Institute of Applied Mathematics, Vilnius University, Vilnius, Lithuania

    Remigijus Leipus

  2. Institute of Mathematics, Vilnius University, Vilnius, Lithuania

    Jonas Šiaulys

  3. Department of Statistics and Actuarial - Financial Mathematics, University of the Aegean, Karlovassi, Greece

    Dimitrios Konstantinides

Authors
  1. Remigijus Leipus
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  2. Jonas Šiaulys
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  3. Dimitrios Konstantinides
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Leipus, R., Šiaulys, J., Konstantinides, D. (2023). Product-Convolution of Heavy-Tailed and Related Distributions. In: Closure Properties for Heavy-Tailed and Related Distributions. SpringerBriefs in Statistics. Springer, Cham. https://doi.org/10.1007/978-3-031-34553-1_5

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