Closure properties and heavy tails: random vectors in the presence of dependence
Dimitrios G. Konstantinides, Charalampos D. Passalidis
TL;DR
This work studies when tail properties are preserved under the product of heavy-tailed random factors in the presence of dependence. It develops a unified framework using weak dependence (via a function $h(y)$) to extend classical product-closure results from independent settings to dependent ones across multiple heavy-tailed classes, and introduces a new univariate class $\mathcal{T}$ together with multivariate vector classes $\mathcal{D}_n$ and $\mathcal{P_D}_n$. Key contributions include closure results for numerous tail classes under product and dependence, the novel class $\mathcal{T}$ with its closure properties, and extensive applications to randomly weighted sums and discrete-time risk models, including ruin probabilities and stopped sums. The paper also clarifies the relationships between multivariate tail notions (such as MRV) and new vector classes, providing practical tools for risk management where dependence among factors is important.
Abstract
This paper is organized in three parts closely related to closure properties of heavy-tailed distributions and heavy-tailed random vectors. In the first part we consider two random variables X and Y with distributions F and G respectively. We assume that these random variables satisfy one type of a weak dependence structure. Under some mild conditions, we examine whether their product convolution distribution H belongs in the same distribution class of the distribution F. Namely we establish the closure property with respect to the product convolution, under this specific weak dependence structure, in the classes ERV, C, D, M, OS, OL, PD and K. Further in the second part we introduce a new distribution class, which satisfies some closure properties such as product and mixture.Further, we provide some applications on randomly weighted sums and on discrete-time risk model with dependent insurance and financial risks. Although the multivariate regular variation is well-established distribution, it does not happen in other heavy tailed random vectors. Therefore in the third part we introduced the class of dominatedly varying vectors and positively decreasing random vectors and we study the closure property of the independent scalar product under. Furthermore we study the closure property of the first class under sum and mixture, and we study the distribution of stopped sums where the summands are random vectors which belongs to this class. Some of these results holds and for positively decreasing random vectors.