Convex bodies generated by sublinear expectations of random vectors

Molchanov, Ilya; Turin, Riccardo

doi:10.1016/j.aam.2021.102251

Cited by 1 publication

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References 52 publications

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“…The integrand in (1) is an

ℝ^{d + 1}

‐valued function whose first element is

g false(x false) \in ℝ

, and the remaining d coordinates are the product

g false(x false) x \in ℝ^{d}

; the integral of this function is understood in the coordinate‐wise sense. The lift zonoids have found important applications in several fields of mathematics, ranging from convex geometry (Huang & Slomka, 2019; Huang et al 2019), functional analysis (Kulik & Tymoshkevych, 2013) and theoretical probability (Koshevoy, 2003) to multivariate statistics (Koshevoy & Mosler, 1997, 1998) and finance (Molchanov & Turin, 2021). For a comprehensive account of theory and practice of lift zonoids, we refer to the seminal monograph (Mosler, 2002).…”

Section: Introduction: Lift Zonoids Of Weakly Convergent Measuresmentioning

confidence: 99%

A note on the convergence of lift zonoids of measures

Hendrych

Nagy

2022

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The lift zonoid is a convenient representation of an integrable measure by a convex set in a higher‐dimensional space. It is known that, under appropriate conditions, a uniformly integrable sequence of measures converges weakly if and only if the corresponding sequence of lift zonoids converges in the Hausdorff metric. We provide a new proof of this essential result. Our proof technique allows us to eliminate the unnecessary conditions previously considered in the literature. As a by‐product, we obtain a characterization of uniform integrability, and a simple sufficient condition for tightness, of a sequence of integrable measures in terms of their lift zonoids.

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“…The integrand in (1) is an