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Ordering Functions of Random Vectors, with Application to Partial Sums

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Abstract

It is known that the sums of the components of two random vectors (X 1,X 2,…,X n ) and (Y 1,Y 2,…,Y n ) ordered in the multivariate (s 1,s 2,…,s n )-increasing convex order are ordered in the univariate (s 1+s 2+⋯+s n )-increasing convex order. More generally, real-valued functions of (X 1,X 2,…,X n ) and (Y 1,Y 2,…,Y n ) are ordered in the same sense as long as these functions possess some specified non-negative cross-derivatives. This note extends these results to multivariate functions. In particular, we consider vectors of partial sums (S 1,S 2,…,S n ) and (T 1,T 2,…,T n ) where S j =X 1+⋯+X j and T j =Y 1+⋯+Y j and we show that these random vectors are ordered in the multivariate (s 1,s 1+s 2,…,s 1+⋯+s n )-increasing convex order. The consequences of these general results for the upper orthant order and the orthant convex order are discussed.

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Acknowledgements

We would like to thank an anonymous referee for his/her detailed report which allowed us to substantially improve the first version of the paper (which initially restricted to the non-negative case). In particular, the ingenious proof of Lemma 2.1 is due to him/her.

The financial support of the Onderzoeksfonds K.U. Leuven (GOA/07: Risk Modeling and Valuation of Insurance and Financial Cash Flows, with Applications to Pricing, Provisioning and Solvency) is gratefully acknowledged by Michel Denuit. Mhamed Mesfioui acknowledges the financial support of the Natural Sciences and Engineering Research Council of Canada.

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Authors and Affiliations

  1. Institut de Statistique, Biostatistique et Sciences Actuarielles (ISBA), Université Catholique de Louvain, 1348, Louvain-la-Neuve, Belgium

    Michel M. Denuit

  2. Département de Mathématiques et D’informatique, Université du Québec à Trois-Rivières, Trois-Rivières, (Québec), Canada, G9A 5H7

    Mhamed Mesfioui

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  1. Michel M. Denuit
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  2. Mhamed Mesfioui
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Correspondence to Michel M. Denuit.

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Denuit, M.M., Mesfioui, M. Ordering Functions of Random Vectors, with Application to Partial Sums. J Theor Probab 26, 474–479 (2013). https://doi.org/10.1007/s10959-012-0402-y

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  • DOI: https://doi.org/10.1007/s10959-012-0402-y

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