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Ordering Functions of Random Vectors, with Application to Partial Sums

Author

Listed:
  • Michel M. Denuit

    (Université Catholique de Louvain)

  • Mhamed Mesfioui

    (Université du Québec à Trois-Rivières)

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.

Suggested Citation

  • Michel M. Denuit & Mhamed Mesfioui, 2013. "Ordering Functions of Random Vectors, with Application to Partial Sums," Journal of Theoretical Probability, Springer, vol. 26(2), pages 474-479, June.
    • Handle: RePEc:spr:jotpro:v:26:y:2013:i:2:d:10.1007_s10959-012-0402-y
    DOI: 10.1007/s10959-012-0402-y
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    Cited by:

    1. Michel M. Denuit & Mhamed Mesfioui, 2016. "Multivariate Higher-Degree Stochastic Increasing Convexity," Journal of Theoretical Probability, Springer, vol. 29(4), pages 1599-1623, December.

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