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1 results

  • Sharp Bounds for Sums of Dependent Risks

  • Part of
  • Giovanni Puccetti, Ludger Rüschendorf
  • Journal:
    Journal of Applied Probability / Volume 50 / Issue 1 / March 2013
    Published online by Cambridge University Press:
    30 January 2018, pp. 42-53
    Print publication:
    March 2013
    • Article
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  • Sharp tail bounds for the sum of d random variables with given marginal distributions and arbitrary dependence structure have been known since Makarov (1981) and Rüschendorf (1982) for d=2 and, in some examples, for d≥3. Based on a duality result, dual bounds have been introduced in Embrechts and Puccetti (2006b). In the homogeneous case, F1=···=Fn, with monotone density, sharp tail bounds were recently found in Wang and Wang (2011). In this paper we establish the sharpness of the dual bounds in the homogeneous case under general conditions which include, in particular, the case of monotone densities and concave densities. We derive the corresponding optimal couplings and also give an effective method to calculate the sharp bounds.