Conditional expectations given the sum of independent random variables with regularly varying densities

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      <subfield code="a">Conditional expectations given the sum of independent random variables with regularly varying densities</subfield>
      <subfield code="c">Michel Denuit, Patricia Ortega-Jiménez and Christian-Yann Robert</subfield>
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      <subfield code="a">The paper examines the conditional expectation of one component of a sum, a quantity that plays a central role in actuarial applications such as risk-sharing rules and capital allocation. While the monotonicity of this function is usually linked to log-concave densities, that assumption excludes heavy-tailed distributions. The study investigates cases where regularly varying tails cause this conditional expectation to lose its monotonic behavior and describes its asymptotic properties when the total sum becomes large. The analysis is further extended to zero-inflated distributions, sums involving more than two random variables, and dependence structures modeled through Farlie-Gumbel-Morgenstern copulas. The paper concludes with implications for risk sharing and capital allocation, supported by numerous numerical examples.</subfield>
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