Conditional expectations given the sum of independent random variables with regularly varying densities
(2024) , 42 pages
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- Abstract
- Stochastic monotonicity of two independent random variables X and Y given the value of their sum S = X + Y has been linked to log-concave densities since Efron (1965). However, the log-concavity assumption is not realistic in some applications because it excludes heavy- tailed distributions. This paper considers random variables with regularly varying densities to illustrate how heavy tails can lead to a non-monotonic behavior for the conditional expectation mX(s) = E[X|S = s], which turns out to be problematic in risk sharing or signal processing (including industry loss warranties or parametric insurance, for instance). This paper first aims to identify situations where a non-monotonic behavior appears according to the tail-heaviness of X and Y . Secondly the paper aims to study the asymptotic behavior of mX (s) as the value s of the sum gets large. The analysis is then extended to zero-augmented probability distributions, commonly encountered in applications to insurance and to sums of more than two random variables. Consequences for signal processing and risk sharing are discussed. Many numerical examples illustrate the results.
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Denuit, M., Ortega Jiménez, P., & Robert, C. Y. (2024). Conditional expectations given the sum of independent random variables with regularly varying densities (LIDAM Discussion Paper ISBA 2024/06). https://hdl.handle.net/2078.5/30646