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distribution theorem function return subexponential lemma portfolio result variable proof probability asset order subexponential distributions application condition limit assumption measure reinsurance geluk value embrecht theory fragility variation dependence class company i.i.d ci xi reinsurance companies vries / insurance market section insurance index domain statement stock attraction behavior correlation level distribution function copula claim economic balance example tailed corollary reinsurer combination theorem 2 distribution function f subexponential distribution function tail function f system 2 x 2 liability goldie subexponential distribution investment finance convolution equivalent journal risk drivers liability sides i =1 p f m g lemmas 3 failure subexponentiality theorem 1 value distribution model equity industry