On two extensions of the canonical Feller–Spitzer distribution

We introduce two extensions of the canonical Feller–Spitzer distribution from the class of Bessel densities, which comprise two distinct stochastically decreasing one-parameter families of positive absolutely continuous infinitely divisible distributions with monotone densities, whose upper tails exhibit a power decay. The densities of the members of the first class are expressed in terms of the modified Bessel function of the first kind, whereas the members of the second class have the densities of their Lévy measure given by virtue of the same function. The Laplace transforms for both these families possess closed–form representations in terms of specific hypergeometric functions. We obtain the explicit expressions by virtue of the particular parameter value for the moments of the distributions considered and establish the monotonicity of the mean, variance, skewness and excess kurtosis within the families. We derive numerous properties of members of these classes by employing both new and previously known properties of the special functions involved and determine the variance function for the natural exponential family generated by a member of the second class.

Sharp Bounds for Sums of Dependent Risks

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, F 1=···=F n , 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.

Sharp Bounds for Sums of Dependent Risks

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.