Abstract. Let X be a centered random variable with unit variance, zero third moment, and such that IE [X 4 ] ≥ 3. Let {Fn : n ≥ 1} denote a normalized sequence of homogeneous sums of fixed degree d ≥ 2, built from independent copies of X. Under these minimal conditions, we prove that Fn converges in distribution to a standard Gaussian random variable if and only if the corresponding sequence of fourth moments converges to 3. The statement is then extended (mutatis mutandis) to the free probability setting. We shall also discuss the optimality of our conditions in terms of explicit thresholds, as well as establish several connections with the so-called universality phenomenon of probability theory. Both in the classical and free probability frameworks, our results extend and unify previous Fourth Moment Theorems for Gaussian and semicircular approximations. Our techniques are based on a fine combinatorial analysis of higher moments for homogeneous sums.
The management of National Social Security Systems is being challenged more and more by the rapid ageing of the population, especially in the industrialized countries. In order to chase the Pension System sustainability, several countries in Europe are setting up pension reforms linking the retirement age and/or benefits to life expectancy. In this context, the accurate modelling and projection of mortality rates and life expectancy play a central role and represent issues of great interest in recent literature. Our study refers to the Italian mortality experience and considers an indexing mechanism based on the expected residual life to adjust the retirement age and keep costs at an expected budgeted level, in the spirit of sharing the longevity risk between Social Security Systems and retirees. In order to combine fitting and projections performances of selected stochastic mortality models, a model assembling technique is applied to face uncertainty in model selection, while accounting for uncertainty of estimation as well. The resulting proposal is an averaged model that is suitable to discuss about the gender gap in longevity risk and its alleged narrowing over time.
Mixture models for ordinal responses in the tradition of cub models use the uniform distribution to account for uncertainty of respondents. A model is proposed that uses more flexible distributions in the uncertainty component:the discretized Beta distribution allows to account for response styles, in particular the preference for middle or extreme categories. The proposal is compared with traditional cub models in simulation studies and its use is illustrated by two applications.
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