Preservation of ILR and IFR aging classes in sums of dependent random variables

Closure of aging classes with respect to sums is a relevant topic in different areas of applied probability. In the case of independent random variables (i.e., for convolutions) this preservation property has been proved in the literature for a number of classes such as the increasing in likelihood ratio (ILR) and the increasing failure rate (IFR) classes. These results were applied, for example, to sums of life lengths in reliability theory and to sums of incomes/returns in economic/risk studies. However, in many practical situations the independence assumption is unrealistic. In the present paper, we provide conditions such that these closure properties are satisfied by the ILR and IFR classes when the assumption of independence is dropped. The classical copula approach is used to model the dependence structure, but other dependence models, such as relevation transforms and load sharing models, are considered as well. Several illustrative examples and counterexamples show how to use the presented theoretical results and which preservation results do not hold.