On Return Probabilities of Adverse Events Under Dependence and Lessons to Learn for Decision-Making

Considering achieving a goal in each of several time intervals when, in every time interval, an adverse event may lead to a failure raises the question of the return probability of adverse events, so the probability of at least one failure to happen during the time period of interest. Through basic mathematical arguments in tractable cases, we investigate the behavior of the return probability of adverse events in various setups. In the univariate case, we consider the independent and identically distributed setup, the independent setup, the dependent but not necessarily identically distributed setup, and the dependent and identically distributed setup. In the multivariate case, we consider several goals to be achieved in each time period. Besides different setups for the marginal failure probabilities, we study dependence in terms of comonotone blocks and independent blocks and via nested copulas. In case closed form expressions are not available, we derive bounds on the return probability of at least one failure. Our results are interpretable in terms of decision-making, provide insight into what affects such return probabilities and thus may help to develop strategies to lower them.