On probability distributions with accumulating failure rates in periodic random environment

The concept of driving-periodic random environment reflecting an accumulated risk is developed. We suggest that the random environment has effects on accumulating and increasing the risk that some event will happen within any specific period of length c > 0. The waiting time until the event occurs defines a random variable of a special kind whose probability distribution and additional properties are studied. The form of this probability distribution is derived, and some useful representations of related random variables are obtained. Phenomena of this type appear in a series of environmental, maintenance and financial processes. An illustrative example is briefly discussed.