Some stochastics on monotone functions

AbstractA measure of departures of monotonicity of a given function, the Lr-DIP, 1 ⩽ r ⩽ ∞, is introduced. Our analysis is performed to cover two different situations: When the function is known our interest is related to its behavior in a stochastic model. However, in most cases, the knowledge of the function is obtained through a preliminary estimation of the function. In both situations the aim focuses in the obtainment of strong consistency results