Extension of Stein’s Lemmas to General Functions and Distributions

In this paper, we extend the lemmas in Stein (1973, 1981) and others to include situations in which the variables are dependent and non-normally distributed. There is no restriction on the form of the function, which could be linear or nonlinear, provided that the function is differentiable and the expectation of the derivative of the function exists. Thereafter, we give some examples of nonnormal distributions and nonlinear functions to illustrate the theorems developed in the paper to hold, and show that the assertion of Genest (2020) is incorrect. In addition, we discuss applications of using the theorems in decision sciences.