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 whichthe 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.
Stein’s Lemma, dependence, non-normality, differentiability, expectations