Stein's lemma is a beautiful and, at the same time, straightforward result in probability theory and statistics. Primarily developed based on the normal distribution for one- and multidimensional random vectors, directly connecting the equality in the equation with the normality of their distributions. Additionally, it serves as a useful tool for determining portfolio composition in business finance and is a fundamental building block of Stein's method for estimating the error in approximating distributions of random vectors. The central task of this work is to present Stein's lemma for any normal random vector, interpret its significance, and explore some of its alternative practical formulations. Furthermore, with the aim of proving the central limit theorem for the sum of independent identically distributed random variables, a presentation of Stein's method follows.
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