Аннотация:The aim of our work is to
provide a generalization of the Stein equation allowing to use the functions f
for which the classical Stein’s identity is not satisfied. We also show that our
equation permits to characterize the law of X, i.e. this equation holds for all f
in the extended F if and only if Law(Y) = P. In the operator theory framework
one can observe that a so-called density approach is a particular case of our
method. Due to the modification proposed we can estimate a distance between
the target distribution P of a random variable X and a distribution of Y when
supp(Y) is not a subset of supp(X). Moreover, we can write the characteristic
Stein equations for random variables with non-interval support. We illustrate
some advantages of the introduced method by a number of interesting examples.