usually, a limit theorem of probability theory is a theorem that concerns convergence of a sequence of distributions $p_n$ to a distribution $p$. however, there is a number of works where the traditional setup is modified, and the object of study is two sequences of distributions, $p_n$ and $q_n$, and the goal consists in establishing conditions implying the convergence $p_n - q_n ->0 (1)$ in particular problems,$p_n$ and $q_n$ are, as a rule, the distributions of the r.v.'s $f(x_1,...,x_n)$ and $f(y_1,...,y_n)$, where $f(.)$ is a function, and $x_1,x_2$,... and $y_1,y_2$,... are two sequences of r.v.'s. the aim here is rather to show that different random arguments $x_1,...,x_n$ may generate close distributions of $f(x_1,...,x_n)$ , than to prove that the distribution of $f(x_1,...,x_n)$ is close to some fixed distribution (which, above else, may be not true). clearly, such a framework is more general than the traditional one. first, as was mentioned, the distributions $p_n$ and $q_n$, themselves do not have to converge. secondly, the sequences $p_n$ and $q_n$ are not assumed to be tight, and the convergence in $(1)$ covers situations when a part of the probability mass or the whole distributions "move away to infinity'", while the distributions $p_n$ and $q_n$, are approaching each other. we consider a theory on this point, including the very definition of convergence $(1)$, and a particular example of the invariance principle in the general non-classical setup.