On estimation of the remainder in the central limit theorem for sums of functions of independent random variables and sums of the form $\Sigma f(t2^k)$

The result of [2] is generalized to the case of sums of functions of independent identically distributed random variables. The dependence of the remainder on $x$ in the theorem of [1] concerning sums of the form $\Sigma f(t2^k)$ is also investigated. In proofs, limit theorems for differently distributed independent summands are used instead of the characteristic function method.