Estimates of transport distance in the central limit theorem

For distributions of sums of independent random variables bounded in absolute value by unity, it is known that the accuracy of approximation in the central limit theorem is estimated in the Kantorovich-Wasserstein transport metric by an absolute constant. In a recent paper by the speaker (2025), it was shown that the same statement remains valid if the absolute value in the definition of the transport distance price function is replaced by an exponential function. Moreover, it is also argued that the same result is true for a class of distributions introduced by the speaker (1986), whose Laplace transforms behave similarly to the Laplace transforms of the normal law. This class includes convolutions of distributions with both bounded supports and those satisfying the conditions of Bernstein's inequality. This class also contains distributions with sufficiently slowly growing semi-invariants, considered by Statulevicius (1965), and infinitely divisible distributions with spectral measures with bounded supports.