Improved asymptotic expansions are constructed in terms of the Chebyshev-Hermite polynomials in the local form of the central limit theorem for sums of independent identically distributed random variables under the condition of absolute integrability of some positive powers of the the characteristic function of a summand. The influence of the requirements to the order of existing moments on the accuracy of approximation is discussed. Theoretical results are illustrated by the example of a particular shifted exponential distribution.
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