The Chebyshev–Edgeworth correction in the central limit theorem for integer-valued independent summands

We give a short overview of the results related to the refined forms of the central limit theorem, with a focus on independent integer-valued random variables (r.v.'s). In the independent and non-identically distributed (non-i.i.d.) case, an approximation is then developed for the distribution of the sum by means of the Chebyshev–Edgeworth correction containing the moments of the third order.