Remarks on estimates in the total-variation metric

Abstract

Abstract. For sums of independent lattice random variables, the limiting normal approximation is trivial if the total-variation distance is considered. In this paper, we show that the normal distribution can be replaced by a suitably chosen lattice distribution. Mixtures of distributions are approximated by a convolution of the normal and Poisson laws. Construction of an asymptotic expansion is discussed.

A dichotomy for CLT in total variation

“…Although central limit theorems in the total variation have been studied in some special circumstances (see, e.g., [8,14]), it is generally believed that the total variation distance is too strong for normal approximation (see, e.g., Čekanavičius [4], Chen and Leong [7], Fang [10]). For example, the total variation distance between any binomial distribution and any normal distribution is always 1.…”

Normal approximation in total variation for statistics in geometric probability

“…Although CLTs with errors measured in the total variation distance have been studied in some special circumstances (see, e.g., [19,36,2]), it is generally believed that the total variation distance is too strong for quantifying the errors in normal approximation (see, e.g., [9,12,21]). For example, the total variation distance between any discrete distribution and any normal distribution is always 1.…”