Narrow Zones of Local Normal Attraction

Rényi divergence and the central limit theorem

“…In this case, the corresponding exponential series can be explicitly computed. At x = 0, it is absolutely convergent for λ > 1 4 , simply convergent for λ = 1 4 and divergent for λ < 1 4 .…”

“…which we use in the polynomial bound on R n (with factor 1/ √ n). Thus, 1) . Now, using a multinomial formula, write…”

“…Recall that, if χ 2 (X, Z) < ∞, then X has finite moments of any order, and moreover, Ee cX 2 < ∞ for any c < 1 4 . Hence, the normal moments EH k (X) are well defined and finite, so that the representation (8.9) makes sense.…”

“…Using the Lipschitz property of the power function near the point 1, we thus obtain that 1) , so that (11.9) 1) .…”

“…Nonuniform bounds in the normal approximation have been intensively studied in the literature; cf. [1,2,24,36,37]. However, existing results start with weaker hypotheses (e.g., moment assumptions) and either provide a polynomial error of approximation with respect to x (such as 1 1+|x| 3 ), or deal with narrow zones contained in regions |x| = o( √ n).…”