Summary
Let F *ndenote the n th convolution of a distribution function F on R kand suppose that F has zero moments of the first order and finite second order moment matrix. It is well-known that F *n(√n·) converges to a Gaussian d.f. Φ as n→ + t8. These d.f.s determine measures F *n(√nA) and Φ(A) for Borelsets A, We present a method that admits the estimation of the remainder-term F *n(√n A)-Φ (A) when A belongs to a certain class of Borelsets. This class contains all convex sets. If F has finite absolute third order moments then the remainder-term is of the order n −1/2. Also the remainder term's dependence on the dimension k is given. These results strengthen and generalize earlier results in the same direction.
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This paper was first communicated at the Scandinavian mathematical congress in Oslo, August 1968.
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Bergström, H. On the central limit theorem in R k . Z. Wahrscheinlichkeitstheorie verw Gebiete 14, 113–126 (1969). https://doi.org/10.1007/BF00537517
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DOI: https://doi.org/10.1007/BF00537517