Berry–Esseen Bounds and Diophantine Approximation

István Berkes1, Bence Borda1
1A. Rényi Institute of Mathematics, Reáltanoda u. 13-15, 1053, Budapest, Hungary

Tóm tắt

Từ khóa


Tài liệu tham khảo

I. Berkes and B. Borda, On the discrepancy of random subsequences of {nα} (submitted for publication).

P. L. Brockett and J. H. B. Kemperman, On the unimodality of high convolutions. Ann. Probab., 10 (1982), 270–277.

A. S. Fainleib, A generalization of Esseen’s inequality and its application in probabilistic number theory, Izv. Akad. Nauk SSSR Ser. Mat., 32 (1968), 859–879.

W. Feller, An Introduction to Probability Theory and its Applications, vol. II, Wiley (New York, 1971).

P. Hall, Two-sided bounds on the rate of convergence to a stable law Z. Wahrsch. Verw. Gebiete, 57 (1981), 349–364.

I. A. Ibragimov and Yu. V. Linnik, Independent and Stationary Sequences of Random Variables, Wolters-Noordhoff (Groningen, 1971).

L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences, Pure and Applied Mathematics, Wiley (New York–London–Sydney, 1974).

P. Lévy, L’addition des variables aléatoires définies sur une circonference, Bull. Soc. Math. France, 67 (1939), 1–40.

H. Niederreiter and W. Philipp, Berry–Esseen bounds and a theorem of Erdős and Turán on uniform distribution mod 1, Duke Math. J., 40 (1973), 633–649.

A. M. Odlyzko and L. B. Richmond, On the unimodality of high convolutions of discrete distributions, Ann. Probab., 13 (1985), 299–306.

V. Paulauskas, Estimates of the remainder term in limit theorems in the case of stable limit law, Lithuanian Math. J., 14 (1974), 127–146.

V. V. Petrov, Limit Theorems of Probability Theory. Sequences of Independent Random Variables, Clarendon Press (New York, 1995).

K. I. Satybaldina, Absolute estimates of the rate of convergence to stable laws, Theory Probab. Appl., 17 (1972), 726–728.

K. I. Satybaldina, On the estimation of the rate of convergence in a limit theorem with a stable stable limit law, Theory Probab. Appl., 18 (1973), 202–204.

P. Schatte, On the asymptotic uniform distribution of the n-fold convolution mod 1 of a lattice distribution, Math. Nachr., 128 (1986), 233–241.

F. E. Su, Convergence of random walks on the circle generated by an irrational rotation, Trans. Amer. Math. Soc., 350 (1998), 3717–3741.