Abstract
The paper gives estimates of the rate of convergence in the central limit theorems for stochastically continuous cadlag processes proved recently by Bézandry and Fernique (Ann. Inst H. Poincare,28) and Bloznelis and Paulauskas (to appear inStoch. Proc. Appl.).
Similar content being viewed by others
References
V. Bentkus, Estimates of the rate of convergence in the central limit theorem in the space C(S),Sov. Math. Dokl.,26, 349–352 (1982).
V. Bentkus, Lower bounds for the rate of convergence in the central limit theorem in Banach spacesLith. Math. J.,26, 312–319 (1986).
V. Bentkus, Smooth approximations of the norm and differentiable functions with bounded support in Banach space,lith. Math. J.,30, 223–230 (1990).
V. Bentkus, F. Götze, V. Paulauskas and A. Račkauskas,The accuracy of Gaussian approximation in Banach spaces, SFB 343 “Diskrete Structuren in der Mathematik,” Preprint 90-100, Universität Bielefeld (in Russian:Itogi Nauki i Tehniki, ser. Sovr. Probl. Mat., Moscow, VINITI 81, 39–139 (1991); to appear in English inEncyclopedia of Mathematical Sciences, Springer).
P. H. Bezandry and X. Fernique, Sur la proprieté de la limite centrale dansD[0, 1],Ann. Inst. Henri Poincaré,28, 31–46 (1992).
P. Billingsley,Convergence of Probability Measures, Wiley, New York (1968).
M. Bloznelis and V. Paulauskas, The central limit theorem in the spaceD[0, 1]. II,Lith. Math. J.,33, 307–323 (1993).
M. Bloznelis and V. Paulauskas, A note on the central limit theorem for stochastically continuous processes,Stoch. Proc. Appl.,53, 351–361 (1994).
X. Fernique,Les fonctions aléatoires càdlàg, la compacité de leurs lois, Preprint, 17th February (1993).
X. Fernique, Régularite des fonctions aléatoires non gaussiennes,Springer Lecture Notes in Math.,976, 1–74 (1983).
E. Gine, Bounds for the speed of convergence in the central limit theorem in C(S),Z. Wahrsch. verw. Geb.,36, 317–331 (1976).
M. Hahn, Central limit theorem inD[0, 1],Z. Wahrsch. verw. Geb.,44, 89–101 (1978).
N. C. Jain and M. B. Marcus, Continuity of subgaussian processes, in: J. Kuelbs (Ed.),Advances in Probability, Vol. 4, Dekker, New York (1978), pp. 81–196.
M. A. Lifshits, The distribution density of the maximum of a Gaussian process,Theory Probab. Appl.,31, 134–142 (1986).
V. Paulauskas, Rate of convergence in the central limit theorem in C(S),Lith. Math. J.,16, 587–611 (1976).
V. Paulauskas and Ch. Stieve, On the central limit theorem inD[0, 1] and (D[0, 1],H),Lith. Math. J.,30, 267–276 (1990).
V. Paulauskas and D. Juknevičienė, On the rate of convergence in the central limit theorem in the spaceD[0, 1],Lith. Math. J.,28, 229–239 (1988).
V. Paulauskas and A. Račkauskas,Approximation Theory in the Central Limit Theorem. Exact Results in Banach Spaces, Kluwer, London (1989).
A. Račkauskas, On the rate of convergence in the central limit theorem, in:XXV Conference of the Lith. Math. Soc., Abstracts of Communications, Vilnius (1984), pp. 239–240.
B. S. Tsirel'son, The density of the distribution of the maximum of a Gaussian process,Theory Probab. Appl., 847–855 (1975).
Additional information
Research supported by the SFB 343 at Bielefeld, by a Grant from the Lithuanian Government, and by V.P. Grant 94.
Vilnius University, Naugarduko 24; Institute of Mathematics and Informatics, Akademijos 4, 2600 Vilnius, Lithuania. Published in Lietuvos Matematikos Rinkinys, Vol. 37, No. 3, pp. 280–294, July–September, 1997.
Rights and permissions
About this article
Cite this article
Bloznelis, M. On the rate of normal approximation inD[0, 1]. Lith Math J 37, 207–218 (1997). https://doi.org/10.1007/BF02465351
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02465351