TY - JOUR AU - Aistleitner, C AU - Berkes, István TI - On the central limit theorem for f (nkx) JF - PROBABILITY THEORY AND RELATED FIELDS J2 - PROBAB THEORY REL VL - 146 PY - 2010 IS - 1-2 SP - 267 EP - 289 PG - 23 SN - 0178-8051 DO - 10.1007/s00440-008-0190-6 UR - https://m2.mtmt.hu/api/publication/2142559 ID - 2142559 AB - By a classical observation in analysis, lacunary subsequences of the trigonometric system behave like independent random variables: they satisfy the central limit theorem, the law of the iterated logarithm and several related probability limit theorems. For subsequences of the system (f(nx))n≥1 with 2π-periodic fεL2 this phenomenon is generally not valid and the asymptotic behavior of (f(nkx))k≥1 is determined by a complicated interplay between the analytic properties of f (e.g., the behavior of its Fourier coefficients) and the number theoretic properties of nk. By the classical theory, the central limit theorem holds for f (nkx) if nk = 2k, or if nk+1/nk → α with a transcendental α, but it fails e.g., for nk = 2k - 1. The purpose of our paper is to give a necessary and sufficient condition for f (nkx) to satisfy the central limit theorem. We will also study the critical CLT behavior of f (nkx), i.e., the question what happens when the arithmetic condition of the central limit theorem is weakened "infinitesimally". © Springer-Verlag 2008. LA - English DB - MTMT ER - TY - JOUR AU - Berkes, István TI - On the asymptotic behaviour of {perspective}f(nkx) - Main theorems JF - ZEITSCHRIFT FUR WAHRSCHEINLICHKEITSTHEORIE UND VERWANDTE GEBIETE J2 - Z WAHRSC VERW GEBIETE VL - 34 PY - 1976 IS - 4 SP - 319 EP - 345 PG - 27 SN - 0044-3719 DO - 10.1007/BF00535967 UR - https://m2.mtmt.hu/api/publication/2142577 ID - 2142577 N1 - WoS:hiba:A1976BM63700005 2019-03-02 22:35 cím nem egyezik LA - English DB - MTMT ER -