Law of the iterated logarithm for the periodogram

Stochastic Processes and their Applications - Tập 123 - Trang 4065-4089 - 2013
Christophe Cuny1, Florence Merlevède2, Magda Peligrad3
1Laboratoire MAS, Ecole Centrale de Paris, Grande Voie des Vignes, 92295 Chatenay-Malabry cedex, France
2Université Paris-Est, LAMA (UMR 8050), UPEMLV, CNRS, UPEC, F-77454 Marne-La-Vallée, France
3Department of Mathematical Sciences, University of Cincinnati, PO Box 210025, Cincinnati, OH 45221-0025, USA

Tài liệu tham khảo

Bingham, 2012, Szegö’s theorem and its probabilistic descendants, Probab. Surv., 9, 287, 10.1214/11-PS178 Brockwell, 1991 Browder, 1958, On the iteration of transformations in noncompact minimal dynamical systems, Proc. Amer. Math. Soc., 9, 773, 10.1090/S0002-9939-1958-0096975-9 Carleson, 1966, On convergence and growth of partial sums of Fourier series, Acta Math., 116, 135, 10.1007/BF02392815 Cuny, 2004, Weak mixing of random walks on groups, J. Theoret. Probab., 16, 923, 10.1023/B:JOTP.0000012000.54810.d2 Cuny, 2011, Pointwise ergodic theorems with rate with applications to limit theorems for stationary processes, Stoch. Dyn., 1, 135, 10.1142/S0219493711003206 C. Cuny, ASIP for martingales in 2-smooth Banach spaces, applications to stationary processes, 2012. arXiv: 1209.3680. C. Cuny, F. Merlevède, Strong invariance principles with rate for “reverse” martingales and applications, 2012. arXiv: 1209.3677. Dedecker, 2010, Some almost sure results for unbounded functions of intermittent maps and their associated Markov chains, Ann. Inst. Henri Poincaré Probab. Statist., 46, 796, 10.1214/09-AIHP343 Dedecker, 2000, On the functional central limit theorem for stationary processes, Ann. Inst. Henri Poincaré Probab. Statist., 36, 1, 10.1016/S0246-0203(00)00111-4 Diaconis, 1999, Iterated random functions, SIAM Rev., 41, 45, 10.1137/S0036144598338446 Duflo, 1996 Dunford, 1988 Gordin, 1969, The central limit theorem for stationary processes, Sov. Math. Dokl., 10, 1174 M.I. Gordin, B.A. Lifschits, A remark about a Markov process with normal transition operator, in: Third Vilnius Conf. Proba. Stat., Akad. Nauk Litovsk, Vilnius, Vol. 1, 1981, pp. 147–148 (in Russian). Heyde, 1973, Invariance principles for the paw of the iterated logarithm for martingales and processes with stationary increments, Ann. Probab., 1, 428, 10.1214/aop/1176996937 Hunt, 1974, A weighted norm inequality for Fourier series, Bull. Amer. Math. Soc., 80, 274, 10.1090/S0002-9904-1974-13458-0 Merlevède, 2006, Recent advances in invariance principles for stationary sequences, Probab. Surv., 3, 1, 10.1214/154957806100000202 Merlevède, 2012, Strong approximation of partial sums under dependence conditions with application to dynamical systems, Stoch. Proc. Appl., 122, 386, 10.1016/j.spa.2011.08.012 Peligrad, 2010, Central limit theorem for Fourier transforms of stationary processes, Ann. Probab., 38, 2009, 10.1214/10-AOP530 Petersen, 1989, vol. 2 Philipp, 1977, A functional law of the iterated logarithm for empirical distribution functions of weakly dependent random variables, Ann. Probab., 5, 319, 10.1214/aop/1176995795 Pomeau, 1980, Intermittent transition to turbulence in dissipative dynamical systems, Comm. Math. Phys., 74, 189, 10.1007/BF01197757 Rio, 2000, vol. 31 Rootzén, 1976, Gordin’s theorem and the periodogram, J. Appl. Probab., 13, 365, 10.1017/S0021900200094456 Rosenblatt, 1956, A central limit theorem and a strong mixing condition, Proc. Natl. Acad. Sci. USA, 42, 43, 10.1073/pnas.42.1.43 Rosenblatt, 1981, Limit theorems for Fourier transforms of functionals of Gaussian sequences, Z. Wahrscheinlichkeitstheor. Verwandte Geb., 55, 123, 10.1007/BF00535155 Schuster, 1898, On the investigation of hidden periodicities with application to a supposed 26 day period of meteorological phenomena, Terr. Magn. Atmos. Electr., 3, 13, 10.1029/TM003i001p00013 Wiener, 1941, On the ergodic dynamics of almost periodic systems, Amer. J. Math., 63, 794, 10.2307/2371623 Woodroofe, 1992, A central limit theorem for functions of a Markov chain with applications to shifts, Stochastic Process. Appl., 41, 33, 10.1016/0304-4149(92)90145-G Wu, 2005, Fourier transforms of stationary processes, Proc. Amer. Math. Soc., 133, 285, 10.1090/S0002-9939-04-07528-8