Critical LIL behavior of the trigonometric system

Berkes, I.

doi:10.1090/S0002-9947-1993-1099352-2

Critical LIL behavior of the trigonometric system
HTML articles powered by AMS MathViewer

by I. Berkes

Trans. Amer. Math. Soc. 338 (1993), 553-585

DOI: https://doi.org/10.1090/S0002-9947-1993-1099352-2

PDF | Request permission

Abstract:

It is a classical fact that for rapidly increasing $({n_k})$ the sequence $(\cos {n_k}x)$ behaves like a sequence of i.i.d. random variables. Actually, this almost i.i.d. behavior holds if $({n_k})$ grows faster than ${e^{c\sqrt k }}$; below this speed we have strong dependence. While there is a large literature dealing with the almost i.i.d. case, practically nothing is known on what happens at the critical speed ${n_k} \sim {e^{c\sqrt k }}$ (critical behavior) and what is the probabilistic nature of $(\cos {n_k}x)$ in the strongly dependent domain. In our paper we study the critical LIL behavior of $(\cos {n_k}x)$ i.e., we investigate how classical fluctuational theorems like the law of the iterated logarithm and the Kolmogorov-Feller test turn to nonclassical laws in the immediate neighborhood of ${n_k} \sim {e^{c\sqrt k }}$.

References

A treatise on trigonometric series

I. Berkes, On the central limit theorem for lacunary trigonometric series, Anal. Math. 4 (1978), no. 3, 159–180 (English, with Russian summary). MR 514757, DOI 10.1007/BF01908987
I. Berkes, Non-Gaussian limit distributions of lacunary trigonometric series, Canad. J. Math. 43 (1991), no. 5, 948–959. MR 1138574, DOI 10.4153/CJM-1991-052-0
I. Berkes, A note on lacunary trigonometric series, Acta Math. Hungar. 57 (1991), no. 1-2, 181–186. MR 1128854, DOI 10.1007/BF01903816
Uwe Einmahl and David M. Mason, Some results on the almost sure behavior of martingales, Limit theorems in probability and statistics (Pécs, 1989) Colloq. Math. Soc. János Bolyai, vol. 57, North-Holland, Amsterdam, 1990, pp. 185–195. MR 1116787

On the law of the iterated logarithm

58

P. Erdős, On trigonometric sums with gaps, Magyar Tud. Akad. Mat. Kutató Int. Közl. 7 (1962), 37–42 (English, with Russian summary). MR 145264
W. Feller, The general form of the so-called law of the iterated logarithm, Trans. Amer. Math. Soc. 54 (1943), 373–402. MR 9263, DOI 10.1090/S0002-9947-1943-0009263-7
W. Feller, The law of the iterated logarithm for identically distributed random variables, Ann. of Math. (2) 47 (1946), 631–638. MR 17894, DOI 10.2307/1969225
V. F. Gapoškin, Lacunary series and independent functions, Uspehi Mat. Nauk 21 (1966), no. 6 (132), 3–82 (Russian). MR 0206556
Takafumi Murai, On lacunary series, Nagoya Math. J. 85 (1982), 87–154. MR 648419

Almost sure invariance principles for sums of weakly dependent random variables

Walter Philipp and William Stout, Invariance principles for martingales and sums of independent random variables, Math. Z. 192 (1986), no. 2, 253–264. MR 840828, DOI 10.1007/BF01179427
R. Salem and A. Zygmund, On lacunary trigonometric series, Proc. Nat. Acad. Sci. U.S.A. 33 (1947), 333–338. MR 22263, DOI 10.1073/pnas.33.11.333
Shigeru Takahashi, On lacunary trigonometric series, Proc. Japan Acad. 41 (1965), 503–506. MR 196377

On the lacunary Fourier series

19

Shigeru Takahashi, On lacunary trigonometric series. II, Proc. Japan Acad. 44 (1968), 766–770. MR 243261
Shigeru Takahashi, On the law of the iterated logarithm for lacunary trigonometric series, Tohoku Math. J. (2) 24 (1972), 319–329. MR 330905, DOI 10.2748/tmj/1178241542
Shigeru Takahashi, On the law of the iterated logarithm for lacunary trigonometric series. II, Tohoku Math. J. (2) 27 (1975), no. 3, 391–403. MR 440269, DOI 10.2748/tmj/1203529250
Shigeru Takahashi, Almost sure invariance principles for lacunary trigonometric series, Tohoku Math. J. (2) 31 (1979), no. 4, 439–451. MR 558676, DOI 10.2748/tmj/1178229729
A. Zygmund, Trigonometric series. 2nd ed. Vols. I, II, Cambridge University Press, New York, 1959. MR 0107776