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Berkes, I.: Almost sure invariance principles for mixing processes. [To appear in Ann. Probability]
Gapoškin, V.F.: Lacunary series and independent functions (In Russian). Uspehi Mat. Nauk 21, 3–82 (1966). English translation: Russian Math. Surveys 21, 3–82 (1966)
Gapoškin, V.F.: On some sequences of almost independent functions (In Russian). Sibirsk Mat. Ž. 9, 264–279 (1968)
Gapoškin, V.F.: On the central limit theorem for some weakly dependent sequences (In Russian). Teor. Verojatnost. i. Primenen. 15, 666–684 (1970). English translation: Theor. Probability Appl. 15, 649–666 (1970)
Hartman, P.: The divergence of non-harmonic gap series. Duke Math. J. 9, 404–405 (1942)
Ibragimov, I.A.: Asymptotic distribution of values of certain sums. (In Russian). Vestnik Leningrad. Univ., 55–69 (1960)
Kac, M.: Probability methods in some problems of analysis and number theory. Bull. Amer. Math. Soc. 55, 641–665 (1949)
Kac, M., Salem, R., Zygmund, A.: A gap theorem. Trans. Amer. Math. Soc. 63, 235–243 (1948)
Kiefer, J.: On the deviations in the Skorohod-Strassen approximation scheme. Z. Wahrschein-lichkeitstheorie verw. Gebiete 13, 321–332 (1969)
Komlös, J., Major, P., Tusnády, G.: An approximation of partial sums of independent RV's and the sample DF, I. Z. Wahrscheinlichkeitstheorie verw. Gebiete 32, 111–131 (1975). II. Z. Wahrscheinlichkeitstheorie verw. Gebiete 34, 33–58 (1976)
Philipp, W.: Limit theorems for lacunary series and uniform distribution mod 1. Acta Arith. 26, 241–251 (1975)
Philipp, W., Stout, W.F.: Almost sure invariance principles for sums of weakly dependent random variables. Mem. Amer. Math. Soc. No. 161
Strassen, V.: An invariance principle for the law of the iterated logarithm. Z. Wahrscheinlichkeitstheorie verw. Gebiete 3, 211–226 (1964)
Strassen, V.: Almost sure behavior of sums of independent random variables and martingales. Proc. 5-th Berkeley Sympos. Math. Statist. Probab., Vol. II (Part I) 315–343. Univ. of Calif. Press, Berkeley 1967
Takahashi, S.: On the distribution of values of the type ⌆f(q kt). TÔhoku Math. J. 14, 233–243 (1962)
Takahashi, S.: An asymptotic property of a gap sequence. Proc. Japan Acad. 38, 101–104 (1962)
Takahashi, S.: The law of the iterated logarithm for a gap sequence with infinite gaps. TÔhoku Math. J. 15, 281–288 (1963)
Zygmund, A.: Trigonometric series, Vol. I. Cambridge: University Press 1959
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Berkes, I. On the asymptotic behaviour of ⌆f(nkx). Z. Wahrscheinlichkeitstheorie verw Gebiete 34, 319–345 (1976). https://doi.org/10.1007/BF00535967
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DOI: https://doi.org/10.1007/BF00535967