Almost sure invariance principles for sums of 𝐵-valued random variables with applications to random Fourier series and the empirical characteristic process

Marcus, Michael B.; Philipp, Walter

doi:10.1090/S0002-9947-1982-0637029-8

Almost sure invariance principles for sums of -valued random variables with applications to random Fourier series and the empirical characteristic process

Authors: Michael B. Marcus and Walter Philipp
Journal: Trans. Amer. Math. Soc. 269 (1982), 67-90
MSC: Primary 60F17; Secondary 60B12
DOI: https://doi.org/10.1090/S0002-9947-1982-0637029-8
MathSciNet review: 637029
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Abstract: We establish an almost sure approximation of the partial sums of independent, identically distributed random variables with values in a separable Banach space by a suitable -valued Brownian motion under the hypothesis that the partial sums can be ${L^1}$ -closely approximated by finite-dimensional random variables. We show that this hypothesis is satisfied if the given random variables are random Fourier series or related stochastic processes. As an application we obtain an almost sure approximation of the empirical characteristic process by a suitable ${\mathbf{C}}(K)$ -valued Brownian motion whenever the empirical characteristic process satisfies the central limit theorem.

[1] István Berkes and Walter Philipp, Approximation theorems for independent and weakly dependent random vectors, Ann. Probab. 7 (1979), no. 1, 29–54. MR 515811
[2] Leo Breiman, Probability, Addison-Wesley Publishing Company, Reading, Mass.-London-Don Mills, Ont., 1968. MR 0229267
[3] Sándor Csörgő, Limit behaviour of the empirical characteristic function, Ann. Probab. 9 (1981), no. 1, 130–144. MR 606802
[4] Herold Dehling, Limit theorems for sums of weakly dependent Banach space valued random variables, Z. Wahrsch. Verw. Gebiete 63 (1983), no. 3, 393–432. MR 705631, https://doi.org/10.1007/BF00542537
[5] X. Fernique, Continuité et théorème central limite pour les transformées de Fourier des mesures aléatoires du second ordre, Z. Wahrsch. Verw. Gebiete 42 (1978), no. 1, 57–66. MR 486036, https://doi.org/10.1007/BF00534207
[6] E. Giné and M. B. Marcus, On the central limit theorem in , Colloq. Statistical and Dynamical Properties of Gaussian Processes, Lecture Notes in Math., Springer-Verlag, New York, 1980.
[7] J. Hoffmann-Jørgensen, Probability in Banach space, École d’Été de Probabilités de Saint-Flour, VI-1976, Springer-Verlag, Berlin, 1977, pp. 1–186. Lecture Notes in Math., Vol. 598. MR 0461610
[8] Naresh C. Jain and Michael B. Marcus, Central limit theorems for 𝐶(𝑆)-valued random variables, J. Functional Analysis 19 (1975), 216–231. MR 0385994
[9] Naresh C. Jain and Michael B. Marcus, Integrability of infinite sums of independent vector-valued random variables, Trans. Amer. Math. Soc. 212 (1975), 1–36. MR 0385995, https://doi.org/10.1090/S0002-9947-1975-0385995-7
[10] -, Continuity of subgaussian processes, Advances in Probability, Vol. 4, Dekker, New York, 1978.
[11] J. Kuelbs, Sample path behavior for Brownian motion in Banach spaces, Ann. Probability 3 (1975), 247–261. MR 0372937
[12] J. Kuelbs, Kolmogorov’s law of the iterated logarithm for Banach space valued random variables, Illinois J. Math. 21 (1977), no. 4, 784–800. MR 0455061
[13] J. Kuelbs and Walter Philipp, Almost sure invariance principles for partial sums of mixing 𝐵-valued random variables, Ann. Probab. 8 (1980), no. 6, 1003–1036. MR 602377
[14] Michael B. Marcus, Continuity and the central limit theorem for random trigonometric series, Z. Wahrsch. Verw. Gebiete 42 (1978), no. 1, 35–56. MR 486035, https://doi.org/10.1007/BF00534206
[15] Michael B. Marcus, Weak convergence of the empirical characteristic function, Ann. Probab. 9 (1981), no. 2, 194–201. MR 606982
[16] Michael B. Marcus and Gilles Pisier, Random Fourier series with applications to harmonic analysis, Annals of Mathematics Studies, vol. 101, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1981. MR 630532
[17] Walter Philipp, Almost sure invariance principles for sums of 𝐵-valued random variables, Probability in Banach spaces, II (Proc. Second Internat. Conf., Oberwolfach, 1978) Lecture Notes in Math., vol. 709, Springer, Berlin, 1979, pp. 171–193. MR 537701
[18] Walter Philipp, Weak and 𝐿^{𝑝}-invariance principles for sums of 𝐵-valued random variables, Ann. Probab. 8 (1980), no. 1, 68–82. MR 556415
[19] G. Pisier, Le théorème de la limite centrale et la loi du logarithme iteré dans les espaces de Banach, Séminaire Maurey-Schwartz 1975-76, Exposé IV, L'Ecole Polytéchnique, Paris, 1975.
[20] V. V. Jurinskiĭ, The error of Gaussian approximation of convolutions, Teor. Verojatnost. i Primenen. 22 (1977), no. 2, 242–253 (Russian, with English summary). MR 0517490