An almost sure invariance principle for Hilbert space valued martingales

Morrow, Gregory; Philipp, Walter

doi:10.1090/S0002-9947-1982-0664040-3

An almost sure invariance principle for Hilbert space valued martingales
HTML articles powered by AMS MathViewer

by Gregory Morrow and Walter Philipp PDF

Trans. Amer. Math. Soc. 273 (1982), 231-251 Request permission

Abstract:

We obtain an almost sure approximation of a martingale with values in a real separable Hilbert space $H$ by a suitable $H$-valued Brownian motion.

References

István Berkes and Walter Philipp, Approximation theorems for independent and weakly dependent random vectors, Ann. Probab. 7 (1979), no. 1, 29–54. MR 515811
B. M. Brown, Martingale central limit theorems, Ann. Math. Statist. 42 (1971), 59–66. MR 290428, DOI 10.1214/aoms/1177693494
D. L. Burkholder, Distribution function inequalities for martingales, Ann. Probability 1 (1973), 19–42. MR 365692, DOI 10.1214/aop/1176997023
Herold Dehling and Walter Philipp, Almost sure invariance principles for weakly dependent vector-valued random variables, Ann. Probab. 10 (1982), no. 3, 689–701. MR 659538
Xavier Fernique, Intégrabilité des vecteurs gaussiens, C. R. Acad. Sci. Paris Sér. A-B 270 (1970), A1698–A1699 (French). MR 266263
Peter Gänssler and Winfried Stute, Wahrscheinlichkeitstheorie, Springer-Verlag, Berlin-New York, 1977 (German). MR 0501219
V. Goodman, J. Kuelbs, and J. Zinn, Some results on the LIL in Banach space with applications to weighted empirical processes, Ann. Probab. 9 (1981), no. 5, 713–752. MR 628870
Naresh C. Jain, Kumar Jogdeo, and William F. Stout, Upper and lower functions for martingales and mixing processes, Ann. Probability 3 (1975), 119–145. MR 368130, DOI 10.1214/aop/1176996453
N. C. Jain and S. J. Taylor, Local asymptotic laws for Brownian motion, Ann. Probability 1 (1973), 527–549. MR 365732, DOI 10.1214/aop/1176996884
J. Kuelbs and R. Lepage, The law of the iterated logarithm for Brownian motion in a Banach space, Trans. Amer. Math. Soc. 185 (1973), 253–265. MR 370725, DOI 10.1090/S0002-9947-1973-0370725-3
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
Steven Orey and William E. Pruitt, Sample functions of the $N$-parameter Wiener process, Ann. Probability 1 (1973), no. 1, 138–163. MR 346925, DOI 10.1214/aop/1176997030

Almost sure invariance principles for partial sums of weakly dependent random variables

William F. Stout, Almost sure convergence, Probability and Mathematical Statistics, Vol. 24, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1974. MR 0455094
V. Strassen, An invariance principle for the law of the iterated logarithm, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 3 (1964), 211–226 (1964). MR 175194, DOI 10.1007/BF00534910

Almost sure behavior of sums of independent random variables and martingales