Summary
Strassen's original functional law of the iterated logarithm for partial sums and Brownian motion examined convergence and clustering in the sup-norm. Here we address what happens if we use the much larger H-norm. We provide the answer to a query which appeared at the end of Strassen's original paper, and also present several contrasting results which are shown to be essentially best possible.
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[deA] de Acosta, A.: Small deviations in the functional central limit theorem with applications to functional laws of the iterated logarithm. Ann. Probab.11, 78–101 (1983)
[E] Einmahl, U.: Strong invariance principles for partial sums of independent random vectors. Ann. Probab.15, 1419–1440 (1987)
[EG] Einmahl, U., Goodman, V.: Clustering behavior of finite variance partial sum processes. Probab. Theory Relat. Fields102, 547–565 (1995)
[EM] Einmahl, U., Mason, D.M.: Rates of clustering in Strassen's LIL for partial sum processes. Probab. Theory Relat. Fields97, 479–487 (1993)
Grill, K.: A lim inf result in Strassen's law of the iterated logarithm. Probab. Theory Relat. Fields.89, 149–157 (1991)
[G92] Grill, K.: Exact rate of convergence in Strassen's law of the iterated logarithm. J. Theory Probab.5, 197–204 (1992)
Goodman, V., Kuelbs, J.: Rates of clustering in Strassen's LIL for Brownian motion. J. Theory Probab.4, 285–309 (1991)
[KL] Kuelbs, J., Li, W.V.: Gaussian samples approach “smooth points” slowest. J. Funct. Anal.124, 333–348 (1994)
[KLT] Kuelbs, J., Li, W.V., Talagrand, M.: Lim inf results for Gaussian samples and Chung's functional LIL. Ann. Probab.22, 1879–1903 (1994)
[M] Major, P.: An improvement of Strassen's invariance principle. Ann. Probab.7, 55–61 (1979)
Strassen, V.: An invariance principle for the law of the iterated logarithm. Z. Wahrsch. verw. Gebiete3, 211–226 (1964)
Talagrand, M.: On the rate of convergence in Strassen's LIL, Probability in Banach Spaces 8, Progress in Probability30, Birkhäuser, 339–351, 1992
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Supported in part by NSA Grant MDA-904-93-H-3033
Supported in part by NSF Grant DMS-9400024