Goodman, Victor; Kuelbs, James Rates of clustering for some Gaussian self-similar processes. (English) Zbl 0695.60040 Probab. Theory Relat. Fields 88, No. 1, 47-75 (1991). The analogue of Strassen’s functional law of the iterated logarithm is known for many Gaussian processes which have suitable scaling properties, and here we establish rates at which this convergence takes place. We provide a new proof of the best upper bound for the convergence to \({\mathcal K}\) by suitably normalized Brownian motion, and then continue with this method to get similar bounds for the Brownian sheet and other self- similar Gaussian processes. The previous method, which produced these results for Brownian motion in \({\mathbb{R}}^ 1\), was highly dependent on many special properties unavailable when dealing with other Gaussian processes. Cited in 2 ReviewsCited in 19 Documents MSC: 60F17 Functional limit theorems; invariance principles 60G15 Gaussian processes 60J65 Brownian motion Keywords:functional law of the iterated logarithm; Gaussian processes; Brownian motion; Brownian sheet; self-similar Gaussian processes PDFBibTeX XMLCite \textit{V. Goodman} and \textit{J. Kuelbs}, Probab. Theory Relat. Fields 88, No. 1, 47--75 (1991; Zbl 0695.60040) Full Text: DOI References: [1] Bass, R.F.: Probability estimates for multiparameter Brownian processes. Ann. Probab.16, 251-264 (1988) · Zbl 0645.60044 [2] Bolthausen, E.: On the speed of convergence in Strassen’s law of the iterated logarithm. Ann. Probab.6, 668-672 (1978) · Zbl 0391.60036 [3] Borell, C.: The Brunn-Minkowski inequality in Gauss space. Invent. Math.30, 207-216 (1975) · Zbl 0311.60007 [4] Goodman, V., Kuelbs, J.: Rates of convergence for increments of Brownian motion. J. Theor. Probab.1, 27-63 (1988) · Zbl 0651.60039 [5] Goodman, V., Kuelbs, J.: Rates of clustering in Strassen’s LIL for Brownian motion (to appear in the J. Theor. Probab.) · Zbl 0724.60034 [6] Goodman, V., Kuelbs, J., Zinn, J.: Somes results on the LIL in Banach spaces with applications to weighted empirical processes. Ann. Probab.9, 713-742 (1981) · Zbl 0472.60004 [7] Grill, K.: On the rate of convergence in Strassen’s law of the iterated logarithm. Probab. Th. Rel. Fields.74, 585-589 (1987) · Zbl 0592.60022 [8] Gross, L.: Lectures in modern analysis and applications II. (Lect. Notes Math., vol. 140) Berlin Heidelberg New York: Springer 1970 [9] Hochstadt, H.: Integral equations. New York: Wiley 1973 · Zbl 0259.45001 [10] Jain, N.C., Marcus, M.B.: Continuity of subgaussian processes. (Advances in Probability and Related Topics 4.) New York: Dekker 1978 [11] Kuelbs, J.: A strong convergence theorem for Banach space valued random variables. Ann. Probab.4, 744-771 (1976) · Zbl 0365.60034 [12] Mandelbrot, B.R., Van Ness, J.W.: Fractional Brownian motions, fractional noises and applications. SIAM Rev.10, 422-437 (1968) · Zbl 0179.47801 [13] Park, W.J.: A multi-parameter Gaussian process. Ann. Math. Statist.41, 1582-1595 (1970) · Zbl 0279.60030 [14] Talagrand, M.: Sur l’integrabilite’ des vecteurs gaussiens Z. Wahrscheinlichkeitstheor. Verw. Geb.68, 1-8 (1984) · Zbl 0529.60034 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.