Summary
Given independent identically distributed random variables {x n ;n ∈ ℕq| indexed by q-tuples of positive integers and taking values in a separable Banach space B we approximate the rectangular sums \(\{ \sum\limits_{m \leqq n} {x_m ;n \in \mathbb{N}^q } \} \) by a Brownian sheet. We obtain the corresponding result for random variables with values in a separable Hilbert space H while assuming an optimal moment condition. Generalized versions of the functional law of the iterated logarithm are thus derived.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Berkes, I., Philipp, W.: Approximation theorems for independent and weakly dependent random vectors. Ann. Probab. 1, 29–54 (1979)
Breiman, L.: Probability. Reading, Mass.: Addison Wesley 1968
Fernique, X.: Integrabilité des vecteurs Gaussiens, C.R. Acad. Science Paris 270, 1698–1699 (1970)
Gross, L.: Lectures in modern analysis and applications. II, Lecture Notes in Math., 140, New York-Heidelberg-Berlin: Springer 1970
Hardy, G.H., Wright, E.M.: An introduction to the theory of numbers. Oxford 1945
Hartman, Ph., Wintner, A.: On the law of the iterated logarithm. Amer. J. Math. 63, 169–176 (1941)
Kuelbs, J.: The law of the iterated logarithm and related strong convergence theorems for Banach space valued random variables. Ecole d'Eté de Proabilités de Saint-Flour V-1975, Lecture Notes in Math. 539, New York: Springer 1976
Kuelbs, J.: Kolmogorov's law of the iterated logarithm for Banach space valued random variables. Illinois J. Math. 21, 784–800 (1977)
Kuelbs, J., Lepage, R.: The law of the iterated logarithm for Brownian motion in a Banach space. Trans. Amer. Math. Soc., 185, 253–264 (1973)
Kuelbs, J., Philipp, W.: Almost sure invariance invariance principles for partial sums of mixing B-valued random variables. Preprint (1977)
Major, P.: Approximation of partial sums of iid. r.v.s. when the summands have only two moments. Z. Wahrscheinlichkeitstheorie verw. Gebiete 35, 221–229 (1976)
Orey, S., Pruitt, W.E.: Sample functions of the N-parameter Wiener Process. Ann. Probab. 1, 138–163 (1973)
Philipp, W.: Almost sure invariance principles for sums of B-valued random variables. Lecture Notes in Math. 709, New York: Springer 1979
Pyke, R.: Partial sums of matrix arrays and Brownian sheets, Stochastic Analysis. eds. E.F. Harrding and D.G. Kendall. New York: Wiley 1973
Strassen, V.: An invariance principle for the law of the iterated logarithm. Z. Wahrscheinlichkeitstheorie verw. Gebiete 3, 211–226 (1964)
Wichura, M.J.: Some Strassen type laws of the iterated logarithm for multiparameter stochastic processes. Ann. Probab. 1, 272–296 (1973)
Yurinskii, V.V.: A smoothing inequality for estimates of the Levy-Prokhorov distance. Theory Probab. Appl. 20, 1–10 (1975)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Morrow, G.J. Approximation of rectangular sums of B-valued random variables. Z. Wahrscheinlichkeitstheorie verw Gebiete 57, 265–291 (1981). https://doi.org/10.1007/BF00535494
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00535494