The α-dimensional measure of the graph and set of zeros of a Brownian path
- 1 April 1955
- journal article
- research article
- Published by Cambridge University Press (CUP) in Mathematical Proceedings of the Cambridge Philosophical Society
- Vol. 51 (2) , 265-274
- https://doi.org/10.1017/s030500410003019x
Abstract
In a recent joint paper (1) with Prof. Besicovitch we announced the conjecture that for almost all one-dimensional Brownian paths, the set of zeros has dimensional number ½, and zero A½-measure. It is the purpose of this paper to give a proof of this result. In doing so we consider the graph C(ω) of a Brownian path ω as a point set in the plane, and prove that, with probability 1, C(ω) has dimensional number ¾ and zero Λ¾-measure.Keywords
This publication has 8 references indexed in Scilit:
- Processus Stochastiques et Mouvement Brownien.Biometrika, 1966
- On the Complementary Intervals of a Linear Closed Set of Zero Lebesgue MeasureJournal of the London Mathematical Society, 1954
- The dimension of Cartesian product setsMathematical Proceedings of the Cambridge Philosophical Society, 1954
- The Hausdorff α-dimensional measure of Brownian paths in n-spaceMathematical Proceedings of the Cambridge Philosophical Society, 1953
- On Hausdorff's measures and generalized capacities with some of their applications to the theory of functionsJapanese journal of mathematics :transactions and abstracts, 1945
- Sets of Fractional Dimensions (V): on Dimensional Numbers of Some Continuous CurvesJournal of the London Mathematical Society, 1937
- Über den transfiniten Durchmesser ebener PunktmengenMathematische Zeitschrift, 1930
- ber die Gleichverteilung von Zahlen mod. EinsMathematische Annalen, 1916