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Dimension results related to the St. Petersburg game

Kern P., Wedrich L.

Probability and Mathematical Statistics

2014

Vol. 34, Fasc. 1

97--117

Let Sn be the total gain in n repeated St. Petersburg games. It is known that n−1(Sn − n log2 n) converges in distribution along certain geometrically increasing subsequences and its possible limiting random variables can be parametrized as Y (t) with t ∈ [1/2, 1]. We determine the Hausdorff and box-counting dimension of the range and the graph for almost all sample paths of the stochastic process {Y(t)}t∈[1/2, 1]. The results are compared to the fractal dimension of the corresponding limiting objects when gains are given by a deterministic sequence initiated by Hugo Steinhaus.

On Besov regularity of Brownian motions in infinite dimensions

Hytönen T. P., Veraar M. C.

Probability and Mathematical Statistics

2008

Vol. 28, Fasc. 1

143--162

We extend to the vector-valued situation some earlier work of Ciesielski and Roynette on the Besov regularity of the paths of the classical Brownian motion.We also consider a Brownian motion as a Besov space valued random variable. It turns out that a Brownian motion, in this interpretation, is a Gaussian random variable with some pathological properties. We prove estimates for the first moment of the Besov norm of a Brownian motion. To obtain such results we estimate expressions of the form E supn1‖ξn‖, where ξn are independent centered Gaussian random variables with values in a Banach space. Using isoperimetric inequalities we obtain two-sided inequalities in terms of the first moments and the weak variances of ξn.