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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.

Almost sure limit theorems for semi-selfsimilar processes

Fazekas I., Rychlik Z.

Probability and Mathematical Statistics

2005

Vol. 25, Fasc. 2

241--255

An integral analogue of the almost sure limit theorem is presented for semi-selfsimilar processes. In the theorem, instead of a sequence of random elements, a continuous time random process is involved; moreover, instead of the logarithmical average, the integral of delta-measures is considered. Then the theorem is applied to obtain almost sure limit theorems for semistable processes. Discrete versions of the above theorems are proved. In particular, the almost sure functional limit theorem is obtained for semistable random variables.