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Favourite points, cover times and fractals. (English) Zbl 1102.60009
Dembo, Amir et al., Lectures on probability theory and statistics. Ecole d’Eté de Probabilités de Saint-Flour XXXIII – 2003. Lectures given at the 33rd probability summer school, Saint-Flour, France, July 6–23, 2003. Berlin: Springer (ISBN 3-540-26069-2/pbk). Lecture Notes in Mathematics 1869, 5-101 (2005).
This paper dedicates to recent advances in the study of the fractal nature of certain random sets, emphasizing the methods used to obtain such results. The author focuses on some of the fine properties of the sample path of the most basic stochastic processes such as simple random walks, Brownian motion, and symmetric stable processes. It is shown that along the way he also mentions quite a few challenging open research problems. Among the methods that will be detailed here are: Cover time for Markov chains; The dimension of discrete $$\lim\sup$$ random fractals; The truncated second moment method; The KMT strong approximation construction; Ciesielski-Taylor identities.
For the entire collection see [Zbl 1084.60005].

##### MSC:
 60D05 Geometric probability and stochastic geometry 60G17 Sample path properties 60J65 Brownian motion
##### Keywords:
random set; Brownian motion; stable processes; Markov chains