In this chapter, we will consider some of the applications of probability theory to the study of fractals. The most important of these is the possibility of random fractals, that is, random sets (or random measures) that exhibit the irregularities (and regularities) characteristic of the sets and measures that we normally associate with the word “fractal.” The basic idea of self-similarity can be modified slightly to statistical self-similarity. A random set is said to be statistically self-similar if the set is made up of smaller parts, and each part is similar to an instance of the same random set. The parts are not actually similar to the whole, but they are similar to sets that might have been obtained for the whole if the random events had turned out differently. (More technically, the part is similar to a random set whose probability distribution is the same as the distribution of the whole set.)
KeywordsBrownian Motion Hausdorff Dimension Iterate Function System Pointwise Dimension Levy Flight
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