STUDY OF SELF-SIMILAR PROCESSES

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・We studied the limiting processes for occupation times of fractional Brownian motion, which is a typical self-similar stochastic process. We see that the limiting process degenerates with the usual linear normalization but if consider the processes with the log-scale, we do have a meaningful process, which is the inverse of the so-called extremal process.・We generalized the above result to a certain class of Gaussian processes, where the occupation time increases slowly.・We studied the asymptotic behavior of the local times of fractional Brownian motion. As the index times the dimension approaches 1, we see that the one-dimensional marginal distributions of the local time at the origin converge to the exponentioal distribution, and furthermore, the processes, with a suitable time scale, converge to the inverse extremal process.・The invariant set under a family of functions has self-similarity. We studied its topological aspects.