Yuliya Mishura: What can happen between two self-similarities?
Yuliya Mishura (Taras Shevchenko National University of Kyiv) gives a lecture with the title: What can happen between two self-similarities?
Everybody knows that fractional Brownian motion with any Hurst index H is self-similar process with stationary increments. According to geometric terminology of J. P. Kahane, it belongs to helix. Self-similarity and incremental stationarity are very useful when we study the properties of different functionals based on fBm however these properties are rather restrictive. For example, Ornstein-Uhlenbeck process starting from zero time point is neither self-similar nor stationary or with stationary increments. Therefore the goal of the present talk is to consider wider class of Gaussian processes. In geometric terminology of J. P. Kahane, such processes belong to quasi-helix, in our terminology they live between two self-similarities, or belong to the generalized quasi-helix.
We consider three problems concerning such processes:
--asymptotic behavior of maximal functionals;
--representation theorems involving integrals w.r.t. such processes;
--some statistical results.