FRACTAL DIMENSION AND POINT-WISE PROPERTIES OF TRAJECTORIES OF FRACTIONAL PROCESSES

Doctoral thesis (2022)

The topics of this thesis lie at the interference of probability theory with dimensional and harmonic analysis, accentuating the geometric properties of random paths of Gaussian and non-Gaussian ... [more ▼]

The topics of this thesis lie at the interference of probability theory with dimensional and harmonic analysis, accentuating the geometric properties of random paths of Gaussian and non-Gaussian stochastic processes. Such line of research has been rapidly growing in past years, paying off clear local and global properties for random paths associated to various stochastic processes such as Brownian and fractional Brownian motion. In this thesis, we start by studying the level sets associated to fractional Brownian motion using the macroscopic Hausdorff dimension. Then as a preliminary step, we establish some technical points regarding the distribution of the Rosenblatt process for the purpose of studying various geometric properties of its random paths. First, we obtain results concerning the Hausdorff (both classical and macroscopic), packing and intermediate dimensions, and the logarithmic and pixel densities of the image, level and sojourn time sets associated with sample paths of the Rosenblatt process. Second, we study the pointwise regularity of the generalized Rosenblatt and prove the existence of three kinds of local behavior: slow, ordinary and rapid points. In the last chapter, we illustrate several methods to estimate the macroscopic Hausdorff dimension, which played a key role in our results. In particular, we build the potential theoretical methods. Then, relying on this, we show that the macroscopic Hausdorff dimension of the projection of a set E ⊂ R^2 onto almost all straight lines passing through the origin in R^2 depends only on E, that is, they are almost surely independent of the choice of straight line. [less ▲]

A UNIFORM RESULT FOR THE DIMENSION OF FRACTIONAL BROWNIAN MOTION LEVEL SETS

in Statistics and Probability Letters (2020)

Let B={Bt:t≥0} be a real-valued fractional Brownian motion of index H∈(0,1). We prove that the macroscopic Hausdorff dimension of the level sets Lx={t∈R+:Bt=x} is, with probability one, equal to 1−H for ... [more ▼]

Let B={Bt:t≥0} be a real-valued fractional Brownian motion of index H∈(0,1). We prove that the macroscopic Hausdorff dimension of the level sets Lx={t∈R+:Bt=x} is, with probability one, equal to 1−H for all x∈R. [less ▲]