Asymptotic behavior of semistable L\'evy exponents and applications to fractal path properties

TL;DR

This paper establishes precise tail bounds for the Le9vy exponent of operator semistable laws and applies these results to determine fractal dimensions of sample paths of related Le9vy processes.

Contribution

It provides explicit bounds on the Le9vy exponent tails and computes fractal dimensions of paths for operator semi-selfsimilar Le9vy processes.

Findings

Sharp tail bounds for Le9vy exponents

Explicit Hausdorff and packing dimension formulas

Elementary proofs using Le9vy exponent properties

Abstract

This paper proves sharp bounds on the tails of the L\'evy exponent of an operator semistable law on $\mathbb R^d$. These bounds are then applied to explicitly compute the Hausdorff and packing dimensions of the range, graph, and other random sets describing the sample paths of the corresponding operator semi-selfsimilar L\'evy processes. The proofs are elementary, using only the properties of the L\'evy exponent, and certain index formulae.