Stationary Increments Harmonizable Stable Fields: Upper Estimates on Path Behaviour

Abstract

Studying sample path behaviour of stochastic fields/processes is a classical research topic in probability theory and related areas such as fractal geometry. To this end, many methods have been developed for a long time in Gaussian frames. They often rely on some underlying “nice” Hilbertian structure and can also require finiteness of moments of high order. Therefore, they can hardly be transposed to frames of heavy-tailed stable probability distributions. However, in the case of some linear non-anticipative moving average stable fields/processes, such as the linear fractional stable sheet and the linear multi-fractional stable motion, rather new wavelet strategies have already proved to be successful in order to obtain sharp moduli of continuity and other results on sample path behaviour. The main goal of our article is to show that, despite the difficulties inherent in the frequency domain, such kind of a wavelet methodology can be generalized and improved, so that it also becomes fruitful in a general harmonizable stable setting with stationary increments. Let us point out that there are large differences between this harmonizable setting and the moving average stable one. The real-valued harmonizable stable stochastic field X on which we focus is defined on $$\mathbb {R}^d$$ through an arbitrary spectral density belonging to a general and wide class of functions. First, we introduce a wavelet-type random series representation of X and express it as the finite sum $$X=\sum _\eta X^\eta $$ , where the fields $$X^\eta $$ are called the $$\eta $$ -frequency parts, since they extend the usual low-frequency and high-frequency parts. Moreover, we show the continuity of the sample paths of the $$X^\eta $$ ’s and X; also, we discuss the existence and continuity of their partial derivatives of an arbitrary order. Thereafter, we obtain several almost sure upper estimates related to: (a) the anisotropic behaviour of generalized directional increments of the $$X^\eta $$ ’s and X, on an arbitrary fixed compact cube of $$\mathbb {R}^d$$ ; (b) the behaviour at infinity of the $$X^\eta $$ ’s, of X, and of their partial derivatives, when they exist. We mention that all the results on sample paths obtained in the article are valid on the same event of probability 1; furthermore, this event is “universal”, in the sense that it does not depend, in any way, on the spectral density associated with X.