Abstract
Our paper starts from presentation and comparison of three definitions for the selfsimilar field. The interconnection between these definitions has been established. Then we consider the Lamperti scaling transformation for the self-similar field and investigate the connection between the scaling transformation for such field and the shift transformation for the corresponding stationary field. It was also shown that the fractional Brownian sheet has the ergodic scaling transformation. The strong limit theorems for the anisotropic growth of the sample paths of the self-similar field at 0 and at ∞ for the upper and lower functions have been proved. It was obtained the upper bound for growth of the field with ergodic scaling transformation for slowly varying functions. We present some examples of iterated log-type limits for the Gaussian self-similar random fields.
Highlights
A self-similar process is a process invariant by distribution under specific time and/or space scaling
We introduce the notions of upper and lower functions for the sample paths of the random field which are similar to the paper [15] and prove the zero–one law for such functions in the case of growth at 0 and at ∞
In this paper the strong limit theorems for the anisotropic growth of the sample paths of the self-similar fields for the upper and lower functions arising in the zero– one law is proved
Summary
A self-similar process is a process invariant by distribution under specific time and/or space scaling. We introduce the notions of upper and lower functions for the sample paths of the random field which are similar to the paper [15] and prove the zero–one law for such functions in the case of growth at 0 and at ∞. In this paper the strong limit theorems for the anisotropic growth of the sample paths of the self-similar fields for the upper and lower functions arising in the zero– one law is proved. The similar theorems for the self-similar stochastic processes were proved in the paper [8] Application of these theorems to the Gaussian fields allows to obtain the iterated log-type lows.
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