Hausdorff Dimension in a Process with Stable Components--An Interesting Counterexample

Abstract

Let $X_{\alpha_1}(t)$ and $X_{\alpha_2}(t)$ be independent stable processes in $R_1$ of stable index $\alpha_1$ and $\alpha_2$ respectively, where $1 < \alpha_2 < \alpha_1 \leqq 2$. Let $X(t) \equiv (X_{\alpha_1} (t), X_{\alpha_2}(t))$ be a process in $R_2$ formed by allowing $X_{\alpha_1}$ to run on the horizontal axis and $X_{\alpha_2}$ on the vertical axis; $X(t)$ is called a process with stable components. The Blumenthal-Getoor indices of $X(t)$ satisfy $\alpha_2 = \beta" < \beta' = 1 + \alpha_2 - \alpha_2/\alpha_1 < \beta = \alpha_1$. Denote by $\dim E$ the Hausdorff dimension of $E$. It is shown that if $E = \lbrack 0, 1\rbrack$ and $F$ is any fixed Borel set for which $\dim F \leqq 1/\alpha_1$ then (with probability 1) we have $\dim X(E) = \beta' \dim E$ and $\dim X(F) = \beta \dim X(F)$. This shows that the results of Blumenthal and Getoor (1961) for the bounds on $\dim X(E)$ for arbitrary processes $X$ and fixed Borel sets $E$ are the best possible, and that their conjecture that $\dim X(E) = \dim X\lbrack 0, 1\rbrack \cdot \dim E$ is incorrect.