Multifractal analysis of the occupation measures of a kind of stochastic processes
An \(\alpha \)-sequence (\(0<\alpha <1\)) is a sequence \(\{ a_1,a_2,\ldots \}\) of positive numbers such that \(a_i\searrow 0\), \(\sum _ia_i=1\), \(\sum _i\min \{ a_i,\varepsilon \}\approx \varepsilon ^{1-\alpha}\), \(\sup \{ k:a_k\geq \varepsilon \}\approx \varepsilon ^{-\alpha}\) (\(0<\varepsilon <1\)). Consider the stochastic process \(\{ X(t):t\in [0,1]\}\) defined on the probability space \(\Omega =[0,1]^{\mathbb N}\) with the product Borel \(\sigma \)-algebra and product Lebesgue measure \({\mathcal L}^{\mathbb N}\) by \[ X_\omega (t)=\sum _ia_iI_{\{ 0\leq \omega _i<t\}},\quad t\in [0,1],\quad \omega =(\omega _1,\omega _2,\ldots)\in \Omega . \] Let \(\mu \) be the occupation measure \(\mu \) of \(X\) (\(\mu (\cdot)={\mathcal L}(\{ t:X(t)\in \cdot \})\)). For fixed \(x\in [0,1]\), the local density \(d(\mu ,x)\) of \(\mu \) at \(x\) equals \(\alpha \) almost surely (this follows from some earlier results of the author). In this paper, the set of points where \(d(\mu ,x)\neq \alpha \) is investigated. In particular, bounds for the Hausdorff dimension of the level sets \(\{ x : d (\mu ,x)=\beta \}\), \(\alpha \leq \beta \leq 2\alpha \), are obtained.
scientific article; zbMATH DE number 1554301