Information Parameters and Large Deviation Spectrum of Discontinuous Measures

Abstract

Let $\nu$ be a finite Borel measure on $[0,1]^d$. Consider the $L^q$-spectrum of $\nu$: $\tau_\nu(q)=\liminf_{n\to\infty}-n^{-1} \log_b\sum_{Q\in\mathcal{G}_n}\nu (Q)^q$ ($q\ge 0$), where $\mathcal{G}_n$ is the set of $b$-adic cubes of generation $n$ ($b$ integer $\ge 2$). Let $q_\tau=\inf\{q\: \tau_\nu(q)=0\}$ and $H_\tau=\tau_\nu'(q_\tau^-)$. When $\nu$ is a mono-dimensional continuous measure of information dimension $D$, $(q_\tau,H_\tau)=(1,D)$. When $\nu$ is purely discontinuous, its information dimension is $D=0$, but the pair $(q_\tau,H_\tau)$ may be non-trivial and contains relevant information on the distribution of $\nu$. Intrinsic characterizations of $(q_\tau,H_\tau)$ are found, as well as sharp estimates for the large deviation spectrum of $\nu$ on $[0,H_\tau]$. We exhibit the differences between the cases $q_\tau=1$ and $q_\tau\in (0,1)$. We conclude that the large deviation spectrum's properties observed for specific classes of measures are true in general.