A new estimator for information dimension with standard errors and confidence intervals (Q1965875)

The problem how to estimate the information dimension \[ \lim _{r\rightarrow 0} (\log r)^{-1} \int \log \mu (\{u\in R^d:\|x-u\|\leq r\}) d\mu (x) \] of the probability measure \(\mu \) on \(X\subset R^d\), on the basis of a finite sample \(X_1,\ldots ,X_N\), of \(R^d\)-valued observations with distribution \(\mu \) is solved. The estimation algorithm is similar to the Grassberger-Procaccia algorithm for estimating the correlation dimension. Under mixing assumptions on the observations, the new estimator is asymptotically normal.