My first and main comment is that it is an excellent article, as is, and should be accepted. I made a few changes to the text, but the author is welcome to back out those changes out if he feels they don't add to his exposition.
The reason I added the defintion of D in terms of a ratio of logs is that the \(C(r)\sim r^d\) scaling can be ambiguous in the presence of lacunarity. I was thinking of adding a few sentences about lacunarity itself (this is a phenomenon that leads to the \(C(r)\) curve showing periodic oscillations on top of the overall slope, so that the local slope in the log-log plot does not converge as \(r\to 0\)) but then I would have to cite my own paper (J. Theiler. ``Lacunarity in a best estimator of fractal dimension. Physics Letters A 133 (1988) 195-200)) and I've probably done enough self-citing in this reveiw already. Also, "fitting" a line to a wide-enough range of \(r\) on the log-log plot will avoid the lacunarity problem.
Some other changes I thought about adding, but instead will just suggest to the author:
- In discussing delay-time embedding, though it is traditionally called the "Takens embedding", I always cite the Santa Cruz guys as well, since their paper was also influential and actually came out a year earlier: N. H. Packard, J. P. Crutchfield, J. D. Farmer, and R. S. Shaw, "Geometry from a time series," Phys. Rev. Lett. 45,712 (1980).
- I was thinking that the notion of plotting all pairs with distance less than some fixed \(r\) goes by the name "Ruelle plot" but I couldn't find a reference for that so I may be mistaken; in any case, if that's so, I'm sure the author is familiar with it.
- Regarding the dimension of stochastic time series, there is a pair of papers that discuss the notion that "noise" can have a finite correlation dimension. (A. R. Osborne and A. Provenzale, "Finite correlation dimension for stochastic systems with power-law spectra," Physica 35D, 357 (1989); and yet another paper by me which responds to that paper: J. Theiler. ``Some comments on the correlation dimension of \(1/f^\alpha\) noise. Physics Letters A 155 (1991) 480-493.) My paper "derives" the result that O+P discuss, and then shows how it can be dismissed based essentially on the argument that \(1/f^\alpha\) noise is not stationary.
Once again, congratulations to the author for a very nicely written article.