Benoit B Mandelbrot comments on the paper by Blake LeBaron, on page 621 of this issue, by tracing the merits and pitfalls of power-law scaling models from antiquity to the present.

In the Brownian model, even the largest of N successive daily price increments contributes negligibly to the overall sample variance. The resulting 'absent' concentration justifies the role of variance in measuring Brownian volatility. Mandelbrot introduced in 1963 an alternative 'mesofractal...

We showed in an earlier paper (1995a) that negatively correlated fractional Brownian motion (FBM) can be generated as a fractal sum of one kind of micropulses (FSM). That is, FBM of exponent is the limit (in the sense of finite-dimensional distributions) of a certain sequence of processes...

We begin with stochastic processes obtained as sums of "up-and-down" pulses with random moments of birth [tau] and random lifetime w determined by a Poisson random measure. When the pulse amplitude [var epsilon] -- 0, while the pulse density [delta] increases to infinity, one obtains a process...

Having been crafted to welcome a new scientific journal, this paper looks forward but requires no special prerequisite. The argument builds on a technical wrinkle (used earlier but explained here fully for the first time), namely, the author’s grid-bound variant of Brownian motion B(t). While...

A sharp distinction is drawn between general multiplicative multifractals, as originally introduced by the author, and the more familiar but more restricted class defined by Frisch and Parisi and by Halsey et al. All the general multiplicative multifractals are exactly renormalizable, by design....

Using two new methods of geometric analysis, this paper establishes that DLA clusters are definitely not self-similar. Compared to small clusters, the morphology of large clusters (of sizes up to 30 million particles) can be characterized, both visually and quantitatively, as being far more...

In diverse sciences that lack Hamiltonians, the analysis of complex systems is helped by the powerful tools provided by renormalization, fixed points and scaling. As one example, an intrinsic form of exact renormalizability was long used by the author in economics and related fields, most...

The right-hand side of the ƒ(α) curve of the harmonic measure on DLA is undefined. This does not necessarily imply that the harmonic measure and the DLA geometry are not self-similar. We show for off-lattice DLA that the right-hand tail satisfies a different rescaling rule. This Cauchy...