Benoit B. Mandelbrot

The variation of certain speculative prices

The classic model of the temporal variation of speculative prices (Bachelier 1900) assumes that successive changes of a price Z(t) are independent Gaussian random variables. But, even if Z(t) is replaced by log Z(t),this model is contradicted by facts in four ways, at least: (1) Large price changes are much more frequent than predicted by the Gaussian; this reflects the “excessively peaked” (“leptokurtic”) character of price relatives, which has been well-established since at least 1915. (2) Large practically instantaneous price changes occur often, contrary to prediction, and it seems that they must be explained by causal rather than stochastic models. (3) Successive price changes do not “look” independent, but rather exhibit a large number of recognizable patterns, which are, of course, the basis of the technical analysis of stocks. (4) Price records do not look stationary, and statistical expressions such as the sample variance take very different values at different times; this nonstationarity seems to put a precise statistical model of price change out of the question.

Intermittent turbulence in self-similar cascades: divergence of high moments and dimension of the carrier

✦ Abstract. Kolmogorovs “third hypothesis” asserts that in intermittent turbulence the average ε of the dissipation ε, taken over any domain , is ruled by the lognormal probability distribution. This hypothesis will be shown to be logically inconsistent, save under assumptions that are extreme and unlikely. A widely used justification of lognormality due to Yaglom and based on probabilistic argument involving a self-similar cascade, will also be discussed. In this model, lognormality indeed applies strictly when is “an eddy,” typically a three-dimensional box embedded in a self-similar hierarchy, and may perhaps remain a reasonable approximation when consists of a few such eddies. On the other hand, the experimental situation is better described by considering averages taken over essentially one-dimensional domains .

Random Walk Models for the Spike Activity of a Single Neuron

Quantitative methods for the study of the statistical properties of spontaneously occurring spike trains from single neurons have recently been presented. Such measurements suggest a number of descriptive mathematical models. One of these, based on a random walk towards an absorbing barrier, can describe a wide range of neuronal activity in terms of two parameters. These parameters are readily associated with known physiological mechanisms.

Self-Affine Fractals and Fractal Dimension

Evaluating a fractal curves approximate length by walking a compass defines a compass exponent. Long ago, I showed that for a self-similar curve (e.g., a model of coastline), the compass exponent coincides with all the other forms of the fractal dimension, e.g., the similarity, box or mass dimensions. Now walk a compass along a self-affine curve, such as a scalar Brownian record B(t). It will be shown that a full description in terms of fractal dimension is complex. Each version of dimension has a local and a global value, separated by a crossover. First finding: the basic methods of evaluating the global fractal dimension yield 1: globally, a self-affine fractal behaves as it if were not fractal. Second finding: the box and mass dimensions are 1.5, but the compass dimension is D = 2. More generally, for a fractional Bownian record BH(t), (e.g., a model of vertical cuts of relief), the global fractal dimensions are 1, several local fractal dimensions are 2-H, and the compass dimension is 1/H. This 1/H is the fractal dimension of a self-similar fractal trail, whose definition was already implicit in the definition of the record of BH(t).

New Methods in Statistical Economics

An interesting relationship between the methods in this chapter and renormalization as understood by physicists is described in the Annotation for the physicists that follows this text.

Fractals and Scaling In Finance: Discontinuity, Concentration, Risk

Mandelbrot is world famous for his creation of the new mathematics of fractal geometry. Yet few people know that his original field of applied research was in econometrics and financial models, applying ideas of scaling and self-similarity to arrays of data generated by financial analyses. This book brings together his original papers as well as many original chapters specifically written for this book.

Multifractal Measures, Especially for the Geophysicist

This text is addressed to both the beginner and the seasoned professional, geology being used as the main but not the sole illustration. The goal is to present an alternative approach to multifractals, extending and streamlining the original approach in Mandelbrot (1974). The generalization from fractal sets to multifractal measures involves the passage from geometric objects that are characterized primarily by one number, namely a fractal dimension, to geometric objects that are characterized primarily by a function. The best is to choose the function ρ(α), which is a limit probability distribution that has been plotted suitably, on double logarithmic scales. The quantity α is called Holder exponent. In terms of the alternative function f(α) used in the approach of Frisch-Parisi and of Halsey et al., one has ρ(α) = f(α) − E for measures supported by the Euclidean space of dimension E. When f(α) ≥ 0, f(α) is a fractal dimension. However, one may have f(α) 1 to be a critical dimension for the cuts. An “enhanced multifractal diagram” is drawn, including f(α), a function called τ(q) and D q .

On the geometry of homogeneous turbulence, with stress on the fractal dimension of the iso-surfaces of scalars

This paper studies several geometric aspects of the Poisson and Gaussian random fields approximating Burgers k −2 and Kolmogorov

Fractal aspects of the iteration of z→λz(1-z) for complex λ and z

k^{-\frac{5}{3}}

Limit Theorem on the Self-Normalized Range for Weakly and Strongly Dependent Process

homogeneous turbulence. In particular, simulated sample scalar iso-surfaces (e.g. surfaces of constant temperature or concentration) are exhibited, and their relative degrees of wiggliness are shown to be best characterized by saying that the corresponding fractal dimensions are respectively equal to 3−½ and