The Physics of Wall Street: a brief History of Predicting the Unpredictable

“It was one of his first attempts . . .”: Mandelbrot coined the term fractal in 
Mandelbrot (1975), which was translated into english as Mandelbrot (1977). But Man-
delbrot (1967) is one of the first places where he describes geometrical objects with non-
integer Hausdorff dimension exhibiting self-similarity.
56 “. . . but anti-Semitism in the south was less virulent . . .”: While the compara-
tive claim is true, it should not be taken to mean that anti-Semitism was not rampant in 
vichy france. for more on vichy france during World War II, including french anti-
Semitism during the war, see Paxton (1972), Marrus and Paxton (1995), and Poznanski 
(2001).
57 “. . . except to say that . . .”: these quotes come from the interview that Man-
delbrot did for Web of Stories (Mandelbrot 1998).
58 “In Thomas Pynchon’s novel Gravity’s rainbow . . .”: this is Pynchon (1973).
59 “The normal distribution shows up . . .”: Indeed, an important result of math-
ematical statistics, the central limit theorem, states that if you can model a random vari-
able as the sum of a sufficiently large number of independent and identically distributed 
random variables, where the distribution of the random variables in the sum has finite 
mean (average) and variance (volatility), then the random variable must be normally 
distributed, even if the variables in the sum are not normally distributed. this means 
that normal distributions appear all over the place. As we shall see, however, Mandelbrot 
argued that for financial markets, one of the assumptions of the central limit theorem 
fails: he argues that the distributions of market returns do not have finite variance. for 
more on the central limit theorem, see Billingsley (1995), casella and Berger (2002), and 
forbes et al. (2011). for more on Mandelbrot’s claims, see Mandelbrot (1997) and Man-
delbrot and Hudson (2004).
59
“. . . the law of large numbers for probability distributions . . .”: It is actually 
more general than the other version of the law of large numbers, which governs how 
probabilities for simple games like coin flips relate to frequency. the law of large num-
bers for probability distributions can be used to prove the other version, as can be seen 
by thinking about the coin-tossing example.

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Notes 
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59
“Not all probability distributions satisfy the law of large numbers . . .”: the more 
precise version of this claim is that not all distributions have finite mean — and indeed, 
cauchy distributions do not have finite mean. for more on cauchy distributions and the 
law of large numbers, see casella and Berger (2002), Billingsley (1995), and forbes et al. 
(2011).
61 “But then ‘a storm’ would come through . . .”: Mandelbrot describes this aspect 
of his wartime experience in Mandelbrot (1998).
62 “This is a general property of fractals . . .”: there are many connections be-
tween fractals and fat-tailed distributions. that certain features of fractals exhibit fat 
tails is one such connection; another is that (some) fat-tailed distributions themselves 
exhibit self-similarity, in the form of power-law scaling in their tails. Mandelbrot was a 
central figure in identifying and exploring these relationships. See Mandelbrot (1997).
63 “Known as the Butcher of Lyon . . .”: for more on Barbie, see Bower (1984) and 
McKale (2012).
65 “. . . ‘there was no great distinction . . .’”: this quote is from Mandelbrot (1998).
66 “. . . and economist named Vilfredo Pareto”: the definitive collection on Pareto 
and his influence is the three-volume Wood and Mcclure (1999); see also cirillo (1979).
68 “. . . it appeared that there was no ‘average’ rate of return”: In other words, 
it seemed that neither mean nor variance was defined for the distributions of cotton 
prices. As described below, Mandelbrot would later argue that the distributions of rates 
of return for financial markets do have finite means, but not variances. However, it can 
often be difficult to calculate the mean for a Lévy-stable distribution — in cases where 
variance is undefined, the average value calculated from any finite data set takes a long 
time to converge to the mean — which accounts for why Mandelbrot and Houthakker 
originally believed that the mean did not exist.
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