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130 5. Scaling in Financial Data and in Physics

The convergence towards a Gaussian can also be studied from the char-

acteristic function (5.52). One can expand its logarithm to second order

in z,

(z) ∼−

cos(πµ/2)

(µ − µ

)

+ ... . (5.56)

Fourier transformation implies that the Gaussian behavior of the charac-

teristic function for small z translates into Gaussian large-|x| tails in the

probability distribution. On the other hand,

(z) ∼−a|z|

for|z|→∞, (5.57)

which implies L´evy behavior for small |x|. One would therefore conclude that

the convergence towards a Gaussian, for the distribution of a sum of many

variables, should predominantly take place from the tails. Also, depending

on the cutoﬀ variable and due to the stability of the L´evy distributions, the

convergence can be extremely slow. As shown in Fig. 5.11, such a distribution

describes ﬁnancial data extremely well.

Notice that one could also use a hard-cutoﬀ truncation scheme, such as

[71, 74]

(x)=L

(x)Θ(α

−1

−|x|) . (5.58)

While it has the advantage of being deﬁned directly in variable space and

avoiding complicated Fourier transforms, the hard cutoﬀ produces smooth

distributions only after the addition of many random variables.

Student-t Distribution

A (symmetric) Student-t distribution is deﬁned in variable space by

(x)=

Γ [(1 + µ)/2]

√

πΓ (µ/2)

+ x

)

(1+µ)/2

. (5.59)

A is a scale parameter, Γ (x) is the Gamma function, and the deﬁnition of

the index µ is consistent with Sect. 5.4.3. A priori, there is no restriction on

the value of µ>0. For large arguments, the distribution decays with a power

law

(x) ∼

|x|

1+µ

for |x|A, (5.60)

that is, formally in the same way as would do a L´evy distribution. Its char-

acteristic function is

(z)=

1−µ/2

Γ (µ/2)

(Az)

µ/2

(Az) , (5.61)

≈

√

Γ (µ/2)





(µ−1)/2

−Az

for z →∞, (5.62)

≈ 1+

1 −





−

Γ (1 +

)

Γ (1 −

)





for z → 0 . (5.63)

5.5 Scaling, L´evy Distributions, and L´evy Flights in Nature 131

Interestingly, for µ>2, the dominant term in the expansion of the charac-

teristic function for small z is identical in form to that of a similar expansion

of a Gaussian distribution while, for µ<2, it is identical in form to that of

a small-z expansion of a stable L´evy distribution. When µ<2, the distribu-

tion of a sum of many Student-t distributed random variables with index µ

will converge to a stable L´evy distribution with the same index, according to

the generalized central limit theorem. For example, for µ = 1, the Student-t

distribution reduces to the Lorentz–Cauchy distribution

(x)=L

(x)=

+ x

, (5.64)

which also is a stable L´evy distribution. For µ>2 the central limit theorem

requires the distribution of a sum of many Student-t distributed random

variables to converge to a Gaussian distribution.

The Student-t distribution is named after the pseudonym “Student” of

the English statistician W. S. Gosset and arises naturally when dividing a

normally distributed random variable by a χ

-distributed random variable

[44].

5.5 Scaling, L´evy Distributions,

and L´evy Flights in Nature

Although the na¨ıve interpretation of the central limit theorem seems to sug-

gest that the Gaussian distribution is the universal attractor for distributions

of random processes in nature, distributions with power-law tails arise in

many circumstances. It is much harder, however, to ﬁnd situations where the

actual diﬀusion process is non-Brownian, and close to a L´evy ﬂight [75]–[77].

5.5.1 Criticality and Self-Organized Criticality,

Diﬀusion and Superdiﬀusion

The asymptotic behavior of a L´evy distribution is, (5.44),

p(x) ∼|x|

−(1+µ)

. (5.65)

The classical example for the occurrence of such distributions is provided by

the critical point of second-order phase transitions [78], such as the transition

from a paramagnet to a ferromagnet, as the temperature of, say, iron is

lowered through the Curie temperature T

. At a critical point, there are

power-law singularities in almost all physical quantities, e.g., the speciﬁc heat,

the susceptibility, etc. The reason for these power-law singularities are critical

ﬂuctuations of the ordered phase (ferromagnetic in the above example) in

the disordered (paramagnetic) phase above the transition temperature, resp.

132 5. Scaling in Financial Data and in Physics

vice versa below the critical temperature, as a consequence of the interplay

between entropy and interactions. In general, for T = T

, there is a typical size

ξ (the correlation length) of the ordered domains. At the critical point T = T

however, ξ →∞, and there is no longer a typical length scale. This means that

ordered domains occur on all length scales, and are distributed according to a

power-law distribution (5.65). The same holds for the distribution of cluster

sizes in percolation. The divergence of the correlation length is the origin of

the critical singularities in the physical quantities.

Critical points need ﬁne tuning. One must be extremely close to the criti-

cal point in order to observe the power-law behavior discussed, which usually

requires an enormous experimental eﬀort in the laboratory for an accurate

control of temperature, pressure, etc. Such ﬁne tuning by a gifted experimen-

talist is certainly not done in nature. Still, there are many situations where

power-law distributions are observed. Examples are given by earthquakes

where the frequency of earthquakes of a certain magnitude on the Richter

scale, i.e., a certain release of energy, varies as N (E) ∼ E

−1.5

, avalanches,

traﬃc jams, and many more [79].

To explain this phenomenon, a theory of self-organized criticality has been

developed [79]. The idea is that open, driven, dissipative systems (notice that

physical systems at the critical point are usually in equilibrium!) may spon-

taneously approach a critical state. An example was thought to be provided

by sandpiles. Imagine pouring sand on some surface. A sandpile will build

up with its slope becoming steeper and steeper as sand is added. At some

stage, one will reach a critical angle where the friction provided by the grains

surrounding a given grain is just suﬃcient to compensate gravity. As sand is

added to the top of a pile, the critical slope will be exceeded at some places,

and some grains will start sliding down the side of the pile. As a consequence,

at some lower position, the critical slope will be exceeded, and more grains

will slide. An avalanche forms. It was conjectured, and supported by nu-

merical simulations [79], that the avalanches formed in this way will possess

power-law distributions. (Unfortunately, it appears that real sandpiles have

diﬀerent properties.)

Both with critical phenomena, and with self-organized criticality, one

looks at statistical properties of a system. Can one observe true “anomalous”

distributions, corresponding to L´evy ﬂights, in nature? Diﬀusion processes

can be classiﬁed according to the long-time limit of the variance

lim

t→∞

(t)

⎧

⎨

⎩

0 : subdiﬀusive

D : diﬀusive

∞ : superdiﬀusive .

(5.66)

A numerical simulation of superdiﬀusion, modeled as a L´evy ﬂight in two

dimensions with µ =3/2, is shown in Fig. 5.15. For comparison, Brownian

motion was shown in Fig. 3.8. One again notices the long straight lines, corre-

sponding more to ﬂights of the particle than to the short-distance hops associ-

ated with diﬀusion. This is a pictorial representation of superdiﬀusive motion.

5.5 Scaling, L´evy Distributions, and L´evy Flights in Nature 133

Fig. 5.15. Computer simulation of a two-dimensional L´evy ﬂight with µ =3/2.

Reprinted from J. Klafter, G. Zumofen, and M. F. Shlesinger: in L´evy Flights and

Related Topics, ed. by M. F. Shlesinger et al. (Springer-Verlag, Berlin 1995)

1995

Springer-Verlag

Of course, this is in direct correspondence with the continuity/discontinuity

observed in the 1D versions, cf. Figs. 1.3 and 5.14.

5.5.2 Micelles

Micelles are long, wormlike molecules in a liquid environment. Unlike poly-

mers, they break up at random positions along the chains at random times,

and recombine later with the same or a diﬀerent strand. The distribution of

chain lengths  is

mic

() ≈ exp(−/

) . (5.67)

Any ﬁxed-length chain performs ordinary Brownian diﬀusion with a diﬀusion

coeﬃcient which depends on its length as

D()=D



−2β

. (5.68)

In order to observe a L´evy ﬂight, one can attach ﬂuorescent tracer mole-

cules to such micelles [80]. Due to break-up and recombination, any given

tracer molecule will sometimes be attached to a short chain, and sometimes

to a long chain. When the chain is short, which according to (5.67) happens

frequently, it will diﬀuse rapidly, cf. (5.68). On long chains, it will diﬀuse

slowly.

Using photobleaching techniques, one can observe the apparent diﬀusion

of the tracer molecules and evaluate its statistical properties [80]. One in-

deed ﬁnds the superdiﬀusive behavior associated with L´evy ﬂights, namely a

134 5. Scaling in Financial Data and in Physics

characteristic function

trac

(q, t)=exp(−D|q|

t) ,µ=2/β ≤ 2 , (5.69)

where the precise value of µ depends somewhat on experimental conditions.

Notice, however, that the true physical diﬀusion process underlying this

example is still Brownian (diﬀusion of the micelles). It is the length depen-

dence of the diﬀusion constant [again typical of Brownian diﬀusion, remember

Einstein’s formula (3.25)] which conveys an apparent superdiﬀusive character

to the motion of the tracer particles when their support is ignored.

5.5.3 Fluid Dynamics

The transport of tracer particles in ﬂuid ﬂows is usually governed both by

advection and “normal” diﬀusion processes. Normal diﬀusion arises from the

disordered motion of the tracer particles, hit by particles from the ﬂuid, cf.

Sect. 3.3. Advection of the tracer particles by the ﬂow, i.e., tracer parti-

cles being swept along by the ﬂow, leads to enhanced diﬀusion, but not to

superdiﬀusion. It can be described as an ordinary random walk, but with

diﬀusion rates enhanced over those typical for “normal” diﬀusion. For long

times, therefore, the transport in real ﬂuid ﬂows is normally diﬀusive.

The short-time limit may be diﬀerent in some situations. If there are

vortices in the system, the tracer particles may stick to the vortices and

their transport may become subdiﬀusive. On the other hand, in ﬂows with

coherent jets, the tracer particles may move ballistically for long distances.

This process may eventually lead to superdiﬀusion, and to L´evy ﬂights.

One can now set up an experiment where the ﬂow pattern is composed

both of coherent jets and vortices, i.e., sticking and ballistic ﬂights of the

tracer particles. The experimental setup is shown in Fig. 5.16 [81, 82]. A 38

weight% mixture of water and glycerol (viscosity 0.03 cm

/s) is contained in

an annular tank rotating with a frequency of 1.5 s

−1

. In addition, ﬂuid is

pumped into the tank through a ring of holes (labeled I in Fig. 5.16) and

out of the tank through another ring of holes (O). The radial ﬂow couples

to the Coriolis force to produce a strong azimuthal jet, in the direction op-

posite to the rotation of the tank. Above the forcing rings, there are strong

velocity gradients, and the shear layer becomes unstable. As a result, a chain

of vortices forms above the outer ring of holes (a similar vortex chain above

the inner ring is inhibited artiﬁcially). In a reference frame rotating with the

vortices, they appear sandwiched between two azimuthal jets going in oppo-

site directions. Such pattern of jets and vortices is shown in the lower panel

of Fig. 5.16. Depending on perturbations generated by deliberate axial inho-

mogeneities of the pattern of the radial ﬂow, diﬀerent regimes of azimuthal

ﬂow can be realized [81].

When a 60 degrees sector has a radial ﬂow less than half of that of the

ﬂow between the remaining source and sink holes, a “time-periodic” regime

5.5 Scaling, L´evy Distributions, and L´evy Flights in Nature 135

Fig. 5.16. Setup of an experiment with coexisting jets and vortices. Upper panel:

the rotating annulus. I and O label two rings of holes for pumping and extracting

ﬂuid in the sense of the arrows. As explained in the text, under such conditions

both jets and vortices form in the tank. Lower panel : streaks formed by 90-second-

long trajectories of about 30 tracer particles reveal the presence of six vortices

sandwiched between two azimuthal jets. The picture has been taken in a reference

frame corotating with the vortex chain. By courtesy of H. Swinney. Reprinted with

permission from Elsevier Science from T. H. Solomon et al.: Physica D 76,70

(1994).

1994 Elsevier Science

136 5. Scaling in Financial Data and in Physics

is established. One can then map out the trajectories of passive tracer parti-

cles. The motion of these particles, and of the supporting liquid, has periods

of ﬂight, where the particles are simply swept along ballistically by the az-

imuthal jets, of capture and release by the vortices, and diﬀusion. These

processes can be analyzed separately, and probability density functions for

various processes can be derived. They generally scale as power laws of their

variables. For example, the probability density function of times where the

tracer particles stick to vortices behaves as

(t) ∼ t

−(1+µ

)

,µ

∼ 0.6 ±0.3 . (5.70)

The probability density distribution of the ﬂight times behaves as

(t) ∼ t

−(1+µ

)

,µ

∼ 1.3 ±0.2 , (5.71)

that is, it carries a diﬀerent exponent. Figure 5.17 shows the results of such

an experiment leading to these power laws. Yet another exponent is measured

by the distribution of the ﬂight lengths 

Fig. 5.17. Probability distributions of the sticking times (a) and ﬂight times (b)

for tracer particles moving in a ﬂow pattern composed of two azimuthal jets and

vortices. The straight lines have slopes −1.6 ± 0.3 and −2.3 ± 0.2, respectively

(cf. text). The tracer particles execute L´evy ﬂights. By courtesy of H. Swinney.

Reprinted with permission from Elsevier Science from T. H. Solomon et al.: Physica

D 76, 70 (1994).

1994 Elsevier Science

5.5 Scaling, L´evy Distributions, and L´evy Flights in Nature 137

P () ∼ 

−(1+µ



)

,µ



∼ 1.05 ±0.0.3 . (5.72)

In these experiments, the ﬂuid and the tracer particles in this system there-

fore perform a L´evy ﬂight. Pictures of the traces of individual particles are

available in the literature [81, 82], and generally look rather similar to the

computer simulation shown in Fig. 5.15.

5.5.4 The Dynamics of the Human Heart

The human heart beats in a complex rhythm. Let B(i) ≡ ∆t

= t

i+1

−t

de-

note the interval between two successive beats of the heart. Figure 5.18 shows

two sequences of interbeat intervals, one (top) of a healthy individual, the

Fig. 5.18. The time series of intervals between two successive heart beats, (a)for

a healthy subject, (b) for a patient with a heart disease (dilated cardiomyopathy).

By courtesy of C.-K. Peng. Reprinted from C.-K. Peng et al.: in L´evy Flights and

Related Topics, ed. by M. F. Shlesinger et al. (Springer-Verlag, Berlin 1995)

1995

Springer-Verlag

138 5. Scaling in Financial Data and in Physics

other (bottom) of a patient suﬀering from dilated cardiomyopathy [83]. For

reasons of stationarity, one prefers to analyze the probability for a variation

of the interbeat interval (in the same way as ﬁnancial data use returns rather

than prices directly). The surprising ﬁnding then is that both time series lead

to L´evy distributions for the increments I

= B(i +1)− B(i) with the same

index µ ≈ 1.7 (not shown) [83]. The main diﬀerence between the two data

sets, at this level, is the standard deviation, which is visibly reduced by the

disease.

To uncover more diﬀerences in the two time series, a more reﬁned analysis,

whose results are shown in Fig. 5.19, is necessary. The power spectrum of the

time series of increments, S

(f)=|I(f)|

with I(f) the Fourier transform

of I

, for a normal patient has an almost linear dependence on frequency,

S(f ) ∼ f

0.93

. For the suﬀering patient, on the other hand, the power spec-

trum is almost ﬂat at low frequencies, and only shows an increase above a

ﬁnite threshold frequency [83]. To appreciate these facts, note that, for a

purely random signal, S(f ) = const., i.e., white noise. Correlations in the

signal lead to red noise, i.e., a decay of the power spectrum with frequency

S(f ) ∼ f

−β

with 0 <β≤ 1. 1/f noise, typically caused by avalanches, is an

example of this case. On the other hand, with anticorrelations (a positive sig-

nal preferentially followed by a negative one), the power spectrum increases

with frequency, S(f) ∼ f

. This is the case here for a healthy patient. With

the disease, the small-frequency spectrum is almost white, and the typical an-

ticorrelations are observed only at higher beat frequencies. Also, detrended

ﬂuctuation analysis shows diﬀerent patterns for both healthy and diseased

subjects [83].

5.5.5 Amorphous Semiconductors and Glasses

The preceding discussion may be rephrased in terms of a waiting time dis-

tribution between one heartbeat and the following one. Waiting time distri-

butions are observed in a technologically important problem, the photocon-

ductivity of amorphous semiconductors and discotic liquid crystals. These

materials are important for Xerox technology.

In the experiment, electron–hole pairs are excited by an intense laser pulse

at one electrode and swept across the sample by an applied electric ﬁeld. This

will generate a displacement current. Depending on the relative importance of

various transport processes, diﬀerent current–time proﬁles may be observed.

For Gaussian transport, the electron packet broadens, on its way to the other

electrode, due to diﬀusion. A snapshot of the electron density will essentially

show a Gaussian proﬁle. The packet will hit the right electrode after a char-

acteristic transit time t

which shows up as a cutoﬀ in the current proﬁle.

Up to the transit time, the displacement current measured is a constant.

In a strongly disordered material, the transport is dispersive, however.

Now, electrons become trapped by impurity states in the gap of the semi-

conductor. They will be released due to activation. The release rates depend

5.5 Scaling, L´evy Distributions, and L´evy Flights in Nature 139

Fig. 5.19. The power spectrum S

(f) for the time series of increments of the time

between two successive heart beats, (a) for a healthy individual, (b) for a patient

with heart disease. The power spectrum of the healthy subject is characteristic of

time series with anticorrelations over the entire frequency range, while that of the

patient with heart failure is white at low frequencies, and exhibits the anticorrela-

tions only in its high-frequency part. By courtesy of C.-K. Peng. Reprinted from

C.-K. Peng et al.: in L´evy Flights and Related Topics, ed.byM.F.Shlesingeretal.

(Springer-Verlag, Berlin 1995)

1995 Springer-Verlag