130
9 Bubbles, Crashes, Fat Tails and Lévy-Stable Distributions
lim
N →∞
˜
Nr (k) = e
−c 2 k
2 /2
,
(9.23)
where we tacitly assumed that the distributions are centered with c 1 = 0. For the
limiting distribution in real space we find
r (y) =
1
2π
∞
−∞
e
−iky e
−c 2 k
2 /2
=
1
√
2π c 2
e
−y
2 /2c 2 ,
(9.24)
which is a Gaussian with standard deviation given by the second cumulant of the
underlying distribution function D(x). Note that there are very few prerequisites on
the distribution function, only that it can be expressed as a cumulant expansion, which
requires the existence of at least the first and second cumulant. In fact, this covers
all distributions with existing second moment or equivalently second cumulant.
But this is the essence of the central limit theorem, which states that sums of
random numbers drawn from a distribution that has a second moment will converge
towards a Gaussian in case sufficiently many random numbers are drawn and the
spreading of the distribution is counteracted by rescaling, which is just what we did.
We can also describe this behavior by considering an arbitrary distribution function
with existing second moment that is repeatedly convoluted with itself and rescaled.
If that procedure is iterated sufficiently often we obtain a Gaussian distribution function. Once we arrive at a Gaussian, convoluting with itself and rescaling reproduces
the same Gaussian. Basically we map distributions onto other distributions and eventually get stuck at some general distribution, which is a fixed point of the mapping
rule. Note that this is a very pedestrian description of the renormalization group.
This discussion, however, does not cover power laws—they do not have a first or
a second moment. In particular random numbers drawn from a Cauchy or Lorentz
distribution in (9.4) will not converge to a Gaussian distribution. In the next section
we will figure out what they do.
9.8 Lévy-Stable Distributions
It turns out that the process of repeatedly convoluting and rescaling the fat-tailed
power law distributions discussed in Sect. 9.5 converge [13] to so-called Lévy-stable
distributions. They obey the fixed-point condition
L μ (y; N )dy = L μ (x; 1)dx with y = a N x + b N ,
(9.25)
where L μ (x; N ) is L μ (x; 1) convoluted N times with itself but retains the same
form as the original distribution. There are very few explicitly known representations of Lévy-stable distributions in real space known, but the Fourier-transforms of
symmetric distributions with L μ (x; 1) = L μ (−x; 1) have the form
9 Bubbles, Crashes, Fat Tails and Lévy-Stable Distributions
lim
N →∞
˜
Nr (k) = e
−c 2 k
2 /2
,
(9.23)
where we tacitly assumed that the distributions are centered with c 1 = 0. For the
limiting distribution in real space we find
r (y) =
1
2π
∞
−∞
e
−iky e
−c 2 k
2 /2
=
1
√
2π c 2
e
−y
2 /2c 2 ,
(9.24)
which is a Gaussian with standard deviation given by the second cumulant of the
underlying distribution function D(x). Note that there are very few prerequisites on
the distribution function, only that it can be expressed as a cumulant expansion, which
requires the existence of at least the first and second cumulant. In fact, this covers
all distributions with existing second moment or equivalently second cumulant.
But this is the essence of the central limit theorem, which states that sums of
random numbers drawn from a distribution that has a second moment will converge
towards a Gaussian in case sufficiently many random numbers are drawn and the
spreading of the distribution is counteracted by rescaling, which is just what we did.
We can also describe this behavior by considering an arbitrary distribution function
with existing second moment that is repeatedly convoluted with itself and rescaled.
If that procedure is iterated sufficiently often we obtain a Gaussian distribution function. Once we arrive at a Gaussian, convoluting with itself and rescaling reproduces
the same Gaussian. Basically we map distributions onto other distributions and eventually get stuck at some general distribution, which is a fixed point of the mapping
rule. Note that this is a very pedestrian description of the renormalization group.
This discussion, however, does not cover power laws—they do not have a first or
a second moment. In particular random numbers drawn from a Cauchy or Lorentz
distribution in (9.4) will not converge to a Gaussian distribution. In the next section
we will figure out what they do.
9.8 Lévy-Stable Distributions
It turns out that the process of repeatedly convoluting and rescaling the fat-tailed
power law distributions discussed in Sect. 9.5 converge [13] to so-called Lévy-stable
distributions. They obey the fixed-point condition
L μ (y; N )dy = L μ (x; 1)dx with y = a N x + b N ,
(9.25)
where L μ (x; N ) is L μ (x; 1) convoluted N times with itself but retains the same
form as the original distribution. There are very few explicitly known representations of Lévy-stable distributions in real space known, but the Fourier-transforms of
symmetric distributions with L μ (x; 1) = L μ (−x; 1) have the form