128
9 Bubbles, Crashes, Fat Tails and Lévy-Stable Distributions
˜
G(k, σ ) =
∞
−∞
e
ikx G(x, σ )dx = e
−k
2 σ
2 /2
.
(9.13)
From (9.12) we know ˜
N (k)
˜
N (k) = e
−Nk
2 σ
2 /2
= ˜
G(k,
√
N σ ) ,
(9.14)
which has the same form as ˜
G(k, σ ), but has σ replaced by
√
N σ. Its inverse Fourier
transform will yield N (x) = G(x,
√
N σ ).
Now we add a twist: since we know that the standard deviation grows with
√
N ,
rather than calculating the distribution function of x = x 1 + · · · + x n , we calculate
the distribution function of y = (x 1 + · · · + x n )/
√
N , which means that each random
variable x i is divided by
√
N and that means that in the definition of the Gaussian
in (9.10) the standard deviation σ is replaced by σ
√
N and retracing the following
steps we find that the factor
√
N vanishes from (9.14). In other words, using the
scaled variables y = x/
√
N instead of x the sum of random number sampled from
i.i.d Gaussians reproduces the original Gaussian.
Note that we could have used any other distribution D(x), instead of a Gaussian.
But before proceeding further we have to briefly introduce the generating function of
a probability-distribution functions D(x) and its cumulant expansion. The Fourier
transform ˜
D(k) =
e
ikx D(x)dx of D(x) is called the generating function, because
the expansion coefficients of its Taylor series are related to the moments x
m
=
x
m D(x)dx through
˜
D(k) =
∞
m=0
x
m
m!
(ik)
m with x
m
= (−i)
m d
m ˜
D(k)
dk m
k=0
.
(9.15)
Equivalently, we introduce the Taylor-expansion coefficients c m of the logarithm of
the generating function log
˜
D(k)
. The c m are called the cumulants and they allow
us to write ˜
D(k) as
˜
D(k) = exp
∞
m=0
c m
m!
(ik)
m
with c m = (−i)
m
d
m log
˜
D(k)
dk m
k=0
. (9.16)
The cumulants have the very convenient property that convoluting two distribution
functions corresponds to adding their respective cumulants. We already saw this feature for the Gaussians, whose variances are added when convoluting two Gaussians.
Let us now return to calculating the sum of random numbers, drawn from the
distribution function D(x) with Fourier-transform ˜
D(k). The generating function
˜
N (k) of the sum of N samples drawn from D(x) is given by the N th power of
˜
D(k)
9 Bubbles, Crashes, Fat Tails and Lévy-Stable Distributions
˜
G(k, σ ) =
∞
−∞
e
ikx G(x, σ )dx = e
−k
2 σ
2 /2
.
(9.13)
From (9.12) we know ˜
N (k)
˜
N (k) = e
−Nk
2 σ
2 /2
= ˜
G(k,
√
N σ ) ,
(9.14)
which has the same form as ˜
G(k, σ ), but has σ replaced by
√
N σ. Its inverse Fourier
transform will yield N (x) = G(x,
√
N σ ).
Now we add a twist: since we know that the standard deviation grows with
√
N ,
rather than calculating the distribution function of x = x 1 + · · · + x n , we calculate
the distribution function of y = (x 1 + · · · + x n )/
√
N , which means that each random
variable x i is divided by
√
N and that means that in the definition of the Gaussian
in (9.10) the standard deviation σ is replaced by σ
√
N and retracing the following
steps we find that the factor
√
N vanishes from (9.14). In other words, using the
scaled variables y = x/
√
N instead of x the sum of random number sampled from
i.i.d Gaussians reproduces the original Gaussian.
Note that we could have used any other distribution D(x), instead of a Gaussian.
But before proceeding further we have to briefly introduce the generating function of
a probability-distribution functions D(x) and its cumulant expansion. The Fourier
transform ˜
D(k) =
e
ikx D(x)dx of D(x) is called the generating function, because
the expansion coefficients of its Taylor series are related to the moments x
m
=
x
m D(x)dx through
˜
D(k) =
∞
m=0
x
m
m!
(ik)
m with x
m
= (−i)
m d
m ˜
D(k)
dk m
k=0
.
(9.15)
Equivalently, we introduce the Taylor-expansion coefficients c m of the logarithm of
the generating function log
˜
D(k)
. The c m are called the cumulants and they allow
us to write ˜
D(k) as
˜
D(k) = exp
∞
m=0
c m
m!
(ik)
m
with c m = (−i)
m
d
m log
˜
D(k)
dk m
k=0
. (9.16)
The cumulants have the very convenient property that convoluting two distribution
functions corresponds to adding their respective cumulants. We already saw this feature for the Gaussians, whose variances are added when convoluting two Gaussians.
Let us now return to calculating the sum of random numbers, drawn from the
distribution function D(x) with Fourier-transform ˜
D(k). The generating function
˜
N (k) of the sum of N samples drawn from D(x) is given by the N th power of
˜
D(k)