9.9 Extreme-Value Theory
135
T (x) = 1 − C(x) =
∞
x
D(x
)dx
=
x
ˆ
x
−μ ,
(9.39)
such that the expression for the expectation x n value becomes
x n = nμ
∞
ˆ
x
ˆ
x
μ
x μ
1 −
x
ˆ
x
−μ
n−1
dx .
(9.40)
Substituting y = 1 − (x/ ˆ
x)
−μ changes the integration boundaries and we arrive at
x n = n ˆ
x
1
0
(1 − y)
1/μ y
n−1 dy = n ˆ
x B
n,
μ − 1
μ
,
(9.41)
where B(z, w) is the beta function we already encountered in Sect. 7.7. Now we need
to find the asymptotic dependence of the beta function for large arguments n. We
therefore express the beta function through the Gamma function and use Stirling’s
formula (6.1.37 in [15]). To simplify the writing we set z = (μ − 1)/μ) and get
B(n, z) =
(n))(z)
(n + z)
≈ (z)
e
−n n
n
e −(n+z) (n + z) n+z ≈ (z)
n
n
(n + z) n+z ≈ (z)n
−z
,
(9.42)
where we used several times that z n. For the asymptotic dependence of x n
we therefore find
x n ∼ n
1+(μ−1)/μ
∼ n
1/μ
.
(9.43)
The maximum value that we can expect to obtain after n samples thus grows as n
1/μ .
Note that this grows much more rapidly than the logarithmically growing expected
values for the exponential and Gaussian distributions from the first two examples.
In Fig. 9.11 we show the the dependence of the expected maximum value for
several distributions with μ = 0.5, 1, 1.5 and a Gaussian for comparison. Note that,
for Gaussian distributions, large values (high on the vertical axis) are rarely expected,
despite waiting for many samples, whereas those distributions with fat tails, and this
includes the Cauchy distribution with μ = 1, have a much higher expectancy of large
values, even after a moderate number of samples.
It is instructive to investigate what fraction of an asset x with distribution D(x)
is actually distributed in a certain part of the distribution. This is what V. Pareto did
when he analyzed what fraction of the wealth of the Italian population is held by
what fraction of the population. Imagine that someone with a wealth of 100 Billion
US$ holds about 0.6% of the US gross domestic product (GDP) in 2015. The latter
is approximately 18 000 Billion US$.
135
T (x) = 1 − C(x) =
∞
x
D(x
)dx
=
x
ˆ
x
−μ ,
(9.39)
such that the expression for the expectation x n value becomes
x n = nμ
∞
ˆ
x
ˆ
x
μ
x μ
1 −
x
ˆ
x
−μ
n−1
dx .
(9.40)
Substituting y = 1 − (x/ ˆ
x)
−μ changes the integration boundaries and we arrive at
x n = n ˆ
x
1
0
(1 − y)
1/μ y
n−1 dy = n ˆ
x B
n,
μ − 1
μ
,
(9.41)
where B(z, w) is the beta function we already encountered in Sect. 7.7. Now we need
to find the asymptotic dependence of the beta function for large arguments n. We
therefore express the beta function through the Gamma function and use Stirling’s
formula (6.1.37 in [15]). To simplify the writing we set z = (μ − 1)/μ) and get
B(n, z) =
(n))(z)
(n + z)
≈ (z)
e
−n n
n
e −(n+z) (n + z) n+z ≈ (z)
n
n
(n + z) n+z ≈ (z)n
−z
,
(9.42)
where we used several times that z n. For the asymptotic dependence of x n
we therefore find
x n ∼ n
1+(μ−1)/μ
∼ n
1/μ
.
(9.43)
The maximum value that we can expect to obtain after n samples thus grows as n
1/μ .
Note that this grows much more rapidly than the logarithmically growing expected
values for the exponential and Gaussian distributions from the first two examples.
In Fig. 9.11 we show the the dependence of the expected maximum value for
several distributions with μ = 0.5, 1, 1.5 and a Gaussian for comparison. Note that,
for Gaussian distributions, large values (high on the vertical axis) are rarely expected,
despite waiting for many samples, whereas those distributions with fat tails, and this
includes the Cauchy distribution with μ = 1, have a much higher expectancy of large
values, even after a moderate number of samples.
It is instructive to investigate what fraction of an asset x with distribution D(x)
is actually distributed in a certain part of the distribution. This is what V. Pareto did
when he analyzed what fraction of the wealth of the Italian population is held by
what fraction of the population. Imagine that someone with a wealth of 100 Billion
US$ holds about 0.6% of the US gross domestic product (GDP) in 2015. The latter
is approximately 18 000 Billion US$.