Luận văn toán học EXTREME VALUES AND PROBABILITY DISTRIBUTION FUNCTONS ON FINITE DIMENSIONAL SPACES

VIETNAM NATIONAL UNIVERSITY
UNIVERSITY OF SCIENCE
FACULTY OF MATHEMATICS, MECHANICS AND INFORMATICS
Do Dai Chi
EXTREME VALUES AND
PROBABILITY DISTRIBUTION FUNCTONS
ON FINITE DIMENSIONAL SPACES
Undergraduate Thesis
Advanced Undergraduate Program in Mathematics
Hanoi - 2012
VIETNAM NATIONAL UNIVERSITY
UNIVERSITY OF SCIENCE
FACULTY OF MATHEMATICS, MECHANICS AND INFORMATICS
Do Dai Chi
EXTREME VALUES AND
PROBABILITY DISTRIBUTION FUNCTONS
ON FINITE DIMENSIONAL SPACES
Undergraduate Thesis
Advanced Undergraduate Program in Mathematics
Thesis advisor: Assoc.Prof.Dr. Ho Dang Phuc
Hanoi - 2012
Acknowledgments
It would not have been possible to write this undergraduate thesis without the help,
and support, of the kind people around me, to only some of whom it is possible to
give particular mention here.
This thesis would not have been possible without the help, support and patience of
my advisor, Assoc.Prof.Dr. Ho Dang Phuc, not to mention his advice and unsur-
passed knowledge of probability and statistic. The advice, support and friendship
of his have been invaluable on both an academic and a personal level, for which I
am extremely grateful.
I would like to show my gratitude to my teachers at Faculty of Mathematics, Me-

chanics and Informatics, University of Sciences, VietNam National University who
equip me with important mathematics knowledge during first four years at the uni-
versity.
I would like to thank my parents for their personal support and great patience at all
times. My parents have given me their unequivocal support throughout, as always,
for which my mere expression of thanks likewise does not suffice.
Last, but by no means least, I thank my friends in K53-Advanced Math for their
support and encouragement throughout.
i
List of abbreviations and symbols
Here is a glossary of miscellaneous symbols, in case you need a reference guide.
∼ f (x) ∼ g(x) as x → x
0
means that lim
x→x
0
f (x)
g(x)
= 1
d
→ X
n
d
→ X : convergence in distribution.
P
→ X
n
P
→ X convergence in probability.
a.s

→ X
n
a.s
→ X almost surely convergence.
v
→ µ
n
v
→ µ vague convergence.
d
= X
d
= Y: X and Y have the same distribution.
o(1) f (x) = o(g(x)) as x → x
0
means that lim
x→x
0
f (x)
g(x)
= 0.
f
←
The generalized inverse of a monotone function f defined by
f
←
(x) = inf{y : f ( y) ≥ x}.
Λ(x) Gumbel distribution.
Φ
α

(x) Fr
´
echet distribution.
Ψ
α
(x) Weibull distribution.
x
F
x
F
= sup{x ∈ R : F(x) < 1}.
f (x−) f (x−) = lim
y↑x
f (y).
[F > 0] means the set {x : F(x) > 0}.
M
+
(E) The space of nonnegative Radon measures on E.
C (f ) The points at which the function f is continuous.
d.f. Distribution function.
r.v. Random variable.
DOA Domain of attraction.
ii
Contents
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i
List of abbreviations and symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
Chapter 1. Univariate Extreme Value Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1. Limit Probabilities for Maxima . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2. Maximum Domains of Attraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.1. Max-Stable Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3. Extremal Value Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3.1. Extremal Types Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3.2. Generalized Extreme Value Distributions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.4. Domain of Attraction Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.4.1. General Theory of Domains of Attraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.5. Condition for belonging to Extreme Value Domain . . . . . . . . . . . . . . . . . . . . 26
Chapter 2. Multivariate Extreme Value Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.2. Limit Distributions of Multivariate Maxima . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.2.1. Max-infinitely Divisible Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.2.2. Characterizing Max-id Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.3. Multivariate Domain of Attraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.3.1. Max-stability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.4. Basic Properties of Multivariate Extreme Value Distributions . . . . . . . . . . 41
2.5. Standardization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
iii
Chapter A. Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
A.1. Modes of Convergence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
A.2. Inverses of Monotone Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
A.3. Some Convergence Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
iv
Introduction
Extreme value theory developed from an interest in studying the behavior of the
maximum or minimum (extremes) of independent and identically distributed ran-
dom variables. Historically, the study of extremes can be dated back to Nicholas
Bernoulli who studied the mean largest distance from the origin to n points scat-

tered randomly on a straight line of some fixed length (Gumbel.1958 [15]). Extreme
value theory provides important applications in finance, risk management, telecom-
munication, environmental and pollution studies and other fields. In this thesis, we
study the probabilistic approach to extreme value theory. The thesis is divided into
the two chapters, namely,
Chapter 1: Univariate Extreme Value Theory.
Chapter 2: Multivariate Extreme Value Theory.
Chapter 1 introduces the basic concepts related to Univariate Extreme Value Theory.
This chapter concerns with the limit problem of determining the possible limits of
sample extremes and the domain of attraction problem.
Chapter 2 provides basic results in Multivariate Extreme Value Theory. We deal
with the probabilistic aspects of multivariate extreme value theory by including the
possible limits and their domain of attraction.
The main materials of the thesis were taken from the books by M. R. Leadbetter, G.
Lindgren, and H. Rootz
´
en [16], Resnick [18], Embrechts [12] and de Haan and Ana
Ferreira [11]. We have also borrowed extensively from lecture notes of Bikramjit
Dass [9].
v
CHAPTER 1
Univariate Extreme Value Theory
This chapter is primarily concerned with the central result of classical extreme value
theory, the Extremal Types Theorem, which specifies the possible forms for the limit-
ing distribution of maxima in sequences of independent and identically distributed
(i.i.d.) random variables(r.v.s). In the derivation, the possible limiting distributions
are identified with a class having a certain stability property, the so-called max-stable
distributions. It is further shown that this class consists precisely of the three families
known (loosely) as the three extreme value distributions.
1.1. Introduction

The asymptotic theory of sample extremes has been developed in parallel with the
central limit theory, and in fact the two theories bear some resemblance.
Let X
1
, X
2
, . . . , X
n
be i.i.d. random variables. The central limit theory is concerned
with the limit behavior of the partial sums S
n
= X
1
+ X
2
+ ··· + X
n
as n → ∞,
whereas the theory of sample extremes is concerned with the limit behavior of the
sample extremes max(X
1
, X
2
, . . . , X
n
) or min(X
1
, . . . , X
n
) as n → ∞.

We consider some basic theory for sums of independent random variables. This in-
cludes classical results such as the strong law of large numbers and the Central Limit
Theorem. Throughout this chapter X
1
, X
2
, . . . is a sequence of i.i.d. non-degenerate
real random variables defined on a probability space (Ω, F, P) with common distri-
butions function(d.f.) F. We consider the partial sums
S
n
= X
1
+ ···+ X
n
, n ≥ 1.
and of the sample means
X
n
= n
−1
S
n
=
S
n
n
, n ≥ 1.
Let X be random variable and denote the expectation, the variance of X by E(X) =
µ, Var(X) = σ

2
. Firstly, we assume that E(X) = µ < ∞. From the strong law of
1
large numbers, we get
X
n
= n
−1
S
n
a.s
→ µ.
With the additional assumption of Var(X
1
) = σ
2
< ∞, we get the Central Limit
Theorem:
S
n
−nµ
√
nσ
d
→ Z, Z ∼ N(0, 1).
Hence for large n, we can approximate
P(S
n
≤ x) ≈ P


Z ≤
x −nµ
√
nσ

.
Taking an alternative approach, we can deal with the problem of finding possi-
ble limit distributions for (say) sample maxima of independent and identically dis-
tributed random variables.
1.1.1. Limit Probabilities for Maxima
Whereas in above, we introduced ideas on partial sums, in this section we investi-
gate the fluctuations of the sample maxima:
M
n
=
n

i=1
X
i
= max(X
1
, . . . , X
n
), n ≥ 1.
Remark 1.1. Corresponding results for minima can easily be obtained from those for
maxima by using the identity
min(X
1
, . . . , X

n
) = −max(−X
1
, . . . , −X
n
).
We shall therefore only briefly discuss minima explicitly in this work, except where
its joint distribution with M
n
is considered.
We have the exact d.f of the maximum M
n
for x ∈ R, n ∈ N,
P(M
n
≤ x) = P(X
1
≤ x, . . . , X
n
≤ x) =
n
∏
i=1
P(X
i
≤ x) = F
n
(x). (1.1)
Extreme events happen ’near’ the upper end of the support of the distribution. We
denote the right endpoint of F by

x
F
= sup{x ∈ R : F(x) < 1}. (1.2)
That is, F(x) < 1 for all x < x
F
and F(x) = 1 for all x > x
F
. We immediately obtain

P(M
n
≤ x) = F
n
(x) → 0, n → ∞ for all x < x
F
P(M
n
≤ x) = F
n
(x) → 1, n → ∞ in the case x
F
< ∞, ∀x > x
F
2
Therefore the limit distribution lim
n→∞
F
n
(x) is degenerate. Thus M
n

P
→ x
F
as n →
∞ where x
F
< ∞. Since the sequence (M
n
) is non decreasing in n, it converges
almost surely(a.s), no matter whether it is finite or infinite and hence we conclude
that
M
n
a.s
→ x
F
, n → ∞.
This result is quite uninformative for our purpose and does not answer the basic
question in our mind. This difficulty is avoided by allowing a linear renormalization
of the variable M
n
:
M
∗
n
=
M
n
−b
n

a
n
,
for sequences of constants {a
n
> 0} and {b
n
} ∈ R.
Definition 1.1. A univariate distribution function F, belong to the maximum do-
main of attraction of a distribution function G if
1. G is non degenerate distribution.
2. There exist real valued sequence a
n
> 0, b
n
∈ R, such that
P

M
n
−b
n
a
n
≤ x

= F
n
(a
n

x + b
n
)
d
→ G(x). (1.3)
Finding the limit distribution G(x) is called the Extremal Limit Problem. Finding the
F(x) that have sequences of constants as described above leading to G(x) is called
the Domain of Attraction Problem.
For large n, we can approximate P(M
n
≤ x) ≈ G(
x−b
n
a
n
). We denote F ∈ D(G). We
often ignore the term ’maximum’ and abbreviate domain of attraction as DOA.
Now we are faced with certain questions:
1. Given any F, does there exist G such that F ∈ D(G) ?
2. Given any F, if G exist, is it unique?
3. Can we characterize the class of all possible limits G according to definition
1.1?
4. Given a limit G, what properties should F have so that F ∈ D(G)?
5. How can we compute a
n
, b
n
?
The goal of the next section is to answer the above questions.
3

1.2. Maximum Domains of Attraction
Let’s consider probabilities of the form
P

M
n
−b
n
a
n
≤ x

which may be rewritten as
P(M
n
≤ u
n
)
where u
n
= u
n
(x) = a
n
x + b
n
. In order to get more insight into the asymptotic
behavior of M
n
we have to investigate the following aspects:

1. Conditions on F, that ensure the existence of the limit of P(M
n
≤ u
n
) for
n → ∞ and appropriate constants u
n
.
2. Possible limit laws for the (centered and normalized) maxima M
n
(comparable
to the Central Limit Theorem).
Example 1.1. Let X be a standard exponential distribution. Then the distribution
function of X is given by
F
X
(x) = 1 −e
−x
, x > 0.
If X
1
, X
2
, . . . are i.i.d. random variables with common distribution function F, then
P(M
n
≤ x + log n) =

1 −e
−x−log n


n
= (1 −
e
−x
n
)
n
→ exp{−e
−x
} =: Λ(x), x ∈ R.
The limit distribution Λ(x) is called the Gumbel distribution. So we obtain that the
Gumbel distribution is a possible limit distribution according to Definition 1.1.
The following Theorem provides a partial answer to question 1.
Theorem 1.1 (Poisson approximation). For given τ ∈ [0, ∞] and a sequence {u
n
} of real
numbers, the following two conditions are equivalent:
nF(u
n
) → τ as n → ∞, (1.4)
P(M
n
≤ u
n
) → e
−τ
as n → ∞. (1.5)
where
F = 1 − F.

4
Proof. Suppose first that 0 ≤ τ < ∞. If (1.4) holds, then
P(M
n
≤ u
n
) = F
n
(u
n
) = (1 − F(u
n
))
n
=

1 −
τ
n
+ o(
1
n
)

n
,
so that (1.5) follows at once.
Conversely, if (1.5) holds (0 ≤ τ < ∞), we must have
F(u
n

) = 1 − F(u
n
) → 0
(otherwise, F(u
n
k
) would be bounded away from 0 for some subsequence (n
k
) and
P(M
n
k
≤ u
n
k
) = (1 − F(u
n
k
))
n
k
would imply P(M
n
k
≤ u
n
k
) → 0). By taking loga-
rithms in (1.5), we have
−n ln(1 −F(u

n
)) → τ.
Since −ln(1 −x) ∼ x for x → 0, this implies nF(u
n
) = τ + o(1) that giving (1.4).
If τ = ∞ and (1.4) holds but (1.5) does not, there must be a subsequence (n
k
) such
that
P(M
n
k
≤ u
n
k
) → exp{−τ

},
as k → ∞ for some τ

< ∞. But then (1.5) implies (1.4), so that n
k
F(u
n
k
) → τ

< ∞,
contradicting (1.4) with τ = ∞.
Similarly, (1.5) implies (1.4) for τ = ∞.

Example 1.2. We consider the distribution function F
F(x) =

1 −e
1/x
for all x < 0
1 for all x ≥ 0
By theorem 1.1, if {u
n
} such that
ne
1/u
n
n→∞
→ τ > 0,
it follows that
P(M
n
≤ u
n
)
n→∞
→ e
−τ
.
By writing τ = e
−x
(−∞ < x < ∞) and taking u
n
= (log τ −log n)

−1
, it follows that
P(M
n
≤ −(log n + x)
−1
) → exp(−e
−x
),
from which it is readily checked that
P{(log n)
2
[M
n
+
1
log n
] ≤ x + o(1)} → exp(−e
−x
),
giving Gumbel distribution with
a
n
= (log n)
−2
, b
n
= −(log n)
−1
.

5
We denote f (x−) = lim
y↑x
f (y) and p(x) = F(x) − F(x−).
Theorem 1.2. Let F be a d.f. with right endpoint x
F
≤ ∞ and let τ ∈ (0, ∞). There exists
a sequence (u
n
) satisfying nF(u
n
) → τ if and only if
lim
x↑x
F
F(x)
F(x−)
= 1 (1.6)
or equivalently, if and only if
lim
x↑x
F
p(x)
F(x−)
= 0 (1.7)
Hence, by Theorem 1.1, if 0 < ρ < 1, there is a sequence {u
n
} such that P(M
n
≤

u
n
) → ρ if and only if (1.6) (or (1.7)) holds. For ρ = 0 or 1, such a sequence may
always be found.
Proof. We suppose that (1.4) holds for some 0 < τ < ∞ but that, say (1.7), does not.
Then there exists  and a sequence {x
n
} such that x
n
→ x
F
and
p(x
n
) ≥ 2(F(x
n
−)). (1.8)
Now choose a sequence of integers {n
j
} so that 1 −
τ
n
j
is ”close” to the midpoint of
the jump of F at x
j
, i.e. such that
1 −
τ
n

j
≤
F(x
j
−) + F(x
j
)
2
≤ 1 −
τ
n
j
+ 1
.
Clearly we have either
(i) u
n
j
< x
j
for infinitely many values of j, or
(ii) u
n
j
≥ x
j
for infinitely many j-values.
If alternative (i) holds, then for such j,
n
j

F(u
n
j
) ≥ n
j
F(x
j
−). (1.9)
Now, clearly
n
j
F(x
j
−) = τ + n
j

1 −
τ
n
j

−
F(x
j
) + F(x
j
−)
2
+
p(x

j
)
2

≥ τ +
n
j
p(x
j
)
2
−n
j

τ
n
j
−
τ
n
j
+ 1

≥ τ + n
j
F(x
j
−) −
τ
n

j
+ 1
6
by (1.8) so that
(1 −)n
j
F(x
j
−) ≥ τ −
τ
n
j
+ 1
.
Since clearly n
j
→ ∞, it follows that (since τ ∈ (0, ∞) by assumption)
lim
j→∞
sup n
j
F(x
j
−) ≥ τ,
and hence by (1.2),
lim
j→∞
sup n
j
F(u

n
j
) ≥ τ,
which contradicts (1.4). The calculations in case (ii) (u
n
j
≥ x
j
for infinitely many j)
are very similar, with only the obvious changes.
Conversely, suppose that (1.6) holds and let {u
n
}be any sequence such that F(u
n
−) ≤
1 −
τ
n
≤ F(u
n
) (e.g. u
n
= F
−1
(1 −
τ
n
)), from which a simple rearrangement yields
F(u
n

)
F(u
n
−)
τ ≤ nF(u
n
) ≤ τ
from which (1.4) follows since clearly u
n
→ x
F
as n → ∞.
The result applies in particular to discrete distributions with infinite right endpoint.
If the jump heights of the d.f. do not decay sufficiently fast, then a non-degenerate
limit distribution for maxima does not exist.
Example 1.3 (Poisson distribution). Let X be a Poisson r.v.s with expectation λ > 0;
i.e,
P(X = k) = e
−λ
λ
k
k!
, λ > 0, k ∈ N
Then,
F(k)
F(k −1)
= 1 −
F(k) − F(k −1)
F(k −1)
= 1 −

λ
k
k!

∞
∑
r=k
λ
r
r!

−1
= 1 −

1 +
∞
∑
r=k
k!
r!
λ
r−k

−1
.
The latter sum can be estimated as
∞
∑
s=1
λ

s
(k + 1)(k + 2) ···(k + s)
≤
∞
∑
s=1

λ
k

s
=
λ/k
1 −λ/k
, k ≥ s
which tends to 0 as k → ∞, so that
F(k)
F(k−1)
→ 0.
Hence, by virtue of Theorem 1.2 we can see that no non-degenerate distribution can
be limit of normalized maxima taken from a sequence of random variables identi-
cally distributed as X.
7
Example 1.4 (Geometric distribution). We consider the random variable X with ge-
ometric distribution:
P(X = k) = p(1 − p)
k−1
, 0 < p < 1, k ∈ N.
For this distribution, we have
F(k)

F(k −1)
= 1 −(1 − p)
k−1

∞
∑
r=k
(1 − p)
r−1

−1
= 1 − p ∈ (0, 1) .
By the same argument as above, no limit P(M
n
≤ u
n
) → ρ exists except for ρ = 0
or 1, that implies there is no nondegenerate limit distribution for the maxima in the
geometric distribution case.
Example 1.5 (Negative binomial distribution). Let X be the random variable with
P(X = k) = C
k−1
i+k−1
p
i
(1 − p)
k−1
, k ∈ R, 0 < p < 1, i > 0.
Using properties of the binomial coefficients we obtain
F(k)

F(k −1)
= 1 −
F(k) − F(k −1)
F(k −1)
≤ 1 − p ∈ (0, 1),
i.e, no limit P(M
n
≤ u
n
) → ρ exists except for ρ = 0 or 1.
Definition 1.2. Suppose that H : R → R is a non-decreasing function. The general-
ized inverse of H is given by
H
←
(x) = inf{y : H(y) ≥ x}.
Properties of generalized inverse are given in Appendix A.2.
Lemma 1.1. (i) For H as above, if a > 0, b and c are constants, and T(x) = H(ax +
b) −c, then T
←
(y) = a
−1
[H
←
(y + c) − b].
(ii) If F is a non-degenerate d.f., there exist y
1
, y
2
such that F
←

(y
1
) < F
←
(y
2
) are well
defined ( and finite).
Proof. (i) We have
T
←
(y) = inf{x : H(ax + b) −c ≥ y}
= a
−1
[ inf{(ax + b : H(ax + b) ≥ y + c)} − b ]
= a
−1
[H
←
(y + c) − b],
8
as required.
(ii) If F is non-degenerate, there exist x

1
< x

2
such that
0 < F(x


1
) = y
1
< F(x

2
) = y
2
≤ 1.
Clearly x
1
= F
←
(y
1
) and x
2
= F
←
(y
2
) are both well defined. Also F
←
(y
2
) ≥ x

1
and

equality would require F(z) ≥ y
2
for all z > x
1
so that
F(x

1
) = lim
↓0
F(x

1
+ ) = F(x

1
+) ≥ y
2
,
contradicting F(x

1
) = y
1
. Thus F
←
(y
2
) > x


1
≥ x
1
= F
←
(y
1
), as required.
For any function H denote
C (H) = {x ∈ R : H is finite and continuous at x}.
If two r.v.s X and Y have the same distribution, we write
X
d
= Y.
Definition 1.3. Two distribution functions U(x) and V(x) are of the same type if for
some A > 0, B ∈ R
V(x) = U(Ax + B)
for all x.
In terms of random variables, if X has distribution U and Y has distribution V, then
Y
d
=
X − B
A
Example 1.6. Let denote N(0, 1, x) (normal distribution function with mean 0 and
variance 1). Then, it is easy to see that N(µ, σ
2
, x) = N(0, 1,
x−µ
σ

) for σ > 0, µ ∈
R. Then all normalized d.f’s are of the same type called normal type. If X
0,1
has
N(0, 1, x) as its distribution and X
µ,σ
has N(µ, σ
2
, x) as its distribution, then X
µ,σ
d
=
σX
0,1
+ µ.
Now we state the theorem developed by Gnedenko and Khintchin.
Theorem 1.3 (Convergence to types theorem). (a) Suppose U(x) and V(x) are two non-
degenerate distribution functions. Suppose for n ≥ 1, F
n
is a distribution, a
n
≥ 0, b
n
∈
R, α
n
> 0, β
n
∈ R and
F

n
(a
n
x + b
n
)
d
→ U(x), F
n
(α
n
x + β
n
)
d
→ V(x). (1.10)
9
Then as n → ∞
α
n
a
n
→ A > 0,
β
n
−b
n
a
n
→ B ∈ R, (1.11)

and
V(x) = U(Ax + B) (1.12)
An equivalent formulation in term of random variables :
(a’) Let X
n
, n ≥ 1 be random variables with distribution function F
n
and the U, V
are random variables with distribution functions U(x), V(x). If
X
n
−b
n
a
n
d
→ U,
X
n
− β
n
α
n
d
→ V, (1.13)
then (1.11) holds and
V
d
=
U − B

A
. (1.14)
(b) Conversely, if (1.11) holds then either of the two relations in (1.10) (or (1.13) )
implies the other and (1.12) (or (1.14)) holds.
Proof. (b) Suppose (1.11) holds and Y
n
:=
X
n
−b
n
a
n
d
→ U. We must show that
X
n
− β
n
α
n
d
→
U − B
A
.
By Skorohod’s Theorem (see appendix A.3), there exist

Y
n

,

U, n ≥ 1, defined on
([0, 1], B[0, 1], m)(the Lebesgue probability space, m is Lebesgue measure) such that

Y
n
d
= Y
n
,

U
d
= U,

Y
n
a.s
=

U, n ≥ 1.
Put

X
n
:= a
n

Y

n
+ b
n
so

X
n
d
= X
n
. Then
X
n
− β
n
α
n
d
=

X
n
− β
n
α
n
=
a
n
α

n

Y
n
+
b
n
− β
n
α
n
→

U
A
−
B
A
d
=
U − B
A
,
that means
X
n
−β
n
α
n

d
→
U−B
A
.
(a) Using Proposition A.2 (see Appendix A.2) that if G
n
d
→ G, then also G
←
n
d
→ G
←
and the relation in (1.10) can be inverted to give
F
←
n
(y) − b
n
a
n
→ U
←
(y), y ∈ C (U
←
) (1.15)
F
←
n

(y) − β
n
α
n
→ V
←
(y), y ∈ C (V
←
) (1.16)
10
weakly. Since neither U(x) nor V(x) concentrates at one point we can find points
y
1
, y
2
with y
i
∈ C (U
←
) ∩C (V
←
), y
1
< y
2
, for i = 1, 2, satisfying
−∞ < U
←
(y
1

) < U
←
(y
2
) < ∞,
and
−∞ < V
←
(y
1
) < V
←
(y
2
) < ∞.
Therefore from (1.15) we have for i = 1, 2
F
←
n
(y
i
) −b
n
a
n
→ U
←
(y
i
),

F
←
n
(y
i
) − β
n
α
n
→ V
←
(y
i
) (1.17)
and by subtraction
F
←
n
(y
2
) − F
←
n
(y
1
)
a
n
→ U
←

(y
2
) −U
←
(y
1
) > 0,
F
←
n
(y
2
) − F
←
n
(y
1
)
α
n
→ V
←
(y
2
) −V
←
(y
1
) > 0. (1.18)
Divide the first relation in (1.18) by the second to obtain

α
n
a
n
→
U
←
(y
2
) −U
←
(y
1
)
V
←
(y
2
) −V
←
(y
1
)
=: A > 0
Using this and (1.17) we get
F
←
n
(y
1

) −b
n
a
n
→ U
←
(y
1
),
F
←
n
(y
1
) − β
n
a
n
=
F
←
n
(y
1
) − β
n
α
n
.
α

n
a
n
→ V
←
(y
1
)A,
and so subtracting yields
β
n
−b
n
a
n
→ V
←
(y
1
)A −U
←
(y
1
) =: B
This gives (1.11) and (1.12) follows from (b).
Remark 1.2. (a) The answer to question 2 is quite clear from Theorem 1.3. Namely,
F ∈ D(G
1
) and F ∈ D(G
2

) then G
1
and G
2
must be of the same type.
(b) The theorem shows that when
X
n
−b
n
a
n
d
→ U
and U is non-constant, we can always suitable choice of the normalizing con-
stants is
a
n
= F
←
n
(y
2
) − F
←
n
(y
1
),
b

n
= F
←
n
(y
1
).
11
1.2.1. Max-Stable Distributions
In this section we answer the question: What are the possible (non-degenerate) limit
laws for the maxima M
n
when properly normalised and centred?
Definition 1.4. A non-degenerate random distribution function F is max-stable if for
X
1
, X
2
, . . . , X
n
i.i.d. F there exist a
n
> 0, b
n
∈ R such that
M
n
d
= a
n

X
1
+ b
n
.
Example 1.7. If X
1
, X
2
, . . . is a sequence of independent standard exponential Exp(1)
variables, F(x) = 1 −e
−x
for x > 0. Taking a
n
= 1 and b
n
= n, we have
P

M
n
−b
n
a
n
≤ x

= F
n
(x + log n) = [ 1 − e

−(x+log n)
]
n
= [1 −n
−1
e
−x
]
n
→ exp(−e
−x
)
as n → ∞, for each fixed x ∈ R. Hence, with the chosen a
n
and b
n
, the limit distri-
bution of normalized M
n
as n → ∞ is the Gumbel distribution.
Example 1.8. If X
1
, X
2
, . . . is a sequence of independent standard Frechet variables,
F(x) = exp(−
1
x
) for x > 0. For a
n

= n and b
n
= 0.
P

M
n
−b
n
a
n
≤ x

= F
n
(nx) = [exp{−
1
nx
}]
n
= exp(−
n
nx
) = F(x)
as n → ∞, for each fixed x > 0. Hence, the limit in this case - which is an exact result
for all n, because of the max-stability of F - is also the standard Fr´echet distribution.
Example 1.9. If X
1
, X
2

, . . . are a sequence of independent uniform U(0, 1) variables,
F(x) = x for 0 ≤ x ≤ 1. For fixed x < 0, suppose n > −x and let a
n
=
1
n
and b
n
= 1.
Then,
P

M
n
−b
n
a
n
≤ x

= F
n
(n
−1
x + 1)
=

1 +
x
n


n
→ e
x
as n → ∞. Hence, the limit distribution is of Weibull type, that means Weibull
distribution are max-stable.
Theorem 1.4 (Limit property of max-stable laws). The class of all max-stable distribu-
tion functions coincide with the class of all limit laws G for (properly normalised) maxima
of i.i.d. rvs (as given in (1.3)).
12
Proof. 1. If X
1
, X
2
, . . . are i.i.d. G, G is max-stable and M
n
=

n
i=1
X
i
, then
M
n
d
= a
n
X
1

+ b
n
for some a
n
> 0, b
n
∈ R. Then ∀x ∈ R
lim
n→∞
P

M
n
−b
n
a
n
≤ x

= G(x).
2. Now suppose that H is non degenerate and there exist a
n
> 0, b
n
∈ R such that
lim
n→∞
F
n
(a

n
x + b
n
) = H(x).
We claim that H is max-stable. Observe that for all k ∈ N, we have
lim
n→∞
F
nk
(a
n
x + b
n
) = H
k
(x),
lim
n→∞
F
nk
(a
nk
x + b
nk
) = H(x).
By virtue of Convergence to Types Theorem: there exist a
∗
k
> 0, b
∗

k
∈ R such that
lim
n→∞
a
nk
a
n
= a
∗
k
, lim
n→∞
b
nk
−b
n
a
n
= b
∗
k
and
H(x) = H
k
(a
∗
k
x + b
∗

k
).
Therefore if Y
1
, . . . , Y
k
are i.i.d. from H then for all k ∈ N,
Y
1
d
=

n
i=1
Y
i
−b
∗
k
a
∗
k
which implies
n

i=1
Y
i
d
= a

∗
k
Y
1
+ b
∗
k
.
1.3. Extremal Value Distributions
1.3.1. Extremal Types Theorem
The extreme type theorems play a central role of the study of extreme value theory.
In the literature, Fisher and Tippett (1928) were the first who discovered the extreme
type theorems and later these results were proved in complete generality by Gne-
denko (1943). Later Galambos (1987), Leadbetter, Lindgren and Rootzen (1983), and
Resnick (1987) gave excellent reference books on the probabilistic aspect.
13
Theorem 1.5 (Fisher-Tippett(1928), Gnedenko(1943)). Suppose there exist sequence {a
n
>
0} and {b
n
∈ R}, n ≥ 1 such that
M
n
−b
n
a
n
d
→ G

where G is non-degenerate, then G is of one the following three types:
1. Type I, Gumbel : Λ(x) = exp{−e
−x
}, x ∈ R.
2. Type II, Fr´echet : Φ
α
(x) =

0 if x < 0
exp{−x
−α
} if x ≥ 0
for some α > 0.
3. Type III, Weibull : Ψ
α
(x) =

exp{−(−x)
α
} if x < 0
1 if x ≥ 0
for some α > 0
Proof. For t ∈ R Let denote [t] = The greatest integer less than or equal to t. We
proceed in a sequence of steps.
Step(i). From P[
M
n
−b
n
a

n
≤ x] = F
n
(a
n
x + b
n
)
d
→ G(x), we get for any t > 0.
F
[nt]
(a
[nt]
x + b
[nt]
)
d
→ G(x)
and on the other
F
[nt]
(a
n
x + b
n
) = (F
n
(a
n

x + b
n
))
[nt]
n
→ G
t
(x).
Thus G
t
and G are of the same type and the convergence to types theorem applies
the existence of two functions α(t) > 0, β(t) ∈ R, t > 0 such that for all t > 0,
lim
n→∞
a
n
a
[nt]
= α(t), lim
n→∞
b
n
−b
[nt]
a
[nt]
= β(t) (1.19)
and also
G
t

(x) = G(α(t)x + β(t)). (1.20)
Step(ii). We observe that the functionα(t) and β(t) are Lebesgue measurable. For in-
stance, to prove α(·) is measurable, it suffices (since limits of measurable functions
are measurable) to show that the function
t →
a
n
a
[nt]
is measurable for each n. Since an does not depend on t, the previous statement is
true if the function
t → a
[nt]
14
is measurable. Since this function has a countable range {a
j
, j ≥ 1} it suffices to
show
{t > 0 : a
[nt]
= a
j
}
is measurable. But this set equals

k:a
k
=a
j
[

k
n
,
k + 1
n
)
which, being a union of intervals, is certainly a measurable set.
Step(iii). Facts about the Hamel Equation (see [20]). We need to use facts about possi-
ble solutions of functional equations called Hamel’s equation and Cauchy’s equation. If
f (x), x > 0 is finite, measurable and real valued and satisfies the Cauchy equation
f (x + y) = f (x) + f (y), x > 0, y > 0,
then f is necessarily of the form
f (x) = cx, x > 0,
for some c ∈ R. A variant of this is Hamel’s equation. If φ(x), x > 0 is finite,
measurable, real valued and satisfies Hamel’s equation
φ(xy) = φ(x)φ(y), x > 0, y > 0,
then φ is of the form
φ(x) = e
ρ
,
for some ρ ∈ R.
Step(iv). Another useful fact. If F is a non-degenerate distribution function and
F(ax + b) = F(cx + d) ∀x ∈ R,
for some a > 0, c > 0 and b, d are constants, then a = c, and b = d.
Choose y
1
< y
2
and −∞ < x
1

< x
2
< ∞ by (ii) of lemma 1.1 so that x
1
=
F
←
(y
1
), x
2
= F
←
(y
2
). Taking inverses of F(ax + b) and F(cx + d) by (i) of the lemma
1.1, we have
a
−1
(F
←
(y) − b) = c
−1
(F
←
(y) − d)
for all y. Applying this to y
1
and y
2

in turn, we obtain
a
−1
(x
1
−b) = c
−1
(x
1
−d) and a
−1
(x
2
−b) = c
−1
(x
2
−d),
15
from which it follows simply that a = c and b = d.
Step(v). Return to (1.20) and for t > 0, s > 0 we have on the one hand
G
ts
(x) = G(α(ts)x + β(ts))
and on the other
G
ts
(x) = (G
s
(x))

t
= G(α(s)x + β(s))
t
= G(α(t){α(s) x + β(s)}+ β(t) )
= G(α(t)α(s)x + α(t)β(s) + β(t)).
Since G is assumed non degenerate we therefore conclude for t > 0, s > 0
α(ts) = α(t)α(s) (1.21)
β(ts) = α(t)β(s) + β(t) = α(s)β(t) + β(s) (1.22)
the last step following by symmetry. We recognize (1.21) as the famous Hamel func-
tional equation. The only finite measurable, nonnegative solution is of the following
form
α(t) = t
−θ
, θ ∈ R
Step(vi). We will show that
β(t) =

c log t If θ = 0
c(1 −t
θ
) If θ = 0
for some c ∈ R.
If θ = 0, then α(t) = 1 and β(t) satisfies
β(ts) = β(t)β(s).
So exp{β(·)} satisfies the Hamel equation which implies that
exp{β(t)} = t
c
for some c ∈ R and thus β(t) = c log t.
If θ = 0, then
β(ts) = α(t)β(s) + β(t) = α(s)β(t) + β(s).

Fix s
0
= 1 and we get
α(t)β(s
0
) + β(t) = α(s
0
)β(t) + β(s
0
),
16
and solving for β(t) we get
β(t)(1 −α( s
0
)) = β(s
0
)(1 −α(t)).
Note that 1 −α(s
0
) = 0. Thus we conclude
β(t) =
β(s
0
)
1 −α(s
0
)
(1 −α(t)) =: c(1 −t
θ
).

Step(vii). We conclude that
G
t
(x) =

G(x + c log t) If θ = 0 (a)
G(t
θ
x + c(1 −t
θ
)) If θ = 0 (b)
Now we show that θ = 0 corresponds to a limit distribution of type Λ(x), that the
case θ > 0 corresponds to a limit distribution of type Φ
α
and that θ < 0 corresponds
to Ψ
α
.
Consider the case θ = 0. Examine the equation in (a): For fixed x, the function G
t
(x)
is non-increasing in t. So c < 0, since otherwise the right side of (a) would not be
decreasing. If x
0
∈ R such that G(x
0
) = 1, then
1 = G
t
(x

0
) = G(x
0
+ c log t), ∀t > 0,
which implies
G(y) = 1, y ∈ R
and this contradicts G non-degenerate. If x
0
∈ R such that G(x
0
) = 0, then
0 = G
t
(x
0
) = G(x
0
+ c log t), ∀t > 0,
which implies
G(x) = 0, ∀x ∈ R,
again giving a contradiction. We conclude 0 < G(y) < 1, for all y ∈ R.
In (a), set x = 0 and set G(0) = e
−κ
Then
e
−tκ
= G(c log t).
Set y = c log t, and we get
G(y) = exp{−κe
y

c
} = exp{−e
−(
y
|c|
−log κ)
}
which is the type of Λ(x).
We consider the case θ > 0. Examine the equation in (b):
G
t
(x) = G(t
θ
x + c(1 −t
θ
))
= G(t
θ
(x −c) + c)
17
i.e, changing variables
G
t
(x + c) = G(t
θ
x + c).
Set H(x) = G(x + c). Then G and H are of the same type so it suffices to solve for
H. The function H satisfies
H
t

(x) = H( t
θ
x) (1.23)
and H is non-degenerate. Set x = 0 and we get from (1.23)
t log H(0) = log H(0)
for t > 0. So either log H(0) = 0 or −∞; i.e, either H(0) = 0 or 1. However,
H(0) = 1 is impossible since it would imply the existence of x < 0 such that the
left side of (1.23) is decreasing in t while the right side of (1.23) is increasing in t.
Therefore we conclude H(0) = 0. Again from (1.23) we obtain
H
t
(1) = H(t
θ
)
if H(1) = 0, then H ≡ 0 and if H(1) = 1 then H ≡ 1, both statements contradicting
H non-degenerate. Therefore H(1) ∈ (0, 1). Set α = θ
−1
, H(1) = exp{−ρ
−α
}, u = t
θ
so that u
−α
= t. From (1.23) with x = 1 we get for u > 0
H(u) = exp{−ρ
−α
t} = exp{−(ρu)
−α
}
= Ψ

α
(ρu).
The other cases and θ < 0 are handled similarly.
In words, The extreme type theorems say that for a sequence of i.i.d. random vari-
ables with suitable normalizing constants, the limiting distribution of maximum
statistics, if it exists, follows one of three types of extreme value distributions that
labeled I, II and III. Collectively, these three classes of distribution are termed the
extreme value distributions, with types I, II and III widely known as the Gumbel,
Fr´echet and Weibull families respectively. Each family has a location and scale pa-
rameter, band a respectively; additionally, the Fr
´
echet and Weibull families have a
shape parameter α.
Remark 1.3. (a) Though, for modelling purposes the types of Λ, Φ
α
and Ψ
α
are very
different, from a mathematical point of view they are closely linked. Indeed, one
immediately verifes the following properties. Suppose X > 0, then
X ∼ Ψ
α
⇔ −
1
X
∼ Ψ
α
⇔ log X
α
∼ Λ

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