we get
t→∞lim
f(tx) f(t) −1
A(t) = lim
t→∞
D2tρ(xρ−1)
A(t) (D1+D2tρ+D3tρ+ρ0)+ D3tρ+ρ0(xρ+ρ0 −1) A(t) (D1+D2tρ+D3tρ+ρ0)
= xρ−1 ρ .
Since limt→∞A(t) = 0 and |A| ∈RVρ, f ∈2RV0,ρ follows. Now, setting A∗(t) := (ρ+ρ0)D−12 D3tρ0/ρ
yields
t→∞lim 1 A∗(t)
f(tx) f(t) −1
A(t) − xρ−1 ρ
!
= lim
t→∞
D2−1D3tρ0(xρ+ρ0 −1)
ρ ˜a(t) = xρ+ρ0 −1 ρ+ρ0 for any x >0. Note that the resulting limit corresponds to the general form of the limit function H, stated in (2.34), for c1 = 0 and c2 = 1. Since limt→∞A∗(t) = 0 and |A∗(t)| ∈RVρ0, f ∈3RV0,ρ,ρ0 with ρ <0 and ρ0 <0 follows.
Note that by setting g(x) := D0xαf(x) with D0 > 0 and α ∈ R we obtain g ∈3RVα,ρ,ρ0.
2.5 Extended regular variation of second-order
In this section we turn our attention towards classical results established by de Haan and Stadtm¨uller (1996) and consider the convergence rate of extended regularly varying functions. We closely follow the arguments in the Appendix of de Haan and Ferreira (2006). Therefore, we assume thatf ∈ ERVα and there exists a positive function ˜a with limt→∞˜a(t) = 0 such that for any x >0
t→∞lim
f(tx)−f(t)
a(t) −xαα−1
˜
a(t) :=H(x), (2.38)
with some limiting function H 6= 0 which is not a multiple of (xα−1)/α. The reason for the last assumption is as follows. AssumeH(x) :=c(xα−1)/α, then we have for any x >0
t→∞lim
f(tx)−f(t)
a(t) −(1 +c˜a(t))xαα−1
˜
a(t) = 0.
This implies
t→∞lim 1
˜ a(t)
f(tx)−f(t)
a(t)(1 +c˜a(t)) − xα−1 α
= 0.
Hence the limiting relation (2.38) remains rather uninformative if H is a multiple of (xα−1)/α. Possible limits in (2.38) were derived by de Haan and Stadtm¨uller (1996).
Proposition 2.3 (Theorem B.3.1, de Haan and Ferreira (2006)).
Suppose that for some measurable function f and positive functions a and a˜ the limit (2.38) exists for all x >0, where the limit function H is not a multiple of (xα −1)/α. Then, there exist real constants c1, c2 and a parameter ρ ≤0 such that for all x >0,
H(x) =c1 Z x
1
sα−1 Z s
1
uρ−1duds+c2 Z x
1
sα+ρ−1ds, (2.39)
Moreover, for x >0, the auxiliary functions a and ˜a satisfy
t→∞lim
a(tx) a(t) −xα
˜
a(t) =c1xαxρ−1
ρ (2.40)
and
t→∞lim
˜ a(tx)
˜
a(t) =xρ. (2.41)
Note, that if ρ = 0, then c1 6= 0 in order to ensure that H is not a multiple of (xα−1)/α.
The results of de Haan and Stadtm¨uller (1996) are used as motivation for the following definition.
Definition 2.5 (Second-order extended regular variation).
We say f is extended regularly varying of second-order (Notation: f ∈2ERVα,ρ) if for some positive function a and some positive or negative function a˜ with limt→∞˜a(t) = 0 such that f satisfies (2.38) with a limiting function H given by (2.39). The parameter ρ≤0 is called second-order parameter.
Some relations between 2RV and 2ERV, as well as between the corresponding parameters and auxiliary functions in the context of extreme value theory are established in Fraga Alves et al. (2007). Also refer to de Haan and Ferreira (2006). In the following we address this issue and establish relations between the classes 2RV, 3RV and 2ERV. Therefore, recall that considering some measurable, eventually positive functionf :R+→R we have for α >0 the equivalence
f ∈RVα ⇐⇒ f ∈ERVα α >0, as well as
f ∈2RV0,α⇐⇒ f ∈ERVα α ≤0
forα ≤0, provided the auxiliary functionA of f ∈2RV0,α is eventually positive.
Similar relations exists between the three classes 2ERV, 2RV and 3RV.
Theorem 2.12. Letf be measurable, eventually positive function. Then forα >0 we have
f ∈2ERVα,ρ ⇐⇒ f ∈2RVα,ρ. For α ≤0, we have
f ∈2ERVα,ρ ⇐⇒ f ∈3RV0,α,ρ,
provided the auxiliary function A of f ∈3RV0,α,ρ is eventually positive.
Proof. Let f ∈2ERVα,ρ forα >0 and ρ≤0 with some auxiliary function a and
˜
a such that a ∈2RVα,ρ and ˜a ∈ RVρ. Moreover, we know that f ∈ERVα with auxiliary function a. Due to Theorem 2.7 we deduce f ∈ RVα and a possible choice for the auxiliary functiona is given by a(t) =αf(t). Further, due to (2.40) we obtain
t→∞lim
a(tx) a(t) −xα
˜
a(t) = lim
t→∞
f(tx) f(t) −xα
˜
a(t) =c1xαxρ−1
ρ for any x >0.
Thusf ∈2RVα,ρ with auxiliary function A(t) = c−11 a(t) satisfying˜ |A| ∈RVρ. Conversely, letf ∈2RVα,ρ with auxiliary functionA, then for anyx >0, we have
t→∞lim
1 α
f(tx) f(t) −xα
A(t) = lim
t→∞
f(tx)−f(t)
αf(t) − xαα−1
A(t) =xαxρ−1 αρ .
Since
To prove the second assertion of the theorem, let f ∈ 2ERVα,ρ with suitably auxiliary functionsa and ˜a such that
t→∞lim positive sincea is positive and f is eventually positive by assumption. Further, we have for any x >0
In turn, suppose f ∈3RV0,α,ρ with suitable auxiliary functions A and A∗, such that for any x >0 Note that since A and f are both eventually positive, there exists some positive functiona given by a(t) =A(t)f(t).
We conclude this chapter stating uniform inequalities for extended regularly varying functions of second-order.
Proposition 2.4 (de Haan and Ferreira (2006), Theorem B.3.10).
Let f be a measurable function. Suppose for some positive function a, some positive or negative function ˜a with limt→∞˜a(t) = 0 and parameters α ∈R, ρ≤0
for all x > 0, i.e. f ∈2ERVα,ρ. Then, for all ε, δ > 0 there exists t0 = t0(ε, δ)
where for an integrable function h,
ˆh(t) := h(t)−1 t
Z t 0
h(s)ds
and
f¯(t) :=
f(t)−cα−1(tα−1), ρ <0, t−α(f(∞)−f(t)), α < ρ= 0, t−αf(t), α > ρ= 0,
f(t),ˆ α=ρ= 0.
Moreover,
a0(tx) a0(t) −xα
˜
a0(t) −xαxρ−1 ρ
≤εmax(xα+ρ+δ, xα+ρ−δ).
Extreme Value Theory
The purpose of EVT are nondegenerate limit distributions of sample maxima if the sample size tends to infinity. Therefore, letX1, . . . , Xn be an i.i.d. sequence of random variables with common distribution F and X1,n ≤ ... ≤ Xn,n the corresponding order statistics. Then, we can easily deduce that
Xn,n →P xF as n→ ∞,
where xF := sup{x : F(x) < 1}, the right endpoint of F. In particular, the distribution function ofXn,n, given byFn, converges to a degenerate limit. This provides only rough information about the limiting behavior of sample maxima and illustrates the need to normalize Xn,n. Therefore we are seeking for some sequence un := un(x) such that P(Xn,n ≤ un) converges to some non-trivial limit, i.e. a number in (0,1). Intuitively, it is clear that the existence of un will require some conditions on the right tail, ¯F := 1−F, of the underlying distribution.
We start with an important relation between tail behavior of F and limiting behavior ofXn,n, which provides a fundamental concept for the understanding of weak limits of sample maxima.
Theorem 3.1 (Leadbetter et al. (1983), Theorem 1.5.1 ).
For a given τ ∈ [0,∞] and a sequence un of real numbers the following two statements are equivalent
nF¯(un)→τ, (3.1)
P(Xn,n≤un)→e−τ. (3.2)
46
Proof. We follow the arguments in Leadbetter et al. (1983), p.13.
First assume that (3.1) holds with 0≤τ <∞. Then, we have 1−F(un) =τ /n+o(1/n),
which together with
P(Xn,n≤un) =Fn(un) = (1−(1−F(un)))n (3.3) yields (3.2).
Conversely, if (3.2) holds withτ ∈[0,∞), ¯F(un)→0 follows immediately. To see this, observe that otherwise there exists a subsequence unk, such that 1−F(unk) is bounded away from zero. This together with (3.3) results inP(Xn,n ≤un)→0, contradicting the assumptionτ < ∞.
Moreover, taking logarithms in (3.2) and (3.3) leads to nlog(1−(1−F(un)))→ −τ,
which together with ¯F(un)→0 yields n(1−F(un))(1 +o(1))→τ.
Next, assume that (3.1) holds with τ =∞, but P(Xn,n≤un)6→0. Then, there must be a subsequence nk such that P(Xnk,nk ≤unk)→e−τ0 ask → ∞ for some τ0 <∞. As an immediate consequence we obtain 1−F(un)→0, which allows to deduce
nklog(1−(1−F(unk))) → −τ0
by taking logarithm of (3.2) and usingP(Xn,n ≤un) =Fn(un) = (1−(1−F(un)))n. This, however, implies nk(1−F(unk)) →τ0 <∞ contradicting the assumption that (3.1) holds with τ =∞. The converse assertion in case of τ =∞ follows by similar arguments.
Remark. Theorem 3.1 illustrates that the behavior of nF¯(un) will be essential for the existence of nondegenerate limits of normalized maxima. In particular, it already indicates the possible consequences. Indeed, three cases: τ = 0, τ ∈(0,∞) and τ =∞ can be distinguished.
In case τ = ∞, nF¯(un) diverges, which can happen either due to un 6→ xF or if un → xF at a too slow rate. In both cases, we have P(Xn,n ≤ un) → 0, which implies that the normalized maximum still converges to xF in probability.
If τ ∈ [0,∞), convergence in (3.1) obviously implies that un → xF ≤ ∞. If the convergence is too fast, we obtain τ = 0 meaning that the resulting limit
distribution is degenerate.
In the remaining case, τ ∈ (0,∞), un converges to xF at a suitable rate. This is the most favorable situation. Together with the fact that τ :=τ(x), as well as un(x), may depend on x, there is some hope to obtain a nondegenerate limiting distribution of Xn,n normalized by un.
Using Theorem 3.1 a first negative result for the existence ofun can be proven.
Corollary 3.1 (Corollary 1.5.2, Leadbetter et al. (1983)).
Assume xF <∞ and 1−F(xF−)>0. Then for every sequence un with P(Xn,n ≤un)→ρ,
we have ρ∈ {0,1}.
Proof. SupposexF <∞and 1−F(xF−)>0 and assume thatP(Xn,n ≤un)→ρ withρ∈[0,1] for some sequence un. Then, there certainly exists some τ ∈[0,∞]
such that ρ=e−τ. Therefore, by Theorem 3.1 we deduce that n(1−F(un)) →τ.
Under the assumption un < xF for infinitely many values of n this results in τ =∞ and thereforeρ= 0 due to 1−F(un)>1−F(xF−)>0.
In the other case thatun ≥xF for all sufficiently largen, we obtainn(1−F(un)) = 0 such thatτ = 0 and therefore ρ= 1 follows.
Hence, the main message of Corollary 3.1 is that if F has a jump at its finite right endpoint, a nondegenerate limit distribution ofXn,n is not possible, despite normalization. A somewhat similar result which also holds for an infinite right endpoint is due to Leadbetter et al. (1983), Theorem 1.7.13.
Proposition 3.1. Assume xF ≤ ∞ and τ ∈(0,∞). There exists a sequence un satisfying nF¯(un)→τ if and only if
x→xlimF
F¯(x)
F¯(x−) = 1. (3.4)
Remark. Note, that if X is integer-valued with infinite right endpoint, equation (3.4) translates to F¯(n)/F¯(n −1) → 1 as n → ∞. Unfortunately, this rules out many discrete distributions such as Poisson, binomial, negative binomial and geometric.
Example 3.1 (Poisson distribution).
The Poisson distribution with probability mass function p(k) =e−λ λk!k with k ∈N and infinite right endpoint has only jumps at positive integer values k. Therefore we have
p(n)
1−F(n−) = λn/n!
P∞
k=nλk/k! = 1
1 +P∞
k=n+1λn−k(n−k+ 1)! →1 due to
∞
X
k=n+1
λn−kn!
k! =
∞
X
k=1
λk n!
(n+k)! ≤
∞
X
k=1
λkn−k = λn−1
1−λn−1 →0.
Thus,
F¯(n)
F¯(n−1) = 1 + p(n) 1−F(n−1).
This contradicts the condition in Proposition 3.1. Consequently, it is not possible to obtain a nondegenerate limit distribution for normalized maxima of the Poisson distribution.
However, imposing conditions to ensure that the jump heights of the mentioned discrete distributions decay sufficiently fast with the sample size, it is possible to obtain weak limits for the sample maxima. For the Poisson distribution this was established in Anderson et al. (1997) requiring that the intensity is suitably growing with the sample size. Nadarajah and Mitov (2002) have stated adequate conditions for binomial, negative binomial and geometric distribution.
At the end of this section we stress on the fact that P(Xn,n ≤un)→ρ∈(0,1) is necessary for a nondegenerate limit distribution of normalized maxima. Theorem 3.1 illustrates that the existence of such a limit depends precisely on whether or not 1−F(un) ∼ τ /n for τ ∈ (0,∞), where ρ = e−τ. If F is continuous, we can even achieve equality for any τ, n in (3.1) by choosing un = F−1(1−τ /n).
However, the normalizing sequence un is usually chosen under other aspects, for example simplicity.
3.1 Limits of linearly normalized maxima
To obtain concrete nondegenerate limits we may consider linearly normalized sample maxima, i.e. we set un(x) := anx+bn with some real sequences an> 0
and bn ∈R. It turns out that this already leads to a sufficient rich theory. More precisely, we seek to find sequences an>0 and bn∈R such that
Xn,n−bn an
→d G or equivalently
n→∞lim Fn(anx+bn) =G(x)
for every continuity point x of G, where G is some nondegenerate distribution function. Possible limits Gare called Extreme Value Distributions. The so-called tail quantile function U : (1,∞)→(x∗, xF), where x∗ is the left endpoint of the distribution, defined byU := 1/(1−F)←, where F is some distribution function, will play an exceptional role. Note that due to
U(x) = inf
t: 1
1−F(t) ≥x
= inf
t:F(t)≥1− 1 x
=F←(1−1/x), U(x) corresponds to the 1−1/x-quantile of the distribution F.
Another useful tool in order to establish limit theorems is the convergence to types theorem going back to Khinchin (see for instance Feller 1971, Chapter VIII.2, Lemma 1 or Resnick 1987, p.7). More precisely, we say that two distribution functionsG, H are of the same type if for some a≥0,b ∈R
G(x) = H(ax+b) for all x.
Affine transformations, weak convergence and types of distributions are related as follows.
Proposition 3.2 (Convergence to types theorem).
Suppose Xn, Y, Z are random variables, where Y, Z are nondegenerate. Consider some sequences an, αn>0 and bn, βn ∈R and assume that
1
an(Xn−bn)→d Y, 1
αn (Xn−βn)→d Z.
Then
αn
an →a >0, βn−bn
an →b∈R and
Z = (Yd −b)/a.
The class of Extreme Value Distributions is characterized by the next proposition which is taken from Resnick (1987), Proposition 0.3.
Proposition 3.3 (Fisher and Tippett (1928), Gnedenko (1943)).
LetX1. . . , Xn be i.i.d. random variables with common distribution F and suppose that there exist some sequences an>0 and bn∈R such that
P((Xn,n−bn)/an ≤x) =Fn(anx+bn)→G(x), (3.5) weakly asn → ∞, whereGis nondegenerate. ThenGis of the type of the following three classes:
(Fr´echet) (i) Φα(x) =
0, x≤0,
exp (−x−α), x > 0,
α >0. (3.6)
(Gumbel) (ii) Λ(x) = exp −e−x
, x∈R. (3.7)
(Weibull) (iii) Ψα(x) =
exp (−(−x)α), x≤0,
1, x >0,
α >0. (3.8)
Moreover, the normalizing constants may be chosen by (i) an=U(n), bn= 0,
(ii) an = g(U(n)), bn = U(n), with g(t) = RxF
t (1−F(u))du/(1− F(t)) for t < xF,
(iii) an=xF −U(n), bn =xF.
Remark. Note that due to the convergence to types theorem, one can only expect to determine possible limit distributions in (3.5) up to affine transformations.
However, if Fn(anx+bn)→G(ax+b), then adjusting the normalizing constants by αn :=an/a and βn:=bn−ban/a yields Fn(αnx+βn)→G(x).
Jenkinson (1955) established an alternative representation of Extreme Value Distributions.
Theorem 3.2 (de Haan and Ferreira (2006), Theorem 1.1.3).
Possible limits in (3.5) are given by G˜γ(ax+b) with a >0, b ∈R and G˜γ(x) := exp −(1 + ˜γx)−1/˜γ
for 1 + ˜γx >0, where γ˜∈R and in case γ˜= 0, G0(x) := exp (−e−x) for x∈R.
Note, that this parametrization suggests that the class of Extreme Value Distribu-tions is a simple explicit one-parameter family apart from the scale and location parameters.
Proof of Theorem 3.2. We closely follow the arguments in de Haan (1984) and de Haan and Ferreira (2006). In particular, the concept of extended regular variation will be exploited.
We first prove the equivalence
n→∞lim Fn(anx+bn) =G(x) ⇐⇒ lim
t→∞
U(tx)−b(t)
a(t) =G← e−1/x
with a(t) =a[t] and b(t) =b[t] and U = (1/(1−F))←. Subsequently, we derive the explicit form of possible limit distributions G.
SinceF(anx+bn)→1 as n→ ∞ for each continuity point x of G, we have for any suchx, satisfying 0< G(x)<1,
n→∞lim
−log(F(anx+bn)) 1−F(anx+bn) = 1.
Hence we can rewrite the relation Fn(anx+bn)→G(x) forn → ∞as
n→∞lim
1
n(1−F(anx+bn)) =− 1 log(G(x)).
Observe that we have a sequence of monotone functions converging to a monotone limit. This is equivalent to the convergence of the inverted sequence to the inverse of the limit (see for instance de Haan and Ferreira (2006), Lemma 1.1.1). Thus, we have for any x >0
n→∞lim
U(nx)−bn
an =G← e−1/x
. (3.9)
Next, we prove a continuous version of (3.9). Therefore, assumex is a continuity point ofG← e−1/x
. For t≥1, we have U([t]x)−b[t]
a[t] ≤ U(tx)−b[t]
a[t] ≤ U([t]x(1 + 1/[t]))−b[t]
a[t] .
Fort large enough the last expression is less than G← e−1/x0
for any continuity point x0 > xwith G← e−1/x0
> G← e−1/x
. However, since G← is continuous at e−1/x, we have
t→∞lim
U(tx)−b[t]
a[t] =G← e−1/x
for any x >0.
Thus, defining b(t) = b[t] and a(t) =a[t], we obtain
t→∞lim
U(tx)−b(t)
a(t) =G← e−1/x
for any x >0. (3.10) Evaluating (3.10) atx= 1 and subtracting from (3.10) yields
t→∞lim
U(tx)−U(t)
a(t) =G← e−1/x
−G← e−1 . Simultaneously, we have according to Theorem 2.7
t→∞lim
U(tx)−U(t)
a(t) =cx˜γ−1
˜
γ x >0,
for somec6= 0 and ˜γ ∈R. Since G is monotone increasing, c >0 follows. Hence, we get
G← e−1/x
=G← e−1
+cx˜γ−1
˜
γ , c >0, ˜γ ∈R. SettingD(x) := G← e−1/x
leads to D(x) =D(1) +c˜γ−1(x˜γ−1). Inverting the last relation yields
D←(x) =
1 + ˜γx−D(1) c
1/˜γ
. (3.11)
Observe that
D←(x) = 1
−log(G(x)), which together with (3.11) yields
G(x) = exp −
1 + ˜γ
x−D(1) c
−1/˜γ! . Hence, the limit distribution is given byGγ˜(cx+D(1)), where
Gγ˜(x) := exp −(1 + ˜γx)−1/˜γ .
Introducing the location and scale parameter leads to the following definition.
Definition 3.1 (Generalized Extreme Value Distribution (GEVD)).
Let ˜γ ∈R, µ∈R and σ >0. Then G˜γ,µ,σ(x) := exp −
1 + ˜γ
x−µ σ
−1/˜γ!
, with 1 + ˜γ
x−µ σ
>0 defines the Generalized Extreme Value Distribution. Note that any limit distribu-tion of linearly normalized maxima belongs to the class of G˜γ,µ,σ.
Next, we discuss a one-to-one relation between the Gγ˜-parametrization and (Φα,Λ,Ψα)-parametrization. For ˜γ >0, we have
Gγ˜((x−1)/˜γ) = exp(−x−1/˜γ) = Φ1/˜γ(x) for x >0 and 0 else. Further, we get for ˜γ = 0
G0(x) = exp −e−x
= Λ(x) for x∈R and for ˜γ <0
G˜γ(−(1 +x)/˜γ) =
exp −(−x)−1/˜γ
, x <0,
1, x≥0.
Thus G˜γ(−(1 +x)/˜γ) = Ψ−1/˜γ(x) for x ∈ R. Hence, both representations of Extreme Value Distributions together with their types characterize the same set of distributions, thus both are equivalent.
Remark. There exists another approach to derive the explicit form of possible limit distributions of linearly normalized maxima. The idea is to use the equivalence Xn →d X if and only if for all real bounded continuous functions z, E(z(Xn))→ E(z(X)). We refer to Beirlant et al. (2004) for a detailed discussion.