On Robust Tail Index Estimation and Related Topics

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and Related Topics

Dissertation

zur Erlangung des akademischen Grades

des Doktors der Naturwissenschaften (Dr. rer. nat.)

vorgelegt von

Dieter Schell

an der

Mathematisch-Naturwissenschaftliche Sektion Fachbereich Mathematik und Statistik

Konstanz, 2013

Konstanzer Online-Publikations-System (KOPS) URL: http://nbn-resolving.de/urn:nbn:de:bsz:352-0-282868

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This work was financially supported by the German Research Foundation (DFG).

I am deeply grateful to my supervisor Prof. Dr. Jan Beran for his guidance and advice through the course of my studies. Moreover, I would like to express my sincere gratitude for the excellent working conditions at his chair at the University of Konstanz and the opportunity to partake in many international conferences.

Additionally, my thanks go to Prof. Dr. Siegfried Heiler for reviewing this thesis.

Furthermore, I would like to thank my family and friends for their continuous support during these years. Special thanks go to Volker B¨urkel, Klaus Telk- mann, Josip Jakic and Matt O’Neill for proofreading parts of this thesis. Finally, my deepest gratitude goes to Anna Lena Grundler for her enduring patience, encouragement and support.

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In this work we discuss tail index estimation for heavy-tailed distributions with an emphasis on robustness. After a short introduction we provide a theoretical background treating regular variation and its extensions. In particular, we consider second- and third-order properties of regularly varying functions and state uniform approximations. Based on this, classical results of Extreme Value Theory including limit distributions of normalized maxima and necessary and sufficient conditions for maximum domains of attraction (MDA) are discussed. In particular, we present a new two-parametric characterization of limiting distributions of normalized maxima naturally arising from second-order regular variation of the tail quantile function. Additionally we provide an interpretation of MDA-conditions for heavy- tailed distributions and derive their empirical counterparts. Generalized versions of empirical MDA-conditions lead to asymptotic expansions of the tail quantile process and the tail empirical process. Thereafter we discuss different tail index estimators, first considering an approach based on robustification of the Pareto- MLE. We then establish the parametric rate of convergence and quantify the robustness properties of the resulting Huberized Tail Index Estimator by the Influence Function. Subsequently, classical tail index estimators based on relative excesses are considered. In particular these are linked to empirical versions of MDA-conditions. This relation also leads to some new classes of estimators, including p-Quantile Tail Index Estimators and Harmonic Moment Tail Index Estimators (HME). We derive the asymptotic properties of these classes and compare them with the well known Hill estimator. It turns out that the HME outperforms the Hill estimator in certain situations. Moreover, the parametric Huberized Tail Index Estimator shows a competitive behavior in comparison to the Hill estimator for small to moderate sample sizes. These asymptotic results are confirmed by simulations illustrating the finite sample behavior of corresponding estimators.

We conclude by discussing the issue of tail index estimation for linear long memory processes with infinite second moments. A unifying characterization of long memory for strict stationary processes is proposed. Moreover, the tail index of a linear long memory process withα-stable innovations is estimated by a modified version of the Huberized Tail Index Estimator and asymptotic properties are derived.

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In dieser Arbeit beschäftigen wir uns mit der Tail-Index Schätzung von Verteilun- gen mit schweren Tails, wobei der Schwerpunkt auf Robustheit liegt. Nach einer kurzen Einführung werden zunächst die theoretischen Grundlagen - reguläre Vari- ation und ihre Erweiterungen - diskutiert. Dabei gehen wir insbesondere auf die Eigenschaften zweiter und dritter Ordnung der regulär variierenden Funktionen ein und geben gleichmäßige Approximationen an. Darauf aufbauend werden klassische Resultate der Extremwerttheorie, unter anderem Grenzverteilungen von normierten Maxima sowie notwendige und hinreichende Bedingungen für Maxi- mumanziehungsbereiche (MDA) behandelt. Des Weiteren schlagen wir eine neue zwei-parametrische Charakterisierung der Grenzverteilungen von normierten Max- ima vor. Diese ergibt sich auf natürliche Weise aus der regulären Variation zweiter Ordnung der Tail-Quantil-Funktion. Zusätzlich arbeiten wir eine Interpretation für die MDA-Bedingungen für Verteilungen mit schweren Tails heraus und leiten ihre empirischen Entsprechungen her. Verallgemeinerungen der empirischen MDA- Bedingungen führen zu asymptotischen Entwicklungen des Tail-Quantil-Prozesses sowie des Empirischen Tail-Prozesses. Im Anschluss wenden wir uns der Tail-Index Schätzung zu, wobei zunächst der Ansatz einer Robustifizierung des Pareto-ML Schätzers vorgestellt wird. Wir zeigen die parametrische Konvergenzrate und bestimmen die Robustheitseigenschaften des resultierenden Huberisierten Tail- Index Schätzers mit Hilfe der Influence Function. Danach werden klassische Tail-Index Schätzer, welche in der Regel auf relativen Überschüssen basieren, diskutiert. Insbesondere werden diese auf die empirischen MDA-Bedingungen zurückgeführt. Darüber hinaus ergeben sich aufgrund dieser Beziehung neue Klassen von Tail-Index Schätzern. Hierbei sind der p-Quantil-Tail-Index Schätzer sowie der Harmonische-Momenten-Tail-Index Schätzer (HME) zu nennen. Nach der Herleitung der asymptotischen Eigenschaften werden diese mit dem wohlbekan- nten Hill Schätzer verglichen. Es stellt sich heraus, dass der HME in bestimmten Situationen besser abschneiden kann als der Hill Schätzer. Weiterhin zeigt sich, dass der parametrische Huberisierte Tail-Index Schätzer in Bezug auf den Hill Schätzer ein vergleichbares Verhalten für kleine und mittlere Stichprobenumfänge aufweist. Diese asymptotischen Resultate werden mit Hilfe von Simulationen untermauert, welche das Verhalten der entsprechenden Schätzer bei endlichen Stichproben illustrieren.

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eine vereinheitlichende Charakterisierung von langfristigen Abhängigkeiten für strikt stationäre Prozesse vor. Außerdem schätzen wir den Tail-Index eines linearen Prozesses mit langfristigen Abhängigkeiten und α-stabilen Innovationen mit Hilfe einer modifizierten Version des Huberisierten Tail-Index Schätzers und leiten die asymptotischen Eigenschaften her.

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Chapter 1 Introduction

Extreme Value Theory (EVT) deals with rare, extreme events and is applied in various fields including insurance, finance, hydrology, telecommunications and meteorology. EVTs unique characteristic is that it provides well-established statis- tical methods allowing inference about the underlying probability distribution even beyond the range of available data, an area usually avoided in classical statistics.

This extrapolation is strongly connected to limiting behavior of extreme values, i.e.

maximum or minimum of a sample. Without loss of generality one focuses on the right tail and therefore considers the limiting behavior of the sample maximum.

However, as the sample size tends to infinity, the sample maximum converges almost surely to the right endpoint of the distribution providing only limited information about the right tail. Therefore limits of linearly normalized maxima are considered. This leads to the corner stone of EVT, the theorem of Fisher and Tippett (1928), which specifies the three possible limit distributions, namely Fr´echet (Ψ), Gumbel (Λ) and Weibull (Φ), for linearly normalized maxima. The set of parent distributions for which the corresponding distribution of a suitably normalized maximum is attracted by one of the three limit distributions is called maximum domain of attraction (MDA). Necessary conditions for a distribution to belong to a specific MDA were established by von Mises (1936), while Gnedenko (1943) derived necessary and sufficient conditions which were completed and streamlined by de Haan (1970). Jenkinson (1955) found a unifying representation of the three limit distributions introducing the Generalized Extreme Value Dis- tribution (GEVD) which possesses only one parameter, the real valued Extreme Value Index (EVI). It is the main parameter of interest and is also called tail index for positive values. The estimation of the tail index is important for many aspects,

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as since it determines the heaviness of the (right) tail and therefore affects quantile estimation as well as existence of moments. According to the importance there is an extensive literature on tail index estimation since the earliest approaches of Pickands (1975) and Hill (1975) (see e.g. Embrechts et al. 1997, Beirlant et al.

2004, de Haan and Ferreira 2006, Resnick 2007 and references therein). Given the preceding discussion, the main objective of this thesis is to discuss different tail index estimators with an emphasis on robustness.

The central analytical tool of univariate EVT is the theory of regular variation initiated by Karamata (1930, 1933). Later, this concept was generalized to extended regular variation and extended regular variation of second-order (refer to de Haan 1970, Geluk and de Haan 1987, de Haan and Stadtm¨uller 1996, de Haan and Ferreira 2006). After covering these results in chapter two, we introduce a slightly modified class of regularly varying functions based on the concept of slow variation with remainder (see Goldie and Smith 1987, Bingham et al. 1987).

According to Resnick (2007), we identify the resulting class as regularly varying functions of second-order and state uniform weighted approximations for functions from this class, relying on already existing approximations for extended regularly varying functions. Furthermore, we disuss regular variation of third-order and illustrate relations between regular and extended regular variation of different orders.

Chapter three contains classical results of EVT, following in large parts chapter two in de Haan and Ferreira (2006). We begin with the mentioned theorem of Fisher and Tippett (1928), followed by the parametrization of the three limit distributions using the GEVD. Necessary and sufficient conditions for the parent distribution to be part of the MDA of a GEVD are established using classical results about extended regular variation (see de Haan and Ferreira 2006).

Subsequently, we directly derive limit theorems of linearly normalized maxima using the concept of regular variation of second-order. This quite naturally yields a two-parametric characterization of possible limiting distributions, which, in fact, is a compromise between the classical (Ψ, Λ, Φ)-characterization and the one-parametric characterization using the GEVD. In particular, this new approach clarifies the essential difference between the so called heavy-tailed distributions forming the MDA of GEVD for a positive EVI or tail index and distributions with light tails, i.e. exponential tails or even tails with a finite right endpoint.

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Hence, it strongly opposes the subsumption of possible limit distributions into a one-parametric GEVD-family. Moreover, the new parametrization leads to more compelling necessary and sufficient MDA-conditions, especially for light- tailed distributions. Further, we provide an interpretation of the MDA conditions for heavy-tailed distributions saying that it is sufficient if the distribution of relative excesses above some high threshold tends to a corresponding Pareto distribution as the threshold grows to infinity. In particular, the parameter of the Pareto distribution corresponds to the tail index. A similar interpretation with respect to quantiles, relying on an alternative MDA-characterization of heavy- tailed distributions, is provided as well. On that basis empirical counterparts of the MDA conditions are derived using intermediate number of relative excesses, Xn−k+i,n/Xn−k,n, withi = 1, . . . , k, where X_1,n ≤ · · · ≤X_n,n are order statistics of an i.i.d. sequence of random variablesX₁, . . . , X_n with a common heavy-tailed distribution and assuming that k → ∞, k/n→ 0 as n → ∞. In particular, we exploit results about vague convergence of the tail empirical process, established in Resnick (2007). The empirical MDA-conditions serve as the point of departure to motivate existing classical tail index estimators as well as to derive new ones in the forthcoming chapter. Furthermore, relying on results about regular variation of second-order, we discuss a weighted uniform approximation of the tail quantile process established by Drees (1998). This general result can be exploited to derive limit distribution of tail index estimators based on relative excesses. The chapter concludes by stating a somewhat similar approximation for the tail empirical process derived in Drees at al. (2006) with a slight modification of their results in order to meet our requirements.

Several tail index estimators are discussed in chapter four. We start with the Huberized Tail Index Estimator introduced by Beran (1997). This parametric M-estimator, based on original data, reduces the influence of deviations from the model distribution (Pareto) at lower quantiles and is therefore a robusti- fied version of the Pareto maximum likelihood estimator. Following Beran and Schell (2012) asymptotic properties of the Huberized Tail Index Estimator are established. We also illustrate a reduced sensitivity to deviations from the model distribution considering the Influence Function. Subsequently, classical tail index estimators based on relative excesses including the Hill estimator (see Hill 1975) are considered. They are typically constructed replacing original observations in

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some estimator of the Pareto-parameter by relative excesses above the random threshold Xn−k,n and claiming that k → ∞, but k/n →0 as the sample size n tends to infinity. This replacement can be motivated by empirical versions of the MDA-condition discussed in chapter three. Before discussing the well known Hill estimator, we introduce the class ofp-Quantile Tail Index Estimators which we believe is not covered in the literature. Asymptotic properties are derived using empirical MDA-conditions as well as exploiting results about the tail quantile process stated at the end of chapter three. Additionally, we discuss the quite general class of Harmonic Moment Tail Index Estimators (HME) (see Henry 2009) and prove asymptotic properties following Beran et al. (2013b).

Since, the asymptotic normality of all considered tail index estimators based on relative excesses is established under the same second-order condition on the parent distribution, we are able to compare their performance using the asymptotic mean squared error (AMSE). However, it turns out that a comparison at the same level k can provide misleading results. Instead, as suggested by de Haan and Peng (1998), we compare the considered tail index estimators respec- tively at their optimal level k₀, which minimizes the corresponding AMSE. In particular, we show that the relative efficiency of two tail index estimators based on relative excesses evaluated at their optimal level is essentially characterized by a weighted product of the squared bias ratio and the variance ratio, where the weights are determined by the second-order parameter ρ ≤ 0. This results was in principle already derived in de Haan and Peng (1998), confirming their statement that there exists no uniform best tail index estimator but it depends on the situation and in particular on the second-order parameter. Additionally, we prove that in particular for large ρ-values, the HME is able to outperform the Hill estimator in terms of the AMSE. Moreover, we demonstrate that, according to the asymptotic results, the Huberized Tail Index Estimator is able to outperform the Hill estimator by means of the AMSE, for small to moderate sample sizes.

In chapter five we illustrate the finite sample performance of tail index estimators conducting a small simulation study. With regard to tail index estimators based on relative excesses it shows that the second-order parameterρ plays a decisive role for the quality of the approximation resulting from asymptotics. The reason for this is that ρdetermines the rate of convergence in corresponding limit theorems.

Moreover, we confirm the asymptotic result that in certain situations the Hill

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estimator can be outperformed by the HME in terms of the mean squared error in the finite sample setting. Further, we show that in contrast to classical tail index estimators, the Huberized Tail Index Estimator exhibits a much more stable finite sample behavior. This can be traced back to its faster convergence rate of √

n, instead of √

k in case of classical tail index estimators. Additionally we demonstrate that the relative efficiency of the Huberized Tail Index Estimator with respect to the Hill estimator remains in reasonable bounds without requiring any additional information about the underlying distribution. In particular, it turns out that for small and moderate sample sizes, the Hill estimator can be even outperformed by the Huberized Tail Index Estimator, despite evaluation at the asymptotic optimal level.

In chapter six tail index estimation in presence of long memory is discussed.

We introduce several existing notions of long memory and propose a unifying characterization using limits of partial sums. In particular, this allows a quantifi- cation of long memory for linear stochastic processes with infinite second moments.

Subsequently, we estimate the tail index of a linear long memory processes with α-stable innovations using a modified version of the Huberized Tail Index Estima- tor. Following Beran et al. (2012) the asymptotic distribution is derived exploting the results of Koul and Surgailis (2001). We also discuss a possibility to test for the equality of tails.

Chapter seven contains some concluding remarks and reveals possible directions for future research.

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Regular Variation and Extensions

In this chapter we introduce the concept of regular variation mainly developed by Karamata around 1930 and initially used to extend some results on Tauberian theory. Feller (1966), introduced regular variation in probability theory using Karamata’s results for stable distributions and their domains of attraction. In turn, Feller’s book inspired de Haan (1970) to work on regular variation and extensions remaining of great importance for probability theory and in particular for EVT.

Further theoretical developments were made by Seneta (1973). In introducing regular variation and its extensions we closely follow the arguments contained in the Appendix of de Haan and Ferreira (2006) and the encyclopaedic volume of Bingham et al. (1987).

2.1 Regular and Slow Variation

Many phenomena in applied mathematics show scale invariance, a striking property of power laws. Note that a positive functionf :R⁺ →Ris scale invariant if there exists some function g such that f(xt) = g(x)f(t) for any x, t > 0. In case f is a power function, given by f(t) := t^α, α ∈ R, we have g(x) = x^α. Often we are merely interested in the behavior of f(x) for x → ∞. Therefore, we relax the conditions onf and define a class of asymptotically scale invariant functions claiming that for allx >0

t→∞lim f(tx)

f(t) =g(x), (2.1)

where g(x) is finite and positive. Possible limiting functionsg are given by the next theorem.

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Theorem 2.1 (Theorem B.1.3, de Haan and Ferreira (2006)).

Suppose f :R⁺ →R is measurable, eventually positive, and (2.1) holds with some positive functiong. Then, the limit is necessarily given by g(x) :=x^α for some α∈R.

Proof. First note that g is measurable since it is a point wise limit of measurable functions. Further, for arbitraryx, y, t >0 we have

f(txy)

f(t) = f(txy) f(ty)

f(ty) f(t) .

Thus, letting t tend to infinity and using (2.1), we can conclude that g satisfies g(xy) =g(x)·g(y) for any x, y ∈R⁺.

This completes the proof since the only measurable, positive and finite valued solution of Cauchy’s functional equation in its multiplicative form is given byx^α for someα ∈R.

The result of the preceding theorem motivates to the following definition.

Definition 2.1 (Regularly varying function).

A measurable, eventually positive, function f : R⁺ → R varies regularly (at infinity) if there exists some α∈R such that for allx >0

t→∞lim f(tx)

f(t) =x^α. (2.2)

We write f ∈RV_α and call α the index of regular variation. If α = 0, we say f is slowly varying. Note that for any f ∈RV_α we have f(x) =x^αL(x) with some

L∈RV₀.

Moreover, a function f is called regularly varying at the origin with exponent α if f˜∈RVα and f˜(x) = f(x⁻¹).

Note that sincef is eventually positive, there exists at0 >0 such thatf(t)>0 for t≥t₀. Thus, we may alter f by settingf(t) =f(t₀)>0 for t∈(0, t₀] in order to obtain a strictly positive function. This obviously does not change its asymptotic behavior in (2.2). In general, regularly varying functions may be altered on finite intervals since they are defined by an asymptotic relation.

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Remark. There are several ways to motivate the concept of regular variation. We state an approach partly based on de Haan and Ferreira (2006) and Bingham et al.

(1987) providing a possible justification of the term ’regular variation’. Therefore, assume that one is interested in the variation of a function at infinity. Hence, we consider the limiting relation

x→∞lim k(x+h)−k(x) = u(h) for all h∈R (2.3) for some measurable function k : R →R and a possible limiting function u. It turns out that, provided the existence, u does not depend on x. Moreover, since (2.3) holds for any h∈R, we obtain

u(h₁+h₂) =u(h₁) +u(h₂) for any h₁, h₂ ∈R. (2.4) The only measurable solution of (2.4) is of the form u(x) =cx for some c∈R. Thus, (2.3) is equivalent to

x→∞lim

k(x+h)−k(x)

h =c for all h∈R, (2.5)

meaning that in the limit k has a constant relative variation. Hence, one could define the class of all measurable functions k satisfying (2.3), or equivalently (2.5), as the class of regularly varying functions at infinity. After transforming the additive relation (2.3) into a multiplicative one by f(t) = expk(log(t)), we immediately obtain the defining equation of regular variation for f with (c=α)

t→∞lim f(tx)

f(t) =x^α for all x >0.

Therefore, the class of regularly varying functions (at infinity) contains transforms of functions with asymptotically constant relative variation, which may be termed as regular variation (at infinity). In particular, this heuristic approach makes the meaning of slow variation more transparent, since due tolim_x→∞l(x+h)−l(x)→0 for l(x) := log(L(e^x)) and L∈RV0, we could say that the transform l of a slowly varying function L does not vary asymptotically.

Example 2.1. The simplest examples of slowly varying functions are positive, measurable functions with positive limits at infinity. Other typical examples are given by log(x) and its iterates. Examples of regularly varying functions can be obtained from slowly varying functions by multiplying with corresponding powers.

Examples of not regularly varying functions are given by 1 + sin(x) or exp[log(x)].

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The next proposition states that the point wise convergence in (2.1) is even uniform on each compact subset of (0,∞) (see for instance Embrechts et al.

(1997), Theorem A3.2).

Proposition 2.1 (Uniform convergence theorem).

Let 0< a < b. Iff ∈RV_α for α∈R, then

t→∞lim f(tx)

f(t) =x^α for all x >0, holds locally uniformly in x

(i) on each [a, b] if α= 0,

(ii) on each (0, b] if α >0 provided f is bounded on each (0, b], (iii) on each [a,∞) if α <0.

Remark. There are several possibilities to weaken the assumption on f in order to be regularly varying (see e.g. de Haan and Ferreira 2006 or Bingham et al. 1987).

However, measurability is a key requirement to obtain the fundamental result of uniform convergence and avoid pathological solutions in (2.1) (see Korevaar et al.

1949, Bingham et al. 1987). Moreover, it seems natural to assume measurability since we often consider integrals of regularly varying functions.

Next, we study the behavior of regularly varying functions when integrated. In this regard an essential result is the following theorem (see Theorem B.1.5 in de Haan and Ferreira 2006).

Theorem 2.2 (Karamata’s Theorem).

Suppose f ∈RV_α. Then, there exists a t₀ >0such that f(t) is positive and locally bounded for t≥t₀. Moreover, if α≥ −1 then Rt

t0f(s)ds ∈RV_α+1 and

t→∞lim

tf(t) Rt

t0f(s)ds =α+ 1. (2.6)

If α <−1 (or if α=−1 and R∞

t f(s)ds <∞), then

t→∞lim

tf(t) R∞

t f(s)ds =−α−1. (2.7)

Conversely, if (2.6) holds with −1 < α <∞, then f ∈ RV_α; if (2.7) holds with

−∞< α <−1, then f ∈RV_α.

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Proof. We follow the arguments stated in de Haan and Ferreira (2006). First, we show that Rt

t0f(s)ds∈RV_α+1 if f ∈RV_α with α > −1. Therefore, observe that f ∈RV_α withα >−1 implies f(2s)≥2⁻¹f(s) for s sufficiently large. Hence, we have

Z 2ⁿ⁺¹ 2ⁿ

f(s)ds= 2 Z 2ⁿ

2ⁿ⁻¹

f(2s)ds≥ Z 2ⁿ

2ⁿ⁻¹

f(s)ds.

Moreover, there exists some n₀ ∈N with t₀ ≥2ⁿ⁰ such that forn ≥n₀ Z ∞

2ⁿ⁰

f(s)ds =

∞

X

n=n0

Z 2ⁿ⁺¹ 2ⁿ

f(s)ds≥

∞

X

n=n0

Z 2ⁿ⁰⁺¹ 2ⁿ⁰

f(s)ds =∞.

Hence, F(t) → ∞ for t → ∞, where F(t) :=Rt

t0f(s)ds. Moreover, we have for any finite ¯t > t₀

Z t t0

f(s)ds∼ Z t

¯t

f(s)ds as t→ ∞.

To see this, note that according to Theorem 2.1 there exists somec > 0 such that f(tx)/f(t)< c for t ≥t₀, x∈[1,2]. Then for t∈[2ⁿt₀,2ⁿ⁺¹t₀] we have

f(t)

f(t₀) = f(t) f(2⁻¹t)

f(2⁻¹t)

f(2⁻²t). . .f(2⁻ⁿt)

f(t₀) < cⁿ⁺¹. Hence, f(t) is locally bounded for t≥t₀ and Rt

t0f(s)ds <∞ for t≥t₀. Therefore, providedt₁, t₁x, tx > t₀, we obtain

F(tx) F(t) =

Rtx

t0 f(s)ds Rt

t0f(s)ds ∼ Rtx

t1xf(s)ds Rt

t1f(s)ds = xRt

t1f(xs)ds Rt

t1f(s)ds ,

as t → ∞. Moreover, for some fixed x > 0 and arbitrary ε > 0, there exists t₁ =t₁(ε) such that

f(tx)<(1 +ε)x^αf(t) for t > t₁. Hence, we conclude that

F(tx)

F(t) <(1 + 2ε)x^α+1

for t sufficiently large. A lower inequality can be derived by similar arguments such thatF ∈RV_α+1 forα >−1 follows.

In case α = −1 and F(t) → ∞ similar arguments apply. If α = −1 and

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limt→∞F(t)<∞, we directly obtain F ∈RV₀.

Next, we prove (2.6), assuming thatf ∈RV_α with α≥ −1 and F ∈RV_α+1. Note that due to Theorem 2.1 we have for allα∈R

F(tx)−F(t) tf(t) =

Rtx

t f(s)ds tf(t) =

Rx

1 f(tu)du f(t) →

Z x 1

u^αdu= x^α+1−1

α+ 1 , ast → ∞.

Now, sinceF ∈RVα+1 we have

t→∞lim

F(tx)−F(t)

F(t) =x^α+1−1.

This yields

t→∞lim F(t)

tf(t) = 1 α+ 1 which corresponds (2.6).

For the proof of (2.7), note that for α < −1 there exists δ > 0 such that f(s)≤2^−1−δf(s) for s sufficiently large. Then, by similar arguments, we obtain

Z ∞ 2ⁿ¹

f(s)ds=

∞

X

n=n1

Z 2ⁿ⁺¹ 2ⁿ

f(s)ds≤

∞

X

n=n1

2^−δ(n−n¹⁾

Z 2ⁿ¹⁺¹ 2ⁿ¹

f(s)ds <∞.

This impliesR∞

t f(s)ds <∞. The rest of the proof is analogous to the one of (2.6).

For the converse statement define the function b(t) := tf(t)/Rt

t0f(s)ds for t > t₀ and note that by (2.6) limt→∞b(t) = 1/(1 +α) for −1< α <∞. Integration of b(t)/t yields

Z t t1

b(s) s ds=

Z t t1

f(s) Rs

t0f(u)du

!

ds = logF(t) +c₁ (2.8)

where t₁, t > t₀ and c₁ =−log(F(t₁)). Since F(t) = tf(t)/b(t), (2.8) leads to f(t) = cb(t)

t exp Z t

t1

b(s) s ds

(2.9) with c=e^−c¹ >0. Hence, we obtain for t, tx > t₁

f(tx)

f(t) = b(tx) x b(t)exp

Z x 1

b(ts) s ds

. (2.10)

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Moreover, due to (2.6), for any ε >0 there exists a t₂ >0 such that for t ≥ t₂ and s≥min(1, x)

α+ 1−ε < b(ts)< α+ 1 +ε. (2.11) This leads to

t→∞lim Z x

1

b(ts)

s ds= (α+ 1) log(x). (2.12)

Together with limt→∞b(t) = 1/(1 +α) we deduce from (2.10) that f ∈RV_α. The proof of the last statement is similar.

Rewriting (2.9) leads to a first representation of regularly varying functions given by

f(t) = c b(t) exp Z t

t1

b(s)−1 s ds

for any t > t₀. (2.13) In particular, this relation offers a possibility to construct functions with desired regular variation.

Theorem 2.3 (Representation theorem).

For some measurable, eventually positive, function f :R⁺ →R we have f ∈RVα

if and only if there exist measurable functions a:R⁺→R and c:R⁺ →R with

t→∞lim c(t) =c₀ (0< c₀ <∞) and lim

t→∞a(t) = α∈R (2.14) and f can be represented by

f(t) = c(t) exp Z t

t0

a(s) s ds

for t > t₀. (2.15) for somet₀ >0.

Remark. Note that we have some flexibility in the choice of t₀. Different choices of t₀ will alter the functions c and a on finite intervals without changing the asymptotic behavior of f. Hence, the functions c and a in the representation of f are not unique.

Proof. We first prove that (2.15), together with (2.14), implies f ∈RV_α. We have f(tx)

f(t) = c(tx) c(t) exp

Z tx t

a(s) s ds

for t, tx > t₀.

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Note that by substitutings by tu, we get Z tx

t

a(s) s ds=

Z x 1

a(tu) u du.

Due to (2.14), for any ε > 0 there exists some t₁ such that |a(tu)−α| ≤ ε for t≥t₁ and u≥min(1, x). Hence,

exp Z x

1

a(tu) u du

→exp

α Z x

1

1 udu

=x^α ast → ∞.

Together with the convergence of the function cfor t→ ∞ regular variation off follows.

For the converse, observe that if f ∈ RV_α, t^−αf(t) is slowly varying. Hence, proving the representation fort^−αf(t) with limt→∞c(t) =c₀ and limt→∞a(t) = 0, we obtain (2.15), replacinga(s) with a(s) +α andc(t) byt^α₀c(t). Thus, it suffices to prove the assertion for slowly varying functions.

Now, letf ∈ RV₀ and considerh(t) := log(f(e^t)). To prove (2.15) it suffices to show thath can be written as

h(t) = d(t) + Z t

b

ν(u)du, t≥b, (2.16)

with b= log(t₀), d(t) =c(e^t), ν(t) =a(e^t) withd(t)→c₀ and ν(t)→0 as t→ ∞.

In order to proceed, we need the following auxiliary result.

Lemma 2.1 (Seneta (1973)).

If L is positive, measurable, defined on some interval [t₀,∞) and L(tx)

L(x) →1 (x→ ∞) ∀t > t₀,

thenLis bounded on all finite intervals far enough to the right. Ifh(t) := log(L(e^t)), h is also bounded on finite intervals far enough to the right.

According to the preceding lemma h is integrable on finite intervals far enough to the right, being bounded and measurable thereon. Hence for large enough values of uwe can state

h(t) = Z t+1

t

(h(t)−h(s))ds+ Z t

u

(h(s+ 1)−h(s))ds+ Z u+1

u

h(s)ds (t ≥u).

The last term is constant, say c. Further, note that for ν(t) := h(t + 1) − h(t), we have ν(t) → 0 as t → ∞. The first term on the right hand side is R1

0 (h(t)−h(t+s))ds, which tends to 0 as t → ∞ by the uniform convergence theorem. Thus, (2.16) follows withd(t) =c+R1

0 (h(t)−h(t+s))ds.

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Next, we summarize important properties of regularly varying functions. A more detailed discussion can be found in de Haan and Ferreira (2006) as well as Bingham et al. (1987).

Theorem 2.4 (Properties of regularly varying functions).

(i) If f ∈RV_α, then limt→∞log(f(t))/log(t) = α. Therefore,

t→∞lim f(t) =











0, α <0,

∞, α >0.

(ii) If f1 ∈ RVα1, f2 ∈ RVα2, then f1 + f2 ∈ RV_max(α₁_,α₂₎. If moreover limt→∞f₂(t) = ∞, then the composition f₁◦f₂ ∈RV_α₁_α₂.

(iii) If f ∈ RV_α with α > 0 (α < 0) then f is asymptotically equivalent to a strictly increasing (decreasing) differentiable function g with derivative g⁰ ∈RVα−1 if α >0 and −g⁰ ∈RVα−1 if α <0.

(iv) If f ∈ RV_α is integrable on finite intervals of R⁺ and α ≥ −1, then F(s) := Rt

0 f(s)ds ∈RV_α+1.

(v) (Potter 1942) Suppose f ∈RV_α. Then, for arbitrary δ₁, δ₂ >0 there exists t₀ =t₀(δ₁, δ₂) such that for t ≥t₀, tx≥t₀,

(1−δ₁)x^αmin(x^δ², x^−δ²)< f(tx)

f(t) <(1 +δ₁)x^αmax(x^δ², x^−δ²). (2.17) Conversely, if f satisfies (2.17), then f ∈RVα.

Proof. We restrict ourselves to prove (i) and (v). Remaining proofs can be found in de Haan and Ferreira (2006).

To prove (i), assume that f ∈RV_α. Then, from the proof of the representation theorem, we have

h(log(t)) = log(f(t)) =d+o(1) +

Z log(t) log(t0)

(α+o(1))dy.

This results in

t→∞lim

log(f(t)) log(t) =α.

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Moreover, since for α > 0, h(t) → ∞ as t → ∞ and for α < 0, h(t) → −∞ the assertion follows.

Potter’s bounds immediately follow from the representation theorem. We prove (v) for f ∈RV₀. From (2.15) we get

f(tx)

f(t) = c(tx) c(t) exp

Z tx x

a(u) u du

= c(tx) c(t) exp

Z x 1

a(ts) s ds

.

Further, due to lim_t→∞c(t) = c, we deduce that for any δ₁ > 0 there exists a t1 =t1(δ1)>0 such that for t, tx > t1

1−δ₁ < c(tx)

c(t) <1 +δ₁.

Additionally, since limt→∞a(t) = 0, for anyδ2 >0 there exists at2 =t2(δ2)>0 such that fort, tx > t₂

−δ₂ < a(tx)< δ₂. Therefore for anyx >1 and t≥t₂, we have

−δ₂log(x)<

Z x 1

a(ts)

s ds < δ₂log(x).

Moreover, we have for any x <1 and tx≥t₂ δ₂log(x)<

Z x 1

a(ts)

s ds <−δ₂log(x).

Hence, by settingt₀ =t₀(δ₁, δ₂) = max(t₁(δ₁), t₂(δ₂)) we obtain for any δ₁, δ₂ >0 (1−δ₁) min(x^δ², x^−δ²)< f(tx)

f(t) <(1 +δ₁) max(x^δ², x^−δ²).

for any t, tx > t₀.

Closely related to Potter’s bounds are weighted uniform approximations of deviation of regularly varying functions from corresponding power laws.

Theorem 2.5 (de Haan and Ferreira (2006), Drees (1998)).

If f ∈RV_α, for each ε, δ >0 there is a t₀ =t₀(ε, δ) such that for t, tx≥t₀,

f(tx) f(t) −x^α

≤εmax x^α+δ, x^α−δ

. (2.18)

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Proof. We follow de Haan and Ferreira (2006), page 369-370. Again, since for anyf ∈RV_α we have f(x) =x^αL(x) with L is slowly varying, it suffices to prove the assertion for α = 0. First, note that for δ > 0, max x^δ, x^−δ

= e^−δ|^log(x)|. Therefore, starting from Potter’s bounds we get fort, tx≥t0 and δ > δ2

e^{−δ|log(x)|} (1−δ₁)e^−δ²^|^log(x)|−1

≤e^{−δ|log(x)|}

f(tx) f(t) −1

(2.19)

≤e^−(δ−δ²^)|^log(x)| (1 +δ₁)−e^−δ²^|log(x)|

. We show that the left-hand side of (2.19) tends to zero, whenδ1, δ2 →0 uniformly for x > 0. The proof for the right-hand side is similar. Assume first that δ₂|log(x)| →0. Then, the second term tends to zero, and due to e^−δ|^log(x)| ≤ 1 the left hand side of (2.19) tends to zero. If δ₂|log(x)| → c ∈ (0,∞] then δ|log(x)| → ∞ such that the first term tends to zero, while the second term is bounded.

For an alternative proof see Cheng and Jiang (2001).

The last theorem implicitly states that for tlarge enough the term L(tx)/L(t)−1, as a function in x, can be bounded by any power function x^δ. In other words the deviation from pure power law behavior is scaled out asymptotically.

Another interesting property of slowly varying functions is stated by the next proposition.

Proposition 2.2. (de Bruyn conjugate, Beirlant et al. (2004), Proposition 2.5) If L∈RV₀, then there exists L^∗ ∈RV₀, the de Bruyn conjugate of L, such that

L(x)L^∗(xL(x))→1, x→ ∞.

The de Bruyn conjugate is asymptotically unique in the sense that if any L˜ ∈RV₀ satisfiesL(x) ˜L(xL(x))→1 as x→ ∞, then L^∗ ∼L. Furthermore˜ (L^∗)^∗ ∼L.

2.2 Extended regular variation

The definition of regular variation can be modified as follows: A function f : R⁺ →R is said to be regularly varying if there exists a positive function a such that for all x >0 the limit

t→∞lim f(tx)

a(t)

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exists and is positive. An obvious generalization is to consider measurable functions f :R⁺ → R such that there exists a positive function a such that for all x > 0 the limit

t→∞lim

f(tx)−f(t) a(t)

exists and is not constant. It turns out, that similar to regularly varying functions only specific limits are possible.

Theorem 2.6 (Theorem 1.9, Geluk and de Haan (1987)).

Assume f :R⁺→R is measurable and a is a positive function. Then, provided the existence of a non-constant limit, we have

t→∞lim

f(tx)−f(t)

a(t) =c· x^ρ−1

ρ , x >0, (2.20)

for some ρ ∈ R, c6= 0 (for ρ = 0 the right-hand side corresponds to c·log(x)).

Moreover, we have a ∈RV_ρ. Proof. Define first

ψ(x) := lim

t→∞

f(tx)−f(t) a(t) .

By assumption ψ(x) exists for all x > 0 and is not constant. Hence, there exists some x₀ > 0 such that ψ(x₀) 6= 0. Therefore we can choose a(t) = (f(tx0)−f(t))/ψ(x0). Without loss of generality we may assumea to be measur-

able. Moreover, for arbitrary y >0, we have

t→∞lim a(ty)

a(t) = lim

t→∞

f(tx₀y)−f(t)

a(t) − f(ty)−f(t) a(t)

f(tx₀y)−f(ty) a(ty)

= ψ(x₀y)−ψ(y) ψ(x₀) . Thus, A(y) := limt→∞ a(ty)

a(t) exists and is non-negative for all y >0. Since for any t, x, y >0

a(txy)

a(t) = a(txy) a(tx)

a(tx) a(t) , taking the limit with respect tot yields

A(xy) =A(x)·A(y) for all x, y >0.

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Hence,Asatisfies Cauchy’s functional equation. Observe that sinceais measurable A is measurable as well. Moreover, since A is non-negative, A(t) = t^ρ for some ρ∈Rfollows. Note thatA(t) = 0 can be ruled out by contradiction. Consequently, we obtaina ∈RVρ. Further, due to

t→∞lim

f(txy)−f(t)

a(t) − f(ty)−f(t)

a(t) = lim

t→∞

f(txy)−f(ty) a(ty) · lim

t→∞

a(ty) a(t) , we get

ψ(xy)−ψ(y) =y^ρψ(x) for all x, y >0. (2.21) Ifρ= 0, this results again in Cauchy’s functional equation and therefore ψ(y) = c·log(x) for somec6= 0,x >0 follows. Forρ6= 0, interchangingx andy in (2.21) and subtract the resulting equation from (2.21) yields

ψ(x)(1−y^ρ) = ψ(y)(1−x^ρ) for any x, y >0.

Hence, we conclude that ψ(x)/(1−x^ρ) is constant. In particular, we may write ψ(x) = cρ⁻¹(x^ρ−1) for x >0 and some c6= 0.

Theorem 2.6 motivates the following definition. Note that considering (2.20) we can restrict our attention to the casec >0, since replacing f by −f changes the sign of c.

Definition 2.2 (Extended regular variation, de Haan and Ferreira (2006)).

A measurable function f :R⁺ →R is said to be of extended regular variation if there exists a function a:R⁺ →R⁺ such that for some α∈R and all x >0,

t→∞lim

f(tx)−f(t)

a(t) = x^α−1

α . (2.22)

We use the notation f ∈ERV_α. The function a is called an auxiliary function for f. For α = 0, the right-hand side of (2.22) reads as log(x).

The following result establishes a link between extended regular variation and regular variation, provided α6= 0.

Supposef ∈ERVα with α6= 0 and c >0, i.e.

t→∞lim

f(tx)−f(t)

a(t) =c·x^α−1 α .

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1. If α >0 then limt→∞f(t)/a(t) =c/α and hence f ∈RV_α. 2. If α <0 then f(∞) := limx→∞f(x) exists,

t→∞lim

f(∞)−f(t)

a(t) =−c α, and hence f(∞)−f(x)∈RV_α.

Proof. Note that we may assume a∈RV_α. It suffices to prove the assertion for some functiong such that

f(t)−g(t) =o(a(t)). (2.23) First, we show that from

t→∞lim

g(tx)−g(t)

a(t) =c· x^α−1

α for x >0, (2.24) g ∈RV_α follows. Therefore, takey >1 and define the sequence (t_n)n∈N by t₁ := 1 and t_n+1 :=t_ny. Then, by (2.24) we obtain

n→∞lim

g(t_n+2)−g(t_n+1) a(t_n+1)

a(t_n)

g(t_n+1)−g(t_n) = 1 and therefore

n→∞lim

g(t_n+2)−g(t_n+1) g(t_n+1)−g(t_n) =y^α.

For α >0 we haveg(t_n)→ ∞ for n → ∞. Moreover, for any ε >0 there exists n₀ such that for any n > n₀

g(tn+2)−g(tn0+1) =

n

X

k=n0

(g(tk+2)−g(tk+1))

< y^α(1 +ε)

n

X

k=n0

(g(t_k+1)−g(t_k))

=y^α(1 +ε) (g(t_n+1)−g(t_n₀)). Moreover, a similar lower inequality holds. Thus, we can deduce that

n→∞lim

g(t_n+1) g(tn) =y^α. Therefore we get

a(t_n)∼ g(t_n+1)−g(t_n) cα⁻¹(y^α−1) ∼ c

αg(t_n).

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Moreover, forx >1 g(tnx)

g(t_n) −1 = g(tnx)−g(tn)

g(t_n) ∼ g(tnx)−g(tn)

cα⁻¹a(t_n) →x^α−1 (n→ ∞).

In order to prove regular variation ofg for any t >0, choose for any s >0 some sequencen(s)∈Nsuch that t_n(s) ≤s < t_n(s)+1. Then we obtain

g(sx)

g(s) ≤ g(t_n(s)+1x) g(t_n(s)+1)

g(t_n(s)+1)

g(t_n(s)) →x^αy^α, n→ ∞, and

g(sx)

g(s) ≥ g(t_n(s)x) g(t_n(s))

g(t_n(s))

g(t_n(s)+1) →x^αy^−α, n → ∞.

Sincey >1 is arbitrary, g ∈RV_α follows. Rewriting (2.24), we obtain

t→∞lim

g(tx)−g(t)

a(t) = lim

t→∞

g(t) a(t)

g(tx) g(t) −1

= c

α(x^α−1).

This immediately yieldsg(t)/a(t)→c/αfort→ ∞. Using (2.23) we end up with f(t)∼ca(t)/α for t→ ∞ and thereforef ∈RVα.

Note that the proof of the second case (α <0) is similar and can be found in de Haan and Ferreira (2006).

Remark. Note that ERV₀ corresponds to the class Π introduced by de Haan (1970). We refer to de Haan and Ferreira (2006) for a more extensive treatment of the class Π.

Next, we state a result corresponding to Potter’s bounds for regularly varying functions. Therefore, a slight modification of the auxiliary functiona becomes necessary.

Suppose f ∈ ERV_α with auxiliary function a. Then for all ε, δ > 0 there is a t0 =t0(ε, δ) such that for t, tx≥t0,

f(tx)−f(t)

a₀(t) − x^α−1 α

≤εx^αmax x^δ, x^−δ , where

a₀(t) :=











αf(t), α >0,

f(t)−t⁻¹Rt

0 f(s)ds, α = 0,

−α(f(∞)−f(t)), α <0.

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Proof. For α > 0, we conclude from Theorem 2.7 that f ∈ RV_α and therefore a₀(t) :=αf(t)∈RV_α. Moreover, we have

f(tx)−f(t)

a₀(t) −x^α−1 α

= 1 α

a₀(tx) a₀(t) −x^α

.

From Theorem 2.5, we deduce that for each ˜ε, δ >0, there exists somet₀ >0 such that for any t, tx > t₀

a₀(tx) a₀(t) −x^α

≤ε˜max x^α+δ, x^α−δ . Thus, setting ε=α⁻¹ε >˜ 0 yields

f(tx)−f(t)

a₀(t) − x^α−1 α

≤εx^αmax x^δ, x^−δ .

For α≤0 the proof is similar using Theorem 2.7 in conjunction with Theorem B.2.17 in de Haan and Ferreira (2006).

2.3 Second-order regular variation

The class of slowly varying functions contains members with rather diverse limiting behavior for x→ ∞. There are functions converging to some positive constant, diverge (slow enough) for instance f(x) = log(x), as well as functions converging to zero (slow enough) likef(x) = 1/log(x). In order to impose some structure on this class we quantify the convergence rate ofL(tx)/L(t) to 1 as t→ ∞. More precisely, considering someL∈RV₀, we seek for some positive function ˜A with limt→∞A(t) = 0 such that˜

ψ(x) := lim

t→∞

L(tx) L(t) −1

A(t)˜ (2.25)

exists for anyx >0 and is not constant. It turns out that ˜Ais necessarily regularly varying with index ρ≤0 and that only specific limits ψ are possible. This leads to the following definition.

Definition 2.3 (Second-order slow variation).

A slowly varying function L is said to be second-order slowly varying if there exists an auxiliary function A with |A| ∈RV_ρ (ρ≤0), such that

t→∞lim

L(tx) L(t) −1

A(t) = x^ρ−1

ρ for any x >0, (2.26)

where for ρ= 0 the right-hand side of (2.26) reads as log(x).

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We show that|A| ∈RV_ρ as well asψ(x) = (x^ρ−1)/ρfollows from (2.25) following the arguments in the proof of Theorem 1.9 in Geluk and de Haan (1987).

Since ψ is not constant there exists some x₀ > 0 such that ψ(x₀) 6= 0. Hence we can choose ˜A(t) := (L(tx0)/L(t)−1)/ψ(x0). In particular, we may assume without loss of generality ˜A to be measurable. Moreover, we have for anyx, y >0

t→∞lim

L(txy)−L(t)

L(t) ˜A(t) − L(ty)−L(t)

L(t) ˜A(t) = lim

t→∞

L(txy)−L(ty)

L(t) Ã(t) · L(ty) Ã(ty) L(ty) Ã(ty)

= lim

t→∞

L(txy)−L(ty)

L(ty) ˜A(ty) · L(ty)

L(t) · A(ty)˜ A(t)˜ , which yields

ψ(xy)−ψ(y) =ψ(x) lim

t→∞

A(ty)˜

A(t)˜ . (2.27)

This implies the existence of φ(y) := limt→∞A(ty)/˜ A(t) for any˜ y > 0. In particular, note that φ is measurable since ˜A is measurable. Hence, we conclude that eitherφ(y) =y^ρ for some ρ ∈R or φ(y) = 0 for y > 0. However,φ(y) = 0 implies a constant limit ψ contradicting the assumption above. Moreover, for A˜∈ RVρ and ρ >0 we have limt→∞A(t) =˜ ∞. Therefore ˜A ∈RVρ with ρ ≤ 0 follows. From (2.27) we obtain the functional equation

ψ(xy)−ψ(y) =ψ(x)y^ρ for all x, y >0 (2.28) where ρ ≤ 0. If ρ = 0, this again yields Cauchy’s functional equation in its additive form. It follows ψ(y) =c·log(y) for somec6= 0 andy >0.

Ifρ <0, interchanging xand y in (2.28) and considering the difference of (2.28) and the resulting relation yields

ψ(x)(1−y^ρ) = ψ(y)1−x^ρ for all x, y >0.

This implies that ψ(x)/(1−x^ρ) is constant for any x > 0 and x 6= 1. Hence we can conclude that ψ(x) := c(x^ρ −1)/ρ for x > 0 with c 6= 0 and ρ < 0.

Note, that dividing by ρ allows a smooth transition from ρ < 0 to ρ = 0, since limρ→0(x^ρ −1)/ρ = log(x). Moreover, it should be mentioned that we have subsumed the constant c 6= 0 and the function ˜A ∈ RV_ρ, measuring the convergence rate ofL(tx)/L(t) to 1 ast→ ∞, into the functionAin the definition above. Since the sign ofcis unknown, we can only conclude that |A| ∈RV_ρ. If A possesses eventually a positive signL(tx)/L(t) approaches 1 from above, otherwise from below.

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Remark. Goldie and Smith (1987) introduced the class of slowly varying functions of second-order terming it slow variation with remainder (see also Bingham et al.

1987 and Aljanˇci´c et al. 1974). Our definition corresponds to the one stated in Goldie and Smith (1987), denoted by (SR2). It should be mentioned that in contrast to ERV the auxiliary function A˜ of a slowly varying function of second-order has a clear meaning as a rate function tending to zero and possessing a ultimately constant sign indicating the ’direction’ from which L(tx)/L(t) approaches 1.

We can easily generalize the definition of slow variation of second-order to regular variation of second-order taking into account that anyf ∈RV_α can be represented asf(x) = x^αL(x), where L∈RV₀. This gives rise to the following definition.

Definition 2.4 (Second-order regular variation).

Suppose f ∈RV_α and there exists some auxiliary function A with limt→∞A(t) = 0 and ultimately constant sign, such that

t→∞lim

f(tx) f(t) −x^α

A(t) =x^αx^ρ−1

ρ (2.29)

holds for all x > 0 with ρ ≤ 0. Then, we call f second-order regularly varying with second-order parameter ρ. Notation: f ∈ 2RVα,ρ. In particular, we have

|A| ∈RV_ρ. For ρ= 0 the right hand side of (2.29) reads as x^αlog(x).

Remark. Note that2RV_0,ρ is the class of slowly varying functions of second-order.

In particular, it turns that 2RV0,0 is closely related to the class Π introduced by de Haan (1970). A quite extensive treatment of Π can be found in de Haan and Ferreira (2006).

Example 2.2. A simple example for f ∈2RV0,0 is given byf(t) := log(t), where a possible auxiliary function is given by A(t) = 1/log(t). Note that we also have f ∈Π with possible auxiliary function a(t) = 1, illustrating thatlimt→∞a(t)→0 does not necessarily hold in Π (or ERV₀).

Example 2.3. As an example for f ∈ 2RV_0,ρ (ρ 6= 0) we consider functions of the form f(x) := D₁ +D₂x^ρ with D₁ > 0 and D₂ 6= 0. Note that since limx→∞f(x) = D₁ >0 we have f ∈RV₀. Setting A(t) := ρD₁⁻¹D₂t^ρ, we obtain for any x >0

t→∞lim

f(tx) f(t) −1

A(t) = lim

t→∞

D₁+D₂(tx)^ρ−D₁−D₂t^ρ

(D1+D2t^ρ)ρD⁻¹₁ D2t^ρ = x^ρ−1 ρ .

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Hence f ∈ 2RV_0,ρ with possible auxiliary function A. Note that |A| ∈RV_ρ and limt→∞A(t) = 0.

Next, we detail some relations betweenRV,ERV and 2RV.

Theorem 2.9. Let f :R⁺ →R be some measurable, eventually positive function.

Then, we have the following relations:

(i) For α >0 we have f ∈ERV_α ⇐⇒ f ∈RV_α.

(ii) Iff ∈ERV_α, then f ∈2RV_0,α with an ultimately positive auxiliary function A. The converse is also true.

Proof. We start with (i). For f ∈ERV_α with α >0 we obtain by Theorem 2.7 thatf ∈RV_α. For the converse statement suppose f ∈RV_α with α >0. Then, for any x >0,

t→∞lim f(tx)

f(t) =x^α ⇐⇒ lim

t→∞

f(tx) f(t) −1

α = lim

t→∞

f(tx)−f(t)

αf(t) = x^α−1 α . Since, by definition,f is eventually positive, there exists some positive function a with a(t)∼αf(t) such that

t→∞lim

f(tx)−f(t)

a(t) = x^α−1 α . Hence, f ∈ERV_α.

Next, we prove (ii) for α = 0. Therefore we require the following auxiliary result.

Corollary 2.1 (Corollary B.2.13, de Haan and Ferreira (2006)).

If f ∈ERV0, then limt→∞f(t) =f(∞)≤ ∞ exists. If the limit is infinite, then f ∈RV₀. If the limit is finite, then f(∞)−f(t)∈RV₀. Moreover,

a(t) =o(f(t)) as t → ∞.

and when f(∞)<∞,

a(t) =o(f(∞)−f(t)) as t → ∞.

According to Corollary 2.1 limt→∞f(t) :=f(∞)≤ ∞ exists if f ∈ERV₀. Since f is ultimately positive we deduce 0≤ f(∞)≤ ∞. Additionally, if f(∞) <∞, f(∞)−f(t) ∈ RV₀ follows. Hence, we can exclude f(∞) = 0, since otherwise

(33)

−f ∈RV₀, contradicting the assumption that f is ultimately positive.

Thus, for f ∈ ERV₀ and f ultimately positive, we have 0 < f(∞) ≤ ∞. For 0 < f(∞) < ∞ we immediately obtain f ∈ RV₀. For f(∞) = ∞, we have f ∈RV0 by Corollary 2.1. Further, we have for any x >0

t→∞lim

f(tx)−f(t)

a(t) = log(x).

This is equivalent to

t→∞lim

f(tx) f(t) −1

f(t)⁻¹a(t) = log(x).

Hence, f ∈2RV_0,0 with a possible auxiliary function A(t) :=f(t)a(t). Note that A∈ RV₀, since f, a∈ RV₀. For the converse note that due to f ∈2RV_0,0 with auxiliary functionA we have

t→∞lim

f(tx) f(t) −1

A(t) = log(x).

This yields

t→∞lim

f(tx)−f(t)

f(t)A(t) = log(x).

Sincef and A are both ultimately positive, there exists some positive function a with a(t)∼f(t)A(t) such that

t→∞lim

f(tx)−f(t)

a(t) = log(x).

Thusf ∈ERV₀ follows.

Next, we prove (ii) for α <0. For f ∈ERV_α with α <0 and positive auxiliary functiona∈RVα we have

t→∞lim

f(tx)−f(t)

a(t) = x^α−1

α . (2.30)

This implies limt→∞f(tx)−f(t) = 0 or equivalently limt→∞f(tx)/f(t) = 1, i.e.

f ∈RV₀. Moreover, we conclude from (2.30) that

t→∞lim

f(tx) f(t) −1

f(t)⁻¹a(t) = x^α−1

α for any x >0.

On Robust Tail Index Estimation and Related Topics

and Related Topics

Dissertation

zur Erlangung des akademischen Grades

des Doktors der Naturwissenschaften (Dr. rer. nat.)

vorgelegt von

Dieter Schell

an der

Mathematisch-Naturwissenschaftliche Sektion Fachbereich Mathematik und Statistik

Konstanz, 2013

Contents

Chapter 1 Introduction

Regular Variation and Extensions

2.1 Regular and Slow Variation

2.2 Extended regular variation

2.3 Second-order regular variation