A new extension of the class of regularly varying functions

(1)

HAL Id: hal-01181346

https://hal.archives-ouvertes.fr/hal-01181346

Preprint submitted on 30 Jul 2015

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

A new extension of the class of regularly varying functions

Meitner Cadena, Marie Kratz

To cite this version:

Meitner Cadena, Marie Kratz. A new extension of the class of regularly varying functions. 2015.

�hal-01181346�

(2)

A new extension of the class of regularly varying functions

Meitner Cadena^∗and Marie Kratz^†

Abstract

We define a new class of positive and Lebesgue measurable functions in terms of their asymptotic behavior, which includes the class of regularly varying functions.

We also characterize it by transformations, corresponding to generalized moments when these functions are random variables. We study the properties and extensions of classical theorems for this class.

Keywords: asymptotic behavior; Karamata’s representation theorem; Karamata’s the- orem; Karamata’s tauberian theorem; measurable functions; Peter and Paul distribu- tion; regularly varying function

AMS classification: Primary: 26A12; 26A42; 40E5; secondary 28A10; 60G70

Introduction

The class of regularly varying functions has been introduced in the 30s by Karamata, who defined the notion of slowly varying (SV) and regularly varying (RV) functions, describing a specific asymptotic behavior of these functions, namely:

Definition.A Lebesgue-measurable functionU:R⁺→R⁺is RV at infinity if, for allt>0,

x→∞lim U(xt)

U(x) =t^ρ for someρ∈R^, ⁽¹⁾

ρbeing called the tail index ofU, and the caseρ=0 corresponding to the notion of SV function.Uis RV at 0⁺if (1) holds, when taking the limit asx→0⁺instead of+∞.

Since then, much literature has been devoted to RV functions (see e.g. [33], [5] and references therein), in particular in Extreme Value Theory (EVT) (see e.g. [20], [18], [14], [31]) where the RV property helps characterizing maximum domains of attraction. The notion of multivariate regular variation has been developed (see e.g. [15], [32], and references therein) and various extensions of the RV class have been proposed. We may cite, in a non exhaustive way, the class of Extended RV (ERV) (which is implicit in the work of Ma- tuszewska [29], and simply allows the limit in (1) to vary), its natural extension, named the

∗UPMC Paris 6 & CREAR, ESSEC Business School; E-mail: meitner.cadena@etu.upmc.fr or b00454799@essec.edu

†ESSEC Business School, CREAR risk research center; E-mail: kratz@essec.edu

(3)

O-Regularly Varying (O-RV) class, defined and studied by Avakumovi´c [2], also analyzed by Karamata [27] (see also e.g. [29], [19], [33], [1], [28], and [11], where relations between ERVandO-RVare analyzed), the Bojanic-Karamata class (see [7]) that is a subclass of the SV class, theΠclasses (see e.g. [3], or [5]), and the Beurling classes, the slowly varying one (see e.g. [5]) that contains the SV class, or the RV one (see [6]). It is worth noticing that the Beurling theory includes the Karamata theory (see [6]).

In this paper, we propose a new extension of the RV class, defined in terms of the asymptotic decay of the functions, and for which the limit in (1) might not exist. This new class not only extends in a simple way main RV properties but also offers broader applications, as e.g. in EVT. We can mention, for instance, new results on maximum domains domain of attraction (see [10]) and the proposition of a new tail index estimator (see [9]).

The aim of this work is to present and characterize fully this new class.

The paper is organized in two main parts. The first section defines this large class of functions, describing it in terms of their asymptotic behaviors, which may violate (1). It provides its algebraic properties, as well as characteristic representation theorems, one being of Karamata type. In the second section, we discuss extensions for this class of functions of other important Karamata theorems. Proofs of the results are given in the appendix.

1 Study of a new class of functions

We focus on the new classMof positive and measurable functions with supportR⁺^{, char-} acterizing their behavior at∞with respect to polynomial functions. A number of properties of this class are studied and characterizations are provided. Further, variants of this class, considering asymptotic behaviors of exponential type instead of polynomial one, provide other classes, denoted byM∞andM−∞, having similar properties and characterizations asMdoes.

Let us introduce a few notations.

When considering limits, we will discriminate between two main cases, namely when the limit is finite or infinite (±∞), and when it does not exist.

The notation a.s. (almost surely) in (in)equalities concerning measurable functions is omitted. Moreover, for any random variable (rv)X, we denote its distribution byFX(x)= P(X≤x), and its tail of distribution byF_X =1−F_X. The subscriptXwill be omitted when no possible confusion.

RV (RV_ρ respectively) denotes indifferently the class of regularly varying functions (with tail indexρ, respectively) or the property of regularly varying function (with tail indexρ).

Finally recall the notations min(a,b)=a∧band max(a,b)=a∨bthat will be used,bxcfor the largest integer not greater thanxanddxefor the lowest integer greater or equal than x, and log(x) represents the natural logarithm ofx.

1.1 The classM

We introduce a new classMthat we define as follows.

(4)

Definition 1.1. M is the class of positive and measurable functions U with supportR⁺^, bounded on finite intervals, such that

∃ρ∈R,∀ε>0, lim

x→∞

U(x)

x^ρ+² =0 and lim

x→∞

U(x)

x^ρ−² = ∞. (2)

OnM, we can define specific properties.

Properties 1.1.

(i) For any U∈M,ρdefined in (2) is unique, and denoted byρU. (ii) If U,V∈Ms.t.ρU>ρV, then lim

x→∞

V(x) U(x)=0.

(iii) For any U,V ∈M and any a≥0, aU+V ∈MwithρaU+V =ρU∨ρV. (iv) If U∈MwithρU defined in (2), then1/U∈Mwithρ1/U= −ρU.

(v) Let U∈MwithρU defined in(2). IfρU< −1, then U is integrable onR⁺, whereas, if ρU> −1, U is not integrable onR⁺^.

Note that in the caseρU= −1, we can find examples of functions U which are inte- grable or not.

(vi) Sufficient condition forUto belong toM:Let U be a positive and measurable func- tion with supportR⁺, bounded on finite intervals. Then

−∞ < lim

x→∞

log (U(x))

log(x) < ∞ =⇒ U∈M.

To simplify the notation, when no confusion is possible, we will denoteρU byρ. Remark 1.1. Link to the notion of stochastic dominance

Let X and Y be rv’s with distributions FX and FY, respectively, with supportR⁺^{. We say} that X is smaller than Y in the usual stochastic order (see e.g. [34], pp. 3) if

FX(x)≤FY(x) for all x∈R⁺^. ⁽³⁾ This relation is also interpreted as the first-order stochastic dominance of X over Y , as FX≥ FY (see e.g. [22], pp. 289).

Let X , Y be rv’s such that FX =U and FY =V , where U,V ∈M andρU >ρV. Then Prop- erties 1.1, (ii), implies that there exists x0>0such that, for any x≥x0, V(x)<U(x), hence that(3)is satisfied at infinity, i.e. that X strictly dominates Y at infinity.

Furthermore, the previous proof shows that a relation like(3)is satisfied at infinity for any functions U and V inMsatisfyingρU>ρV. It means that the notion of first-order stochas- tic dominance or stochastic order confined to rv’s can be extended to functions inM. In this way, we can say that ifρU>ρV, then U strictly dominates V at infinity.

(5)

Now let us define, for any positive and measurable functionUwith supportR⁺^, κU:=sup

½

r:r∈R ^and Z _∞

1

x^r−1U(x)d x< ∞

¾

. (4)

Note thatκUmay take values±∞.

Definition 1.2. For U∈M,κU defined in(4)is called theM-index of U . Remark 1.2.

1. If the function U considered in(4)is bounded on finite intervals, then the integral involved can be computed on any interval[a,∞)with a>1.

2. When assuming U=F , F being a continuous distribution, the integral in (4) reduces (by changing the order of integration), for r>0, to an expression of moment of a rv:

Z _∞

1

x^r⁻¹F(x)d x=1 r

Z _∞

1

¡x^r−1¢

d F(x)=1 r

Z _∞

1

x^rd F(x)−F(1) r . 3. We haveκU≥0for any tail U=F of a distribution F .

Indeed, suppose there exists F such thatκ_F <0. Let us denoteκ_F byκ. Sinceκ<

κ/2<0, we have by definition ofκthat Z _∞

1

x^κ^/2⁻¹F(x)d x= ∞. But, since F≤1and κ/2−1< −1, we can also write that

Z _∞

1

x^κ^/2−1F(x)d x≤ Z _∞

1

x^κ^/2−1d x< ∞. Hence the contradiction.

4. A similar statement to Properties 1.1, (iii), has been proved for RV functions (see [5], pp. 16).

Let us develop a simple example, also useful for the proofs.

Example1.1. Letα∈R^{and U}αthe function defined on(0,∞)by U_α(x) :=

½ 1, 0<x<1 x^α, x≥1 .

Then U_α∈MwithρU_α=αdefined in (2), and itsM-index satisfiesκU_α= −α.

To check thatU_α∈M, it is enough to find aρU_α, since its unicity follows by Properties 1.1, (i). ChoosingρU_α=α, we obtain, for any²>0, that

xlim→∞

U_α(x) x^ρ^U^α^+² = lim

x→∞

1

x^²=0 and lim

x→∞

U_α(x) x^ρ^U^α^−² = lim

x→∞x^²= ∞.

HenceU_αsatisfies (2) withρU_α=α. Now, noticing that

Z _∞

1

x^s−1U_α(x)d x= Z _∞

1

x^s+α−1d x< ∞ ⇐⇒ s+α<0

(6)

thenκU_αdefined in (4) satisfiesκU_α= −α. 2 As a consequence of the definition of theM-indexκonM, we can prove that Proper- ties 1.1, (vi), is not only a sufficient but also a necessary condition, obtaining then a first characterization ofM.

Theorem 1.1. First characterization ofM

Let U be a positive measurable function with supportR⁺and bounded on finite intervals.

Then

U∈MwithρU= −τ ⇐⇒ lim

x→∞

log (U(x))

log(x) = −τ, (5)

whereρUis defined in(2).

Example1.2. The function U defined by U(x)=x^sin(x)does not belong toMsince the limit expressed in (5) does not exist .

Other properties onMcan be deduced from Theorem 1.1, namely:

Properties 1.2. For U , V ∈MwithρU andρV defined in(2), respectively, we have:

(i) The product U V ∈MwithρU V =ρU+ρV.

(ii) IfρU≤ρV < −1orρU < −1<0≤ρV, then the convolution U∗V ∈M withρU∗V = ρV. If−1<ρU≤ρV, then U∗V∈M withρU∗V =ρU+ρV+1.

(iii) If lim

x→∞V(x)= ∞, then U◦V∈M withρU◦V =ρUρV.

Remark 1.3. A similar statement to Properties 1.2, (ii), has been proved when restricting the functions U and V to RV probability density functions, showing first lim

x→∞

U∗V(x) U(x)+V(x)= 1(see [4], Theorem 1.1). In contrast, we propose a direct proof, under the condition of inte- grability of the function ofMhaving the lowestρ.

When U and V are tails of distributions belonging to RV, with the same tail index, Feller ([18], Proposition, pp. 278-279) proved that the tail of the convolution of1−U and1−V also belongs to this class and has the same tail index as U and V .

We can give a second way to characterizeMusingκU defined in (4).

Theorem 1.2. Second characterization ofM

If U is a positive measurable function with supportR⁺, bounded on finite intervals, then U∈Mwith associatedρU ⇐⇒ κU= −ρU (6) whereρUsatisfies(2)andκU satisfies(4).

Here is another characterization ofM, of Karamata type.

(7)

Theorem 1.3. Representation Theorem of Karamata type forM

(i) Let U∈Mwith finiteρU defined in (2). There exist b>1and functionsα,βand² satisfying, as x→ ∞,

α(x)/ log(x) → 0 , ²(x) → 1 , β(x) → ρU, (7) such that, for x≥b,

U(x)=exp

½

α(x)+²(x) Z x

b

β(t) t d t

¾

. (8)

(ii) Conversely, if there exists a positive measurable function U with supportR⁺, bounded on finite intervals, satisfying (8) for some b>1and functionsα,β, and²satisfying (7), then U∈Mwith finiteρU defined in (2).

Remark 1.4.

1. Another way to express(8)is the following:

U(x)=exp

½

α(x)+²(x) log(x) x

Z x b β(t)d t

¾

. (9)

2. The functionαdefined in Theorem 1.3 is not necessarily bounded, contrarily to the case of Karamata representation for RV functions.

Example 1.3. Let U ∈M withM-indexκU. If there exists c>0such that U <c, then κU≥0.

Indeed, since we have lim

x→∞

log (1/U(x)) log(x) ≥ lim

x→∞

log (1/c)

log(x) =0, applying Theorem 1.1 allows

one to conclude. 2

1.2 Extension of the classM

We extend the classMintroducing two other classes of functions.

Definition 1.3. M∞andM−∞are the classes of positive measurable functions U with sup- portR⁺, bounded on finite intervals, defined as

M∞:=

½

U : ∀ρ∈R^{, lim}

x→∞

U(x) x^ρ =0

¾

, (10)

and

M−∞:=

½

U : ∀ρ∈R, lim

x→∞

U(x) x^ρ = ∞

¾

. (11)

Notice that it would be enough to considerρ<0 (ρ>0, respectively) in (10) ((11), respectively), and thatM∞,M−∞andMare disjoint.

We denote byM±∞the unionM∞∪M−∞.

We obtain similar properties forM∞andM−∞, as the ones given forM, namely:

(8)

Properties 1.3.

(i) U∈M∞ ⇐⇒ 1/U∈M−∞.

(ii) If(U,V)∈M−∞×MorM−∞×M∞orM×M∞, then lim

x→∞

V(x) U(x)=0.

(iii) If U,V∈M∞(M−∞respectively), then U+V∈M∞(M−∞respectively).

The indexκUdefined in (4) may also be used to analyzeM∞andM−∞. It can take infinite values, as can be seen in the following example.

Example1.4. Consider U defined onR⁺ ^{by U}^{(x) :}=e^−x. Then U ∈M∞ withκU = ∞. Choosing U(x)=e^xleads to U∈M−∞withκU= −∞.

A first characterization ofM∞andM−∞can be provided, as done forMin Theorem 1.1.

Theorem 1.4. First characterization ofM∞andM−∞

Let U be a positive measurable function with supportR⁺, bounded on finite intervals. Then we have

U∈M∞ ⇐⇒ lim

x→∞

log (U(x))

log(x) = −∞ (12)

and

U∈M−∞ ⇐⇒ lim

x→∞

log (U(x))

log(x) = ∞. (13)

Remark 1.5. Link to a result from Daley and Goldie.

If we restrictM∪M±∞to tails of distributions, then combining Theorems 1.1 and 1.4 and Theorem 2 in [13] provides another characterization, namely

U∈M∪M±∞ ⇐⇒ XU∈M^DG,

where XU is a rv with tail U andM^DG is the set of non-negative rv’s X having the property introduced by Daley and Goldie (see [13], Definition 1.(a)) that

κ(X∧Y)=κ(X)+κ(Y)

for independent rv’s X and Y . We notice thatκ(X)defined in [13] (called there the mo- ment index) and applied to rv’s, coincides with theM-index of U , when U is the tail of the distribution of X .

An application of Theorem 1.4 provides properties as in Properties 1.2, namely:

Properties 1.4.

(i) If(U,V)∈M∞×M∞orM±∞×M orM−∞×M−∞, then U·V ∈M∞orM±∞or M−∞, respectively.

(ii) If(U,V)∈M∞×MwithρV ≥0orρV < −1, then U∗V ∈MwithρU∗V =ρV. If(U,V)∈M∞×M∞, then U∗V∈M∞.

If(U,V)∈M−∞×MorM−∞×M±∞, then U∗V∈M−∞.

(9)

(iii) If U ∈M±∞and V ∈Msuch that lim

x→∞V(x)= ∞or V ∈M−∞, then U◦V∈M±∞. Looking for extending Theorems 1.2-1.3 toM∞andM−∞provides the next results.

Theorem 1.5.

Let U be a positive measurable function with supportR⁺, bounded on finite intervals, with κUdefined in (4).

(i) (a) U∈M∞ =⇒ κU= ∞.

(b) U continuous, lim

x→∞U(x)=0, andκU= ∞ =⇒ U∈M∞. (ii) (a) U∈M−∞ =⇒ κU= −∞.

(b) U continuous and non-decreasing, andκU= −∞ =⇒ U∈M−∞. Remark 1.6.

1. In (i)-(b), the conditionκU= ∞might appear intuitively sufficient to prove that U∈ M∞. This is not true, as we can see with the following example showing for instance that the continuity assumption is needed. Indeed, we can check that the function U defined onR⁺by

U(x) :=

( 1/x if x∈ S

n∈N\{0}

(n;n+1/nⁿ) e^−x otherwise,

satisfiesκU= ∞and lim

x→∞U(x)=0, but is not continuous and does not belong toM∞. 2. The proof of (i)-(b) is based on an integration by parts, isolating the term t^rU(t). The continuity of U is needed, otherwise we would end up with an infinite number of jumps of the type U(t⁺)−U(t⁻)(6=0)onR⁺^.

Theorem 1.6. Representation Theorem of Karamata Type forM∞andM−∞

(i) If U∈M∞, then there exist b>1and a positive measurable functionαsatisfying α(x)/ log(x) →

x→∞∞, (14)

such that,∀x≥b,

U(x)=exp {−α(x)} . (15)

(ii) If U∈M−∞, then there exist b>1and a positive measurable functionαsatisfying (14) such that,∀x≥b,

U(x)=exp {α(x)} . (16)

(iii) Conversely, if there exists a positive function U with supportR⁺, bounded on finite intervals, satisfying (15) or (16), respectively, for some positive functionαsatisfying (14), then U∈M∞or U∈M−∞, respectively.

(10)

1.3 On the complement set ofM∪M±∞

Considering measurable functionsU:R⁺→R⁺, we have, applying Theorems 1.1 and 1.4, thatU belongs toM,M∞orM−∞if and only if lim

x→∞

log (U(x))

log(x) exists, finite or infinite.

Using the notions (see for instance [5], pp. 73) oflower orderofU, defined by µ(U) := lim

x→∞

log (U(x))

log(x) , (17)

andupper orderofU, defined by

ν(U) := lim

x→∞

log (U(x))

log(x) , (18)

we can rewrite this characterization simply byµ(U)=ν(U).

Hence, the complement set ofM∪M±∞in the set of functionsU:R⁺→R⁺, denoted by O, can be written as

O:={U:R⁺→R⁺ ^:µ(U)<ν(U)}.

This set is nonempty:O6= ;, as we are going to see through examples.

Examples of functionsU satisfyingµ(U)<ν(U) are not well-known. A non explicit one was given by Daley (see [12], pp. 34) when considering rv’s with discrete support (see [13], pp. 831). We will provide a couple of explicit parametric examples of functions in O which include tails of distributions with discrete support. These functions can be extended easily to continuous positive functions not necessarily monotone, for instance adapting polynomials given by Karamata (see [25], pp. 70-71). These examples are more detailed in Appendix A.3.

Example1.5.

Letα>0,β∈R^{such that}β6= −1, and xa>1. Let us consider the increasing series defined by xn=x_a^(1+α)ⁿ, n≥1, well-defined because xa>1. Note that xn→ ∞as n→ ∞.

The function U defined by U(x) :=

( 1, 0≤x<x1

x^α_n⁽¹^+β⁾, x∈[xn,xn+1), ∀n≥1, (19) belongs toO, with











µ(U)=α(1+β)

1+α and ν(U)=α(1+β), if1+β>0 µ(U)=α(1+β) and ν(U)=α(1+β)

1+α , if1+β<0.

Moreover, if1+β<0, then U is a tail of distribution whose associated rv has moments lower than−α(1+β)±

(1+α).

(11)

Example1.6.

Let c>0andα∈Rsuch thatα6=0. Let(xn)n∈Nbe defined by x1=1and x_n+1=2^xⁿ^/c, n≥1, well-defined for c>0. Note that xn→ ∞as n→ ∞.

The function U defined by U(x) :=

½ 1 0≤x<x1

2^α^xⁿ x_n≤x<x_n+1, ∀n≥1, belongs toO, with







µ(U)=αc and ν(U)= ∞, ifα>0 µ(U)= −∞ and ν(U)=αc, ifα<0.

Moreover, ifα<0, then U is a tail of distribution whose associated rv has moments lower than−αc.

2 Extension of RV results

In this section, well-known results and fundamental in Extreme Value Theory, as Kara- mata’s relations and Karamata’s Tauberian Theorem, are discussed onM. A key tool for the extension of these standard results toM is the characterizations ofMgiven in Theo- rems 1.1 and 1.2.

First notice the relation between the classM introduced in the previous section and the class RV defined in (1).

Proposition 2.1. RV_ρ(ρ∈R) is a strict subset ofM.

The proof of this claim comes from the Karamata relation (see [26]) given, for all RV func- tionUwith indexρ∈R, by

x→∞lim

log (U(x))

log(x) =ρ, (20)

which implies, using Properties 1.1, (vi), thatU ∈M withM-indexκU= −ρ. Moreover, RV6=M, noticing that, fort>0, lim

x→∞

U(t x)

U(x) does not necessarily exist, whereas it does for a RV functionU. For instance the function defined onR⁺^by^U^(x)=2+sin(x), is not RV, but lim

x→∞

log (U(x))

log(x) =0, henceU∈M. 2.1 Karamata’s Theorem

We will focus on Karamata’s well-known theorem developed for RV (see [23] and e.g. [14], Theorem 1.2.1) to analyze its extension toM. Let us recall it, borrowing the version given in [14].

Theorem 2.1. Karamata’s Theorem ([23]; e.g. [14], Theorem 1.2.1) Suppose U:R⁺→R⁺is Lebesgue-summable on finite intervals. Then

(12)

(K1)

U∈RV_ρ,ρ> −1 ⇐⇒ lim

x→∞

xU(x) Rx

0 U(t)d t =ρ+1>0.

(K2)

U∈RV_ρ,ρ< −1 ⇐⇒ lim

x→∞

xU(x) R_∞

x U(t)d t = −ρ−1>0.

(K3) (i) U∈RV₋₁ =⇒ lim

x→∞

xU(x) Rx

0U(t)d t =0.

(ii) U∈RV₋₁and Z _∞

0 U(t)d t< ∞ =⇒ lim

x→∞

xU(x) R_∞

x U(t)d t =0.

Remark 2.1. The converse of (K3), (i), is false in general. A counterexample can be given by the Peter and Paul distribution which satisfies lim

x→∞

xU(x) R_∞

x U(t)d t =0but is not RV₋₁. We return to this in more detail in §2.1.2.

Theorem 2.1 is based on the existence of certain limits. We can extend some of the results toM, even when theses limits do not exist, replacing them by more general expressions.

2.1.1 Karamata’s Theorem onM

Let us introduce the following conditions, in order to state the generalization of the Kara- mata Theorem toM:

(C1r) x^rU(x) Rx

b t^r⁻¹U(t)d t ∈MwithM-index 0,i.e. lim

x→∞

Ãlog¡Rx

b t^r−1U(t)d t¢

log(x) −log (U(x)) log(x)

!

=r.

(C2r) x^rU(x) R_∞

x t^r⁻¹U(t)d t ∈M withM-index 0,i.e. lim

x→∞

Ãlog¡R_∞

x t^r−1U(t)d t¢

log(x) −log (U(x)) log(x)

!

=r.

Theorem 2.2. Generalization of the Karamata Theorem toM

Let U:R⁺→R⁺be a Lebesgue-summable on finite intervals, and b>0. We have, for r∈R^, (K1^∗)

U∈MwithM-index(−ρ)such thatρ+r>0 ⇐⇒











xlim→∞

log¡Rx

b t^r⁻¹U(t)d t¢

log(x) =ρ+r>0 U satisfies(C1r).

(K2^∗)

U∈MwithM-index(−ρ)such thatρ+r<0 ⇐⇒











x→∞lim

log¡R_∞

x t^r⁻¹U(t)d t¢

log(x) =ρ+r<0 U satisfies(C2r).

(13)

(K3^∗)

U∈MwithM-index(−ρ)such thatρ+r=0 ⇐⇒











x→∞lim log¡Rx

b t^r⁻¹U(t)d t¢

log(x) =ρ+r=0 U satisfies(C1r).

This theorem provides then a fourth characterization ofM.

Note that ifr=1, we can assumeb≥0, as in the original Karamata’s Theorem.

Remark 2.2.

1. Note that (K3^∗) provides an equivalence contrarily to (K3).

2. Assuming that U satisfies the conditions(C2r)and Z _∞

1

t^rU(t)d t< ∞, (21)

we can propose a characterization of U∈M withM-index(r+1), namely U∈MwithM-index(r+1) ⇐⇒ lim

x→∞

log¡R_∞

x t^rU(t)d t¢ log(x) =0.

This is the generalization of (K3) in Theorem 2.1, providing not only a necessary con- dition but also a sufficient one for U to belong toM, under the conditions(C2r)and (21).

2.1.2 Illustration using Peter and Paul distribution

The Peter and Paul distribution is a typical example of a function which is not RV. It is defined by (see e.g. [21], pp. 440, [17], pp. 50, [16], pp. 82, or [30], pp. 101)

F(x) :=1− X

k≥1: 2^k>x

2^−k, x>0. (22)

Let us illustrate the characterization theorems when applied to the Peter and Paul distribution; we do it for instance for Theorems 1.1 and 2.2, proving that this distribution belongs toM.

Proposition 2.2.

The Peter and Paul distribution does not belong to RV, but toMwithM-index1.

This proposition can be proved using Theorem 1.1 or Theorem 2.2. To illustrate the application of these two theorems, we develop the proof here and not in the appendix.

(14)

(i) Application of Theorem 1.1

Forx∈[2ⁿ; 2ⁿ⁺¹) (n≥0), we have, using (22),F(x)= X

k≥n+1

2⁻^k =2⁻ⁿ, from which

we deduce that n n+1≤ −

log³ F(x)´

log(x) <1, hence lim

x→∞

log³ F(x)´

log(x) = −1, which by The- orem 1.1 is equivalent to

F∈M with M−index 1.

(ii) Application of Theorem 2.2 Let us prove that

x→∞lim log³Rx

b F(t)d t´ log(x) =0.

Suppose 2ⁿ≤x<2ⁿ⁺¹and considera∈Nsuch thata<n. Choose w.l.o.g.b=2^a. Then the Peter and Paul distribution (22) satisfies

Z x b

F(t)d t=

n−1

X

k=a

Z 2^k+1 2^k

F(t)d t+ Z x

2ⁿ

F(t)d t=

n−1

X

k=a

2⁻^k(2^k⁺¹−2^k)+(x−2ⁿ)2⁻ⁿ=n−a+x2⁻ⁿ−1.

Hence

log(n−a+x2⁻ⁿ−1) (n+1) log(2) ≤

log³ Rx

b F(t)d t´

log(x) ≤log(n−a+x2⁻ⁿ−1) nlog(2) ,

and, since 1≤2⁻ⁿx<2, we obtain lim

x→∞

log³Rx

Moreover, we have

x→∞lim log

µ

xF(x) Rx

bF(t)d t

¶

log(x) =1+ lim

x→∞

log³ F(x)´ log(x) − lim

x→∞

log³ Rx

Theorem 2.2 allows one then to conclude that F∈MwithM-index 1. 2 Note that the original Karamata Theorem (Theorem 2.1) does not allow one to prove that the Peter and Paul distribution is RV or not, since the converse of (i) in (K3) does not hold, contrarily to Theorem 2.2. Indeed, although we can prove that

xlim→∞

x F(x) Rx

b F(t)d t = lim

x,n→∞

x2⁻ⁿ

n−a+x2⁻ⁿ−1=0, Theorem 2.1 does not imply thatFis RV₋1.

(15)

2.2 Karamata’s Tauberian Theorem

Let us recall Karamata’s well-known Tauberian Theorem which deals on Laplace-Stieltjes (L-S) transforms and RV functions.

The L-S transform of a positive, right continuous functionU with supportR⁺and with local bounded variation, is defined by

Ub(s) := Z

(0;∞)

e⁻^xsdU(x), s>0. (23)

Theorem 2.3. Karamata’s Tauberian Theorem (see [24])

If U is a non-decreasing right continuous function with supportR⁺and satisfying U(0⁺)= 0, with finite L-S transformU , then, forb α>0,

U∈RV_α at infinity ⇐⇒ Ub∈RV_α at0⁺.

Now we present the main result of this subsection which extends only partly the Karamata Tauberian Theorem toM.

Theorem 2.4.

Let U be a continuous function with supportR⁺and local bounded variation, satisfying U(0⁺)=0. Let g be defined onR⁺by g(x)=1/x. Then, for anyα>0,

(i) U∈M withM-index(−α) =⇒ Ub◦g∈M withM-index(−α).

(ii)

½

Ub◦g∈M withM-index(−α)

and∃η∈[0;α) : x^−ηU(x)concave =⇒ U∈M withM-index(−α).

3 Conclusion

We introduced a new class of positive functions with supportR⁺, denoted byM, strictly larger than the class of RV functions at infinity. We extended toM some well-known results given on RV class, which in particular will help to expand EVT beyond RV. This class satisfies a number of algebraic and characteristic properties, and its membersUare characterized by a unique real number, called theM-indexκU. Extensions toM of the Karamata Theorems were discussed. Four characterizations ofMwere provided, one of them being the extension toMof Karamata’s well-known theorem restricted to RV class.

Furthermore, the casesκU = ∞andκU = −∞were analyzed and their corresponding classes, denoted byM∞andM−∞respectively, were identified and studied, as done for M. The three sets M∞, M−∞and M are disjoint. Explicit examples of functions not belonging toM∪M±∞were given.

Note that any result obtained here can be applied to functions with finite support, i.e.

finite endpointx^∗, by using the change of variabley=1/(x^∗−x) forx<x^∗.

This new class seems promising in terms of applications. Several have already been developed, as the ones mentioned in the introduction (see [10], [9]). Note also a study com- paring the various extensions of the RV class, including this new class (see [8]).

Further investigation will concern a multivariate version ofM.

(16)

Acknowledgments

Meitner Cadena acknowledges the financial support of SWISS LIFE through its ESSEC research program. Partial support from RARE-318984 (an FP7 Marie Curie IRSES Fellow- ship) is also kindly acknowledged.

References

[1] S. ALJAN ˇCI ´C, D. ARANDELOVI ´C, O-regularly varying functions. Publications de l’Institut Mathématique22, (1977) 5-22.

[2] V. AVAKUMOVI ´C, On a O-inverse theorem (in Serbian).Rad Jugoslovenske Akademije Znanosti i Umjetnosti, t. 254 (Razreda Matematiˇcko-Prirodoslovnoga)79, (1936) 167- 186.

[3] N. H. BINGHAM, C. M. GOLDIE, Extensions of Regular Variation, I: Uniformity and Quatifiers.Proceedings London Mathematical Societys3-44, (1982) 473-496.

[4] N. H. BINGHAM, C. M. GOLDIE, E. OMEY, Regularly varying probability densities.

Publications de l’Institut Mathématique80, (2006) 47-57.

[5] N. H. BINGHAM, C. M. GOLDIE, J. L. TEUGELS, Regular Variation.Cambridge Uni- versity Press(1989).

[6] N. H. BINGHAM, A. J. OSTASZEWSKI, Beurling slow and regular variation.Transac- tions of the London Mathematical Society1, (2014) 29-56.

[7] R. BOJANIC, J. KARAMATA, On a Class of Functions of Regular Asymptotic Behavior.

Mathematical Research Centre, U.S. Army, Madison, Wis., Tech. Summary Rep. No.

436, (1963).

[8] M. CADENA, Revisiting extensions of the class of regularly varying functions.

ArXiv:1502.06488v2 [math.CA], (2015).

[9] M. CADENA, A simple estimator for theM-index of functions inM.Hal-01142162 (2015).

[10] M. CADENA, M. KRATZ, New results for tails of probability distributions according to their asymptotic decay.ArXiv, (2015).

[11] D. B. H. CLINE, Intermediate Regular andΠVariation.Proceedings London Mathe- matical Societys3-68, (1994) 594-616.

[12] D. J. DALEY, The Moment Index of Minima.J. Appl. Probab.38, (2001) 33-36.

[13] D. J. DALEY, C. M. GOLDIE, The moment index of minima (II).Stat. & Probab. Letters 76, (2006) 831-837.

[14] L. DEHAAN, On regular variation and its applications to the weak convergence of sample extremes.Mathematical Centre Tracts,32(1970).

(17)

[15] L.DEHAAN, A. FERREIRA, Extreme Value Theory. An Introduction.Springer, (2006).

[16] P. EMBRECHTS, C. KLÜPPELBERG, T. MIKOSCH, Modelling Extremal Events for Insur- ance and Finance.Springer Verlag(1997).

[17] P. EMBRECHTS, E. OMEY, A property of longtailed distributions.J. Appl. Probab.21, (1984) 80-87.

[18] W. FELLER, An introduction to probability theory and its applications. Vol II.J. Wiley

& Sons(1966).

[19] W. FELLER, One-sided Analogues of Karamata’s Regular Variation.L’Enseignement Mathématique15, (1969) 107-121.

[20] B. GNEDENKO, Sur La Distribution Limite Du Terme Maximum D’Une Série Aléa- toire.Ann. Math.44, (1943) 423–453.

[21] C. M. GOLDIE, Subexponential distributions and dominated-variation tails.J. Appl.

Probab.15, (1978) 440-442.

[22] J. HADAR, W. R. RUSSELL, Stochastic Dominance and Diversification.J. Econ. Theory 3, (1971) 288-305.

[23] J. KARAMATA, Sur un mode de croissance régulière des fonctions.Mathematica (Cluj) 4, (1930) 38-53.

[24] J. KARAMATA, Neuer Beweis und Verallgemeinerung der Tauberschen Sätze, welche die Laplacesche und Stieltjessche Transformation betreffen. J. R. A. Math. 1931, (1931) 27-39.

[25] J. KARAMATA, Sur le rapport entre les convergences d’une suite de fonctions et de leurs moments avec application à l’inversion des procédés de sommabilité.Studia Math.3, (1931) 68-76.

[26] J. KARAMATA, Sur un mode de croissance régulière. Théorèmes fondamentaux.Bul- letin SMF61, (1933) 55-62.

[27] J. KARAMATA, Bemerkung über die vorstehende Arbeit des Herrn Avakumovi´c mit, näherer Betrachtung einer Klasse von Funktionen, welche bei den Inversionssätzen vorkommen.Bulletin International de l’Académie Yougoslave29-30, (1935) 117-123.

[28] R. MALLER, A note on Karamata’s generalised regular variation.Journal of the Aus- tralian Mathematical Society24, (1977) 417-424.

[29] W. MATUSZEWSKA, A remark on my paper ‘Regularly increasing functions in connec- tion with the theory ofL^∗φ-spaces’.Studia Mathematica25, (1965) 265-269.

[30] T. MIKOSCH, Non-Life Insurance Mathematics. An Introduction with Stochastic Pro- cesses.Springer(2006).

[31] S. I. RESNICK, Extreme Values, Regular Variation, and Point Processes. Springer- Verlag(1987).

(18)

[32] S. I. RESNICK, On the Foundations of Multivariate Heavy-Tail Analysis. J. Appl.

Probab.41, (2004) 191-212.

[33] E. SENETA, Regularly Varying Functions. Lecture Notes in Mathematics. Springer (1976).

[34] M. SHAKED, J. G. SHANTHIKUMAR, Stochastic Orders.Springer(2007).

A Proofs of results given in Section 1

A.1 Proofs of results concerningM

Proof of Theorem 1.1. The sufficient condition given in Theorem 1.1 comes from Proper- ties 1.1, (vi). So it remains to prove its necessary condition, namely that

x→∞lim −log (U(x))

log(x) = −ρU, (24)

forU∈M with finiteρUdefined in (2).

Let²>0 and defineV by

V(x)=

½ 1, 0<x<1 x^ρ^U^+², x≥1

Applying Example 1.1 withα=ρU+²with²>0 implies thatρV =ρU+², henceρV >ρU. Using Properties 1.1, (ii), provides then that

x→∞lim U(x) V(x) = lim

x→∞

U(x) x^ρ^U^+² =0, so, forn∈N^∗, there existsx0>1 such for allx≥x0,

U(x) x^ρ^U^+²≤ 1

n, i.e. n U(x)≤x^ρ^U^+².

Applying the logarithm function to this last inequality and dividing it by−log(x),x≥x₀, gives−log(n)

log(x)−log(U(x))

log(x) ≥ −ρU−², hence−log(U(x))

log(x) ≥ −ρU−², and then lim

x→∞−log(U(x))

log(x) ≥ −ρU−².

We consider now the function

W(x)=

½ 1, 0<x<1 x^ρ^U^−², x≥1

with²>0 and proceed in the same way to obtain that, for any²>0, lim

x→∞−log(U(x)) log(x) ≤

−ρU+². Hence,∀²>0, we have

−ρU−²≤ lim

x→∞−log(U(x)) log(x) ≤ lim

x→∞−log(U(x))

log(x) ≤ −ρU+² from which the result follows taking²arbitrary.

(19)

Now we introduce a lemma, on which the proof of Theorem 1.2 will be based.

Lemma A.1. Let U∈M with associatedM-indexκUdefined in (4). Then necessarilyκU=

−ρU, whereρU is defined in (2).

Proof of Lemma A.1. LetU∈M withM-indexκU given in (4) andρU defined in (2). By Theorem 1.1, we have lim

x→∞

log(U(x)) log(x) =ρU.

Hence, for all²>0 there existsx₀>1 such that, forx≥x₀,U(x)≤x^ρ^U^+².

Multiplying this last inequality byx^r⁻¹,r∈R, and integrating it on [x0;∞), we obtain Z _∞

x0

x^r⁻¹U(x)d x≤ Z _∞

x0

x^ρ^U^+²+^r⁻¹d x

which is finite ifr< −ρU−². Taking²↓0 then the supremum onr leads toκU= −ρU. Proof of Theorem 1.2.

The necessary condition is proved by Lemma A.1. The sufficient condition follows from the assumption thatρU satisfies (2).

Proof of Theorem 1.3.

• Proof of (i)

ForU∈M, Theorems 1.1 and 1.2 give that

xlim→∞−log(U(x))

log(x) = −ρU=κU withρU defined in (2) andκU in (4). (25) Introducing a functionγsuch that

x→∞lim γ(x)=0, (26)

we can write, for someb>1, applying the L’Hôpital’s rule to the ratio,

x→∞lim



γ(x)+ Rx

b

log(U(t)) log(t)

d t t

log(x)



= lim

x→∞

log(U(x))

log(x) = −κU. (27)

. SupposeκU6=0. Then we deduce from (25) and (27), that

x→∞lim

log(U(x)) γ(x) log(x)+Rx

b

log(U(t))

tlog(t) d t =1. (28)

Hence, defining the function²U(x) := log(U(x)) γ(x) log(x)+Rx

b

log(U(t)) tlog(t) d t

, forx≥b, we can expressU, forx≥b, as

U(x)=exp

½

αU(x)+²U(x) Z x

b

βU(t) t d t

¾

(20)

where αU(x) :=²U(x)γ(x) log(x) and βU(x) :=log(U(x))

log(x) . (29) It is then straightforward to check that the functionsαU, βU and²U satisfy the conditions given in Theorem 1.3. Indeed, by (26) and (28), lim

x→∞

αU(x) log(x) =

x→∞lim ²U(x)γ(x)=0. Using (25), we obtain lim

x→∞βU(x)= lim

x→∞

log(U(x))

log(x) = −κU= ρU. Finally, by (28), we have lim

x→∞²U(x)=1 . . Now supposeκU=0.

We want to prove (8) for some functionsα,β, and²satisfying (7).

Notice that (25) withκU=0 allows one to write that lim

x→∞

log(x U(x)) log(x) =1.

So applying Theorem 1.1 to the functionV defined byV(x)=xU(x), gives that V ∈M withρV = −κV =1. SinceκV 6=0, we can proceed in the same way as previously, and obtain a representation forVof the form (8), namely, ford>1,

∀x≥d,

V(x)=exp

½

αV(x)+²V(x) Z x

d

βV(t) t d t

¾

whereαV, βV,²V satisfy the conditions of Theorem 1.3 andβV = log(V(x)) log(x) (see (29)). Hence we have, forx≥d,

U(x) = V(x) x =exp

½

−log(x)+αV(x)+²V(x) Z x

d

log(t U(t)) t log(t) d t

¾

= exp

½

αV(x)+(²V(x)−1) log(x)−²V(x) log(d)+²V(x) Z x

d

log(U(t)) t log(t) d t

¾ .

Noticing that lim

x→∞

αV(x)+(²V(x)−1) log(x)−²V(x) log(d)

log(x) =0, we obtain that

U satisfies (8) when setting, for x ≥d, αU(x) :=αV(x)+(²V(x)−1) log(x)−

²V(x) log(d),βU(x) :=log(U(x))

log(x) and²U:=²V.

• Proof of (ii)

LetU be a positive function with supportR⁺, bounded on finite intervals. Assume thatUcan be expressed as (8) for some functionsα,β, and²satisfying (7). We are going to check the sufficient condition given in Properties 1.1, (vi), to prove that U∈M.

Sincelog(U(x))

log(x) = α(x) log(x)+²(x)

Rx b β(t)

t d t

log(x) and that, via L’Hôpital’s rule,

xlim→∞

Rx b β(t)

t d t log(x) = lim

x→∞

β(x)/x 1/x = lim

x→∞β(x), then using the limits ofα,β, and²allows one to conclude.

(21)

Proof of Properties 1.1.

• Proof of (i)

Let us prove this property by contradiction.

Suppose there existρandρ⁰, withρ⁰<ρ, both satisfying (2), forU∈M. Choosing

²=(ρ−ρ⁰)/2 in (2) gives

xlim→∞

U(x)

x^ρ⁰^+² =0 and lim

x→∞

U(x) x^ρ−² = lim

x→∞

U(x) x^ρ⁰^+² = ∞, hence the contradiction.

• Proof of (ii)

Choosing²=(ρU−ρV)/2, we can write V(x)

U(x)= V(x) x^ρ^V^+²

x^ρ^V^+²

U(x) = V(x) x^ρ^V^+²

µU(x) x^ρ^U^−²

¶₋1

, from which we deduce (ii).

• Proof of (iii)

LetU,V ∈M,a>0,²>0 and suppose w.l.o.g. thatρU≤ρV. SinceρV−ρU>0, writing aU(x)

x^ρ^V^±² = a x^ρ^V^−ρ^U

U(x)

x^ρ^U^±² gives lim

x→∞

aU(x)+V(x)

x^ρ^V^+² =0 and

x→∞lim

aU(x)+V(x)

x^ρ^V^−² = ∞, we conclude thus thatρaU+V =ρU∨ρV.

• Proof of (iv)

It is straightforward since (2) can be rewritten as

x→∞lim 1/U(x)

x^−ρ^U^−² = ∞ and lim

x→∞

1/U(x) x^−ρ^U^+² =0.

• Proof of (v)

First, let us considerU∈MwithρU< −1.

Choosing²0= −(ρU+1)/2 (>0) in (2) implies that there existC>0 andx0>1 such that, forx≥x0,U(x)≤C x^ρ^U^+²⁰=C x^(ρ^U^−1)/2, from which we deduce that

Z _∞

x₀ U(x)d x< ∞.

We conclude that Z _∞

0

U(x)d x< ∞becauseU is bounded on finite intervals.

Now suppose thatρU> −1.

Choosing²0=(ρU+1)/2 (>0) in (2) gives that forC>0 there existsx₀>1 such that, forx≥x0,U(x)≥C x^(ρ^U^−1)/2

Z _∞

0

U(x)d x≥ Z _∞

x0

U(x)d x≥ ∞.