PUBLICATIONS DE L’INSTITUT MATH´EMATIQUE
Nouvelle s´e rie,tome80(94)(2006),171–192DOI:10.2298/PIM0694171H
REGULARLY V ARYING FUNCTIONS
Anders Hedegaard Jessen and Thomas Mikosch
In memoriam Tatjana Ostrogorski.
Abstract.We consider some elementary functions of the components of a
regularly varying random vector such as linear combinations,products,min-
ima,maxima,order statistics,powers.We give conditions under which these
functions are again regularly varying,possibly with a different index.
1.Introduction
Regular variation is one of the basic concepts which appears in a natural way in different contexts of appl
ied probability theory.Feller’s[21]monograph has certainly contributed to the propagation of regular variation in the context of limit theory for sums of iid random variables.Resnick[50,51,52]popularized the notion of multivariate regular variation for multivariate extreme value theory.Bingham et al.[3]is an encyclopedia where onefinds many analytical results related to one-dimensional regular variation.Kesten[28]and Goldie[22]studied regular variation of the stationary solution to a stochastic recurrence equation.These resultsfind natural applications infinancial time series analysis,see Basrak et al.
[2]and Mikosch[39].Recently,regular variation has become one of the key notions for modelling the behavior of large telecommunications networks,Leland et al.[35],Heath et al.[23],Mikosch et al.[40].
It is the aim of this paper to review some known results on basic functions acting on regularly varying random variables and random vectors such as sums, products,linear combinations,maxima and minima,and powers.These results are often useful in applications related to time series analysis,risk management, insurance and telecommunications.Most of the results belong to the folklore but they are often wide spread over the literature and not always easily accessible.We will give references whenever we are aware of a proved result and give proofs if this is not the case.
2000Mathematics Subject Classification:Primary60G70;Secondary62P05.
Mikosch’s research is also partially supported by the Danish Research Council(SNF)Grant 21-04-0400.Jessen’s research is partly supported by a grant from CODAN Insurance.
171
172JESSEN AND MIKOSCH
We focus on functions offinitely many regularly varying random variables. With a few exceptions(the tail of the marginal distribution of a linear process, functionals with a random index)we will not consider results where an increasing or an infinite number of such random variables or vectors is involved.We exclude distributional limit for partial sums and maxima of iid and strictly stationary sequences,tail probabilities of subadditive functionals acting on a regu-larly varying random uin probabilities)and heavy-tailed large deviation results,tails of solutions to stochastic recurrence equations.
We start by introducing the notion of a multivariate regularly varying vector in Section2.Then we will consider sum-type functionals of regularly varying vectors in Section3.Functionals of product-type are considered in Section4.In Section5 wefinally study order statistics and powers.
2.Regularly varying random vectors
In what follows,we will often need the notion of a regularly varying random vector and its properties;we refer to Resnick[50]and[51,Section5.4.2].This notion was further developed by Tatjana Ostrogorski in a series of papers,see [42,43,44,45,46,47].
Definition2.1.An R d-valued random vector X and its distribution are said to be regularly varying with limiting non-null Radon measureµon the Borelσ-field B(R d0)of R d0=R d {0}if
(2.1)P(x−1X∈·)
P(|X|>x)
v→µ,x→∞.
Here|·|is any norm in R d and v→refers to vague convergence on B(R d0).
Sinceµnecessarily has the propertyµ(t A)=t−αµ(A),t>0,for someα>0 and all Borel sets A in R d0,we say that X is regularly varying with indexαand limiting measureµ,for short X∈RV(α,µ).If the limit measureµis irrelevant we also write X∈RV(α).Relation(2.1)is often used in different equivalent disguis
es. It is equivalent to the sequential definition of regular variation:there exist c n→∞such that n P(c−1n X∈·)v→µ.One can always choose(c n)increasing and such
that n P(|X|>c n)∼1.Another aspect of regular variation can be seen if one switches in(2.1)to a polar coordinate representation.Writing x=x/|x|for any x=0and S d−1={x∈R d:|x|=1}for the unit sphere in R d,relation(2.1)is equivalent to
(2.2)P(|X|>x t, X∈·)
P(|X|>x)
w→t−αP(Θ∈·)for all t>0,
whereΘis a random vector assuming values in S d−1and w→refers to weak conver-gence on the Borelσ-field of S d−1.
Plugging the set S d−1into(2.2),it is straightforward that the norm|X|is regularly varying with indexα.
REGULARLY V ARYING FUNCTIONS173 The special case d=1refers to a regularly varying random variable X with indexα 0:
(2.3)P(X>x)∼p x−αL(x)and P(X −x)∼q x−αL(x),p+q=1, where L is a slowly varying ,L(cx)/L(x)→1as x→∞for every c>0.Condition(2.3)is also referred to as a tail balance condition.The cases p=0or q=0are not excluded.Here and in what follows we write f(x)∼g(x)as x→∞if f(x)/g(x)→1or,if g(x)=0,we interpret this asymptotic relation as f(x)=o(1).
3.Sum-type functions
3.1.Partial sums of random variables.Consider regularly varying ran-dom variables X1,X2,...,possibly with different indices.We write
S n=X1+···+X n,n 1,
for the partial sums.In what follows,we write G=1−G for the right tail of a distribution function G on R.
Lemma3.1.Assume|X1|is regularly varying with indexα 0and distribution function F.Assume X1,...,X n are random variables satisfying
(3.1)lim
x→∞P(X i>x)
F(x)
=c+
i
and lim
x→∞正则化长波方程
P(X i −x)
F(x)
=c−
i
,i=1,...,n,
for some non-negative numbers c±
i
and
lim x→∞P(X i>x,X j>x)
F(x)
=lim
x→∞
P(X i −x,X j>x)
F(x)
=lim
x→∞
P(X i −x,X j −x)
F(x)
=0,i=j.
(3.2) Then
lim x→∞P(S n>x)
F(x)
=c+1+···+c+n and lim
x→∞
P(S n −x)
F(x)
=c−1+···+c−n.
In particular,if the X i’s are independent non-negative regularly varying random variables then
(3.3)P(S n>x)∼P(X1>x)+···+P(X n>x).
The proof of(3.3)can be found in Feller[21,p.278],cf.Embrechts et al. [18,Lemma1.3.1].The general case of possibly dependent non-negative X i’s was proved in Davis and Resnick[14,Lemma2.1];the extension to general X i’s follows along the lines of the proof in[14].Generalizations to the multivariate case are given in Section3.6below.
The conditions in Lemma3.1are sharp in the sense that they cannot be sub-
stantially improved.A condition like(3.1)with not all c±
i ’s vanishing is needed
in order to ensure that at least one summand X i is regularly varying.Condition (3.2)is a so-called asymptotic independence condition.It cannot be avoided as the
174JESSEN AND MIKOSCH
trivial example X2=−X1for a regularly varying X1shows.Then(3.1)holds but
(3.2)does not and S2=0a.s.
A partial converse follows from Embrechts et al.[17].
Lemma3.2.Assume S n=X1+···+X n is regularly varying with indexα 0 and X i are iid non-negative.Then the X i’s are regularly varying with indexαand (3.4)P(S n>x)∼n P(X1>x),n 1.
Relation(3.4)can be taken as the definition of a subexponential distribution. The class of those distributions is larger than the class of regularly varying distri-butions,see Embrechts et al.[18,Sections1.3,1.4and Appendix A3].Lemma3.2 remains valid for subexponential distributions in the sense that subexponentiality of S n implies subexponentiality of X1.This property is referred to as convolution root closure of subexponential distributions.
Proof.Since S n is regularly varying it is subexponential.Then the regular variation of X i follows from the convolution root closure of subexponential dis-tributions,see Proposition A3.18in Embrechts et al.[18].Relation(3.4)is a consequence of(3.3).
An alternative proof is presented in the proof of Proposition4.8in Fa¨y et al.[20].It strongly depends on the regular variation of the X i’s:Karamata’s Tauberian theorem(see Feller[21,XIII,Section5])is used.
In general,one cannot conclude from regular variation of X+Y for independent X and Y that X and Y ar
e regularly varying.For example,if X+Y has a Cauchy distribution,in particular X+Y∈RV(1),then X can be chosen Poisson,see Theorem6.3.1on p.71in Lukacs[37].It follows from Lemma3.12below that Y∈RV(1).
3.2.Weighted sums of iid regularly varying random variables.We assume that(Z i)is an iid sequence of regularly varying random variables with indexα 0and tail balance condition(2.3)(applied to X=Z i).Then it follows from Lemma3.1that for any real constantsψi
P(ψ1Z1+···+ψm Z m>x)∼P(ψ1Z1>x)+···+P(ψm Z1>x).
Then evaluating P(ψi Z1>x)=P(ψ+
i Z+
i
>x)+P(ψ−
i
Z−
i
>x),where x±=
0∨(±x)we conclude the following result which can be found in various Embrechts et al.[18,Lemma A3.26].
Lemma3.3.Let(Z i)be an iid sequence of regularly varying random variables satisfying the tail balance condition(2.3).Then for any real constantsψi and m 1,
(3.5)P(ψ1Z1+···+ψm Z m>x)∼P(|Z1|>x)
m
i=1
p(ψ+
i
)α+q(ψ−
i
)α
.
The converse of Lemma3.3is in general ,regular variation of ψ1Z1+···+ψm Z m with indexα>0for an iid sequence(Z i)does in general
REGULARLY V ARYING FUNCTIONS 175
not imply regular variation of Z 1,an exception being the case m =2with ψi >0,Z i  0a.s.,i =1,2,cf.Jacobsen et al.[27].
3.3.Infinite series of weighted iid regularly varying random vari-ables.The question about the tail behavior of an infinite series
(3.6)X =∞
i =0ψj Z j
for an iid sequence (Z i )of regularly varying random variables with index α>0and real weights occurs for example in the context of extreme value theory for linear processes,including ARMA and FARIMA processes,see Davis and Resnick
[11,12,13],Kl¨u ppelberg and Mikosch [29,30,31],cf.Brockwell and Davis
[5,Section 13.3],Resnick [51,Section 4.5],Embrechts et al.[18,Section 5.5and Chapter 7].
The problem about the regular variation of X is only reasonably posed if the infinite series (3.6)converges a.s.Necessary and sufficient conditions are given by Kolmogorov’s 3-series theorem,cf.Petrov [48,49].For example,if α>2(then var(Z i )<∞),the conditions E (Z 1)=0and  i ψ2i <∞are necessary and sufficient for the vergence of X .
The following conditions from Mikosch and Samorodnitsky [41]are best pos-sible in the sense that the conditions on (ψi )coincide with or are close to the con-ditions in the 3-series theorem.Similar results,partly under stronger conditions,can be found in Lemma 4.24of Resnick [51]for α 1(attributed to Cline [7,8]),Theorem 2.2in Kokoszka and Taqqu [32]for α∈(1,2).
Lemma 3.4.Let (Z i )be an iid sequence of regularly varying random variables with index α>0which sati
sfy the tail balance condition (2.3).Let (ψi )be a sequence of real weights.Assume that one of the following conditions holds:(1)α>2,E (Z 1)=0and  ∞i =0ψ2i <∞.(2)α∈(1,2],E (Z 1)=0and  ∞i =0|ψi |α−ε<∞for some ε>0.(3)α∈(0,1]and  ∞i =0|ψi |α−ε<∞for some ε>0.
Then
(3.7)P (X >x )∼P (|Z 1|>x )∞ i =0
p (ψ+i )α+q (ψ−i )α .
The conditions on (ψj )in the case α∈(0,2]can be slightly relaxed if one knows more about the slowly varying L .In this case the following result from Mikosch and Samorodnitsky [41]holds.
Lemma 3.5.Let (Z i )be an iid sequence of regularly varying random variables with index
α∈(0,2]which satisfy the tail balance condition (2.3).Assume that  ∞i =1|ψi |α<∞,that the infinite series (3.6)converges a.s.and that one of the
following conditions holds:
(1)There exist constants c,x 0>0such that L (x 2) c L (x 1)for all x 0<
x 1<x 2.
176JESSEN AND MIKOSCH
(2)There exist constants c,x 0>0such that L (x 1x 2) c L (x 1)L (x 2)for all
x 1,x 2 x 0>0
Then (3.7)holds.
Condition (2)holds for Pareto-like tails P (Z 1>x )∼c x −α,in particular for α-stable random variables Z i and for student distributed Z i ’s with αdegrees of freedom.It is also satisfied for L (x )=(log k x )β,any real β,where log k is the k th time iterated logarithm.
Classical time series analysis deals with the strictly stationary linear processes
X n =∞
i =0ψi Z n −i ,n ∈Z ,
where (Z i )is an iid white noise sequence,cf.Brockwell and Davis [5].In the case of regularly varying Z i
’s with α>2,var(Z 1)and var(X 1)are finite and there-fore it makes sense to define the autocovariance function γX (h )=cov(X 0,X h )=var(Z 1) i ψi ψi +|h |,h ∈Z .The condition  i ψ2i <∞(which is necessary for the vergence of X n )does not only capture short range dependent sequences
(such as ARMA processes for which γX (h )decays exponentially fast to zero)but also long range dependent sequences (X n )in the sense that  h |γX (h )|=∞.
Thus Lemma 3.4also covers long range dependent sequences.The latter class in-cludes fractional ARIMA processes;cf.Brockwell and Davis [5,Section 13.2],and Samorodnitsky and Taqqu [56].
Notice that (3.7)is the direct analog of (3.5)for the truncated series.The proof of (3.7)is based on (3.5)and the fact that the remainder term  ∞i =m +1ψi Z i is negligible compared to P (|Z 1|>x )when first letting x →∞and then m →∞.More generally,the following result holds:
Lemma 3.6.Let A be a random variable and let Z be positive regularly varying random variable with index α 0.Assume that for every m  0there exist finite positive constants c m >0,random variables A m and B m such that the representa-
tion A d =A m +B m holds and the following three conditions are satisfied:
P (A m >x )∼c m P (Z >x ),
as x →∞,c m →c 0,
as m →∞,lim m →∞lim sup x →∞P (B m >x )P (Z >x )
=0and A m ,B m are independent for every m  1or lim m →∞lim sup x →∞P (|B m |>x )P (Z >x )
=0.Then P (A >x )∼c 0P (Z >x ).
Proof.For every m  1and ε∈(0,1).
P (A >x ) P (A m >x (1−ε))+P (B m >εx ).

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