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) := tα, α ∈ R, we have g(x) = xα. 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) :=xα 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 byxα 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) =xα. (2.2)
We write f ∈RVα and call α the index of regular variation. If α = 0, we say f is slowly varying. Note that for any f ∈RVα we have f(x) =xαL(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.
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) =xα 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 tolimx→∞l(x+h)−l(x)→0 for l(x) := log(L(ex)) 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)].
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 ∈RVα for α∈R, then
t→∞lim f(tx)
f(t) =xα 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 ∈RVα. 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 ∈RVα+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 ∈ RVα; if (2.7) holds with
−∞< α <−1, then f ∈RVα.
Proof. We follow the arguments stated in de Haan and Ferreira (2006). First, we
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 Hence, we conclude that
F(tx)
F(t) <(1 + 2ε)xα+1
for t sufficiently large. A lower inequality can be derived by similar arguments such thatF ∈RVα+1 forα >−1 follows.
In case α = −1 and F(t) → ∞ similar arguments apply. If α = −1 and
limt→∞F(t)<∞, we directly obtain F ∈RV0. f(s)≤2−1−δf(s) for s sufficiently large. Then, by similar arguments, we obtain
Z ∞
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
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 ∈RVα. 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 ∈RVα. We have f(tx)
f(t) = c(tx) c(t) exp
Z tx t
a(s) s ds
for t, tx > t0.
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
=xα ast → ∞.
Together with the convergence of the function cfor t→ ∞ regular variation off follows.
For the converse, observe that if f ∈ RVα, 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) bytα0c(t). Thus, it suffices to prove the assertion for slowly varying functions.
Now, letf ∈ RV0 and considerh(t) := log(f(et)). 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(et), ν(t) =a(et) 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(et)), 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.
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 ∈RVα, 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 ∈ RVmax(α1,α2). If moreover limt→∞f2(t) = ∞, then the composition f1◦f2 ∈RVα1α2.
(iii) If f ∈ RVα 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 ∈ RVα is integrable on finite intervals of R+ and α ≥ −1, then F(s) := Rt
0 f(s)ds ∈RVα+1.
(v) (Potter 1942) Suppose f ∈RVα. Then, for arbitrary δ1, δ2 >0 there exists t0 =t0(δ1, δ2) such that for t ≥t0, tx≥t0,
(1−δ1)xαmin(xδ2, x−δ2)< f(tx)
f(t) <(1 +δ1)xαmax(xδ2, 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 ∈RVα. 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.
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 limt→∞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(xδ2, x−δ2)< f(tx)
f(t) <(1 +δ1) max(xδ2, x−δ2).
for any t, tx > t0.
Closely related to Potter’s bounds are weighted uniform approximations of devia-tion of regularly varying funcdevia-tions from corresponding power laws.
Theorem 2.5 (de Haan and Ferreira (2006), Drees (1998)).
If f ∈RVα, for each ε, δ >0 there is a t0 =t0(ε, δ) such that for t, tx≥t0,
f(tx) f(t) −xα
≤εmax xα+δ, xα−δ
. (2.18)
Proof. We follow de Haan and Ferreira (2006), page 369-370. Again, since for anyf ∈RVα we have f(x) =xαL(x) with L is slowly varying, it suffices to prove the assertion for α = 0. First, note that for δ > 0, max xδ, 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 xδ. 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.