On the order of functions at infinity

J. Math. Anal. Appl. 452 (2017) 109–125

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Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa

On the order of functions at infinity Meitner Cadena a , Marie Kratz b,1 , Edward Omey c a b c

Escuela Politécnica Nacional, Quito, Ecuador ESSEC Business School, CREAR, France KU Leuven at Campus Brussels, Belgium

a r t i c l e

i n f o

Article history: Received 10 May 2016 Available online 27 February 2017 Submitted by U. Stadtmueller Keywords: Karamata’s theorem Karamata’s Tauberian theorem Regular variation Representation theorems

a b s t r a c t We define a new class of positive and measurable functions in terms of their asymptotic behavior at infinity. This new class extends the class of regularly varying functions, for broader applications. We provide different characterizations of the new class and consider integrals, convolutions and Laplace transforms. We give some applications in probability theory. Some natural extensions of the new class are also derived. © 2017 Elsevier Inc. All rights reserved.

0. Introduction The class of regularly varying functions has been introduced in the 30s by Karamata [23–26], who defined the notion of slowly varying and regularly varying functions. Definition. A Lebesgue-measurable function U : R+ → R+ is regularly varying (RV) at infinity with (real) index ρ, U ∈ RVρ , if it satisfies U (xt) = tρ , ∀t > 0. x→∞ U (x) lim

(1)

If ρ = 0, the function is called slowly varying (SV). Since then, much literature has been devoted to RV functions. For a survey of RV and its properties, we refer to Seneta [33], Bingham et al. [3,5] and the references therein, Geluk and de Haan [20]. RV has been extensively used in probability theory and was made popular by Feller [17]. We mention applications in Extreme Value Theory (EVT) (see e.g. Gnedenko [21], Feller [17], de Haan [15,16], Resnick [31]), where RV E-mail addresses: [email protected] (M. Cadena), [email protected] (M. Kratz), [email protected] (E. Omey). 1 Marie Kratz is also member of MAP5, UMR 8145, Univ. Paris Descartes. http://dx.doi.org/10.1016/j.jmaa.2017.02.042 0022-247X/© 2017 Elsevier Inc. All rights reserved.

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was used to characterize maximum domains of attraction. Regular variation was also used in studying sums of random variables and renewal theory. Various extensions of the RV class have been proposed. We may cite, in a non-exhaustive way, the class of Extended RV, which is implicit in the work of Matuszewska [29], the class of O-Regularly Varying functions (O-RV), defined and studied by Karamata [27] and Avakumović [2] (see also e.g. [29,18,33,1,28,12]), the classes Π and Γ (see e.g. Bingham and Goldie [3], de Haan [15]), Beurling classes (see Bingham and Ostaszewski [4]). The notion of multivariate regular variation has been developed in de Haan and Ferreira [16], Omey [30], Resnick [32]. In this paper we propose a new extension of the class RV, defined in terms of the asymptotic behavior of the functions, and for which the limit in (1) might not exist. Hence we present and characterize fully this new class, providing also examples. This new class not only extends in a simple way several main properties of RV, but also offers broader applications, e.g. in EVT, in complex analysis, or in number theory. We can mention, for instance, new results in EVT on maximum domains domain of attraction (see [11] or [9]), and the proposition of a new tail index estimator [8]. Another motivation comes from complex analysis. For an entire function f of the ∞  form f (z) = an z n , |z| < ∞, set U (r) = log M (r) where M (r) = sup {|f (z)| : |z| = r}. Functions of n=0

this type will be studied in Section 2.2, using the notions of upper and lower order of f (see [5], Chap. 7), log U (x) log U (x) and lim . Another field in which our results might also be defined, respectively, as lim x→∞ log x log x x→∞ relevant, belongs to number theory. For instance, in [19] and [35],⎧the authors study⎫ multiplicative arithmetic ⎨ ⎬ functions f and consider the function defined by U (x) = exp p−1 f (p) log p . A major condition in ⎩ ⎭ p≤x

these papers is that log U (x)/ log x converges to a finite constant. Working in our new class of functions might help revisiting these studies. The paper is organized in two main parts. In the first section, we define this large class of functions, describing it in terms of their asymptotic behaviors, which may violate (1). We provide 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. 1. A new class of functions In this section we consider positive and measurable functions U and consider the asymptotic behavior at infinity. We will use the notations a ∨ b = max(a, b) and a ∧ b = min(a, b). The largest integer not greater than x is denoted by [x] and log x is the natural logarithm of x. In what follows we will occasionally consider positive random variables X with distribution function (d.f.) F (x) = FX (x) = P (X ≤ x) and tail F (x) = 1 − F (x). 1.1. The class M 1.1.1. Definition We introduce a new class of functions M as follows. Definition. M is the class of positive and measurable functions U with support R+ , bounded on finite intervals, such that U (x) U (x) = 0 and lim ρ− = ∞. ρ+ x→∞ x x→∞ x

∃ρ ∈ R, ∀ > 0, lim

(2)

The class of functions satisfying (2) will be denoted by M(ρ). The class M is given by M = ∪ρ∈R M(ρ).

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Examples. 1. If U ∈ RVρ , then U ∈ M(ρ). Indeed, let U (x) = xρ L(x) where L ∈ SV . It is well known that for L ∈ SV , we have limx→∞ x L(x) = ∞ and limx→∞ x− L(x) = 0. Hence U ∈ M(ρ). / RV . Another examples given by U (x) = exp([log x]) 2. Let U (x) = 2 + sin x. Clearly U ∈ M(0) and U ∈ / RV (1). or U (x) = x exp(1 + sin x) show that U ∈ M(1) but U ∈ / M. 3. Let U (x) = esin(x) log(x) . Clearly U ∈ 4. Let U (x) = F (x) satisfy (2). Then ρ ≤ 0 and, for  > 0, there exist positive constants x0 , c, d such that, for x ≥ x0 , d xρ− ≤ F (x) ≤ c xρ+ . For positive real values of r, it follows that, for x ≥ x0 , x d

x t

r−1+ρ−

dt ≤

x0

x t

r−1

F (t)dt ≤ c

x0

tr−1+ρ+ dt. x0

∞ xr−1 F (x)dx < ∞ and E[X r ] < ∞.

If r + ρ < 0, then x0 ∞

xr−1 F (x)dx = ∞ and E[X r ] = ∞.

If r + ρ > 0, then x0

1.1.2. Characterizations of M The next result provides a first characterization of the class M. Theorem 1.1 (First characterization of M). Let U be a positive and measurable function with support R+ and bounded on finite intervals. Then U ∈ M(ρ)

if and only if

lim

x→∞

log U (x) = ρ. log x

(3)

Proof. First suppose that (2) holds. Using lim U (x)/xρ+ = 0, we obtain that U (x)/xρ+ is bounded, say x→∞

log U (x) ≤ ρ + . We log x x→∞

by a, for large values of x. For such x, log U (x) ≤ (ρ + ) log x + log a, hence lim sup

log U (x) ≥ ρ − . Since  is arbitrary, we obtain (3). log x For the other implication, take  > 0. We can find x0 such that, ∀x ≥ x0 ,

can prove in a similar way that lim inf

x→∞

(ρ − /2) log x ≤ log U (x) ≤ (ρ + /2) log x. It follows that, ∀x ≥ x0 , xρ−/2 ≤ U (x) ≤ xρ+/2 , hence U ∈ M(ρ).

2

Theorem 1.1 leads to the following representation theorem. Theorem 1.2 (Representation theorem). Let U be a positive and measurable function with support R+ and bounded on finite intervals. Then

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U ∈M

⎧ ⎨ if and only if

U (x) = exp

x α(x) +

⎩

a

⎫ ⎬

β(t) dt , x ≥ a > 0, ⎭ t

(4)

where α(x)/ log x → 0 and β(x) → ρ as x → ∞. Proof. That the representation in (4) implies that U ∈ M(ρ) is trivially true. Now, if U ∈ M(ρ), Theorem 1.1 implies that log U (x) = ρ log x + (x) log x with (x) → 0. x x β(t) (x) − (t) dt + α(x), where β(t) = ρ + (t) and α(x) = dt + We can write ρ log x + (x) log x = t t a

a

β(x) log a. Clearly β(x) → ρ and α(x)/ log x → 0, as x → ∞. Hence the representation (4) of U . 2 Remarks.

1. It is well known that U ∈ RVρ holds iff (4) holds with β(x) → ρ and α(x) → α, a finite constant. So it gives back that RVρ ⊂ M(ρ). Also it is known that U is in the class O − RV of O−varying functions if and only if (4) holds with α(.) and β(.) bounded functions.

 2. The function defined for x > 0 by U (x) = exp (log x)α sin((log x)β ) with 0 < α, β < 1, α + β > 1, belongs to M(0) but U ∈ / O−RV (see [7] or [9]). Another example of a function U such that U ∈ O−RV and U ∈ / M can also be found in [7] or [9]. Let us move to the third characterization of M (ρ). Theorem 1.3 (Third characterization of M). Let U be a positive and measurable function with support R+ and bounded on finite intervals. Then U ∈ M(ρ) if and only if there exist slowly varying functions L1 and U (x) U (x) → 0 and ρ → ∞ as x → ∞. L2 such that ρ x L1 (x) x L2 (x) Proof. First suppose that U ∈ M(ρ), then, via (2) (first limit), we can write U (x) = o(1)xρ+ , ∀ > 0. Using Bingham et al. ([5], Theorem 2.3.6), we obtain that there exists a slowly varying function L1 such that U (x) = o(1)xρ L1 (x) → 0 as x → ∞. Conversely, if U (x) = o(1)xρ L1 (x) for some slowly varying function L1 , then U (x) = o(1)xρ+ , ∀ > 0. If U ∈ M(ρ), then the second limit in (2) implies that V (x) = xρ /U (x) satisfies V (x) = o(1)x , ∀ > 0. We can deduce that there exists a slowly varying function L such that V (x) = o(1)L(x), and therefore that U (x) xρ L2 (x) → ∞ where L2 = 1/L. 2 x→∞

Remarks. xρ L2 (x) U (x) L2 (x) L2 (x) = −→ 0, we have −→ 0. L1 (x) U (x) xρ L1 (x) x→∞ L1 (x) x→∞ 2. Suppose that there is a slowly varying function L such that U (x)/L(x) → 0 and L(x)U (x) → ∞ as x → ∞, then U ∈ M(0). 3. Theorem 1.3 shows that for all  > 0 and C > 0, we can find x0 = x0 (, C) such that 1. Since

Cxρ L2 (x) ≤ U (x) ≤ xρ L1 (x), ∀x ≥ x0 . ∞ We deduce that for ρ < −1,

∞ U (t)dt < ∞, and if ρ > −1, then

1

U (t)dt = ∞. 1

(5)

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Definition. For U , a positive and measurable function with support R+ and bounded on finite interval, define κU by ⎧ ⎨ κU = sup

⎩

∞ r∈R:

xr−1 U (x)dx < ∞ 1

⎫ ⎬ ⎭

.

(6)

We call κU the M-index of U . It may take values ±∞, as noted by Daley (see [13] and [14]). We give another characterization of M by using this index. Theorem 1.4 (Fourth characterization of M). Let U be a positive and measurable function with support R+ and bounded on finite intervals. Then U ∈ M(ρ) if and only if κU = −ρ, where ρ satisfies (2). Proof. First assume that U ∈ M(ρ). Using the right hand side of (5), we can write, for all r such that r < −ρ, ∞

∞ x

r−1

1

U (x)dx ≤

xρ+r−1 L1 (x)dx < ∞. 1

Taking the supremum with respect to r gives κU = −ρ. The converse statement follows directly from the definition (2) of M (see Example 1.5 in [9] or [10]). 2 1.1.3. Properties of functions in the class M In the next result we summarize some of the basic properties of functions in the class M. Proposition 1.1 (Basic properties). (i) For U ∈ M(ρ), the index ρ is unique. (ii) If U ∈ M(ρ), V ∈ M(θ) and ρ > θ, then lim V (x)/U (x) = 0. x→∞

(iii) If U ∈ M(ρ), V ∈ M(θ) and a, b > 0, then aU + bV ∈ M(λ) with λ = ρ ∨ θ. (iv) If U ∈ M(ρ), V ∈ M(θ) and lim V (x) = ∞, then U ◦ V ∈ M (λ) with λ = ρθ. x→∞

(v) If U ∈ M(ρ) and V ∈ M(θ), then the product U V ∈ M(λ) with λ = ρ + θ. (vi) If U ∈ M(ρ), ρ > 0, has an asymptotic inverse V , then V ∈ M(1/ρ). Proof. (i) The property follows from Theorem 1.2. V (x) xθ+ V (x) xρ− V (x) = θ+ = θ+ → 0. (ii) Choosing  = (ρ − θ)/2, we have U (x) x U (x) x U (x) x→∞ aU (x) + bV (x) → a, hence that aU +bV ∈ (iii) Assume w.l.o.g. that ρ ≥ θ. If ρ > θ, then (ii) implies that x→∞ U (x) M(ρ). If ρ = θ, use (2). (iv) The result is straightforward when writing log U (V (x)) log V (x) log(U ◦ V (x)) = . log x log V (x) log x (v) We have

log U (x) log V (x) log(U V (x)) = + → ρ + θ. log x log x log x x→∞

(7)

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(vi) Suppose that U, V are such that U ◦ V (x) ∼ V ◦ U (x) ∼ x as x → ∞. Since ρ > 0, we have lim U (x) = x→∞

lim V (x) = ∞. Using (7), we obtain that

x→∞

1 = lim

x→∞

log U (V (x)) log V (x) log V (x) = ρ lim . x→∞ log V (x) log x log x

2

1.1.4. Applications in probability theory 1. Let X and Y denote positive random variables with d.f. FX and FY respectively. We say that X is stochastically smaller than Y if F X (x) ≤ F Y (x), ∀x ≥ 0 (see [22] or [34]). Now consider F X ∈ M(ρ) and F Y ∈ M(θ). If ρ > θ, then, using Proposition 1.1 (ii), there exists x0 such that F Y (x) < F X (x), ∀x ≥ x0 . It means that X dominates Y at infinity and the class M can be used to “sort” random variables.

If − log F Y (x)/F X (x) ∈ M(δ), the index δ may serve as an indicator for the distance between the two tails. 2. Let F denote a d.f. with tail F and quantile function Q defined as the inverse function of 1/F . Proposition 1.1 (vi) implies that, for ρ > 0, we have F ∈ M(−ρ)

if and only if Q ∈ M(1/ρ).

As an example consider a Pareto distribution with tail F (x) = x−ρ , x ≥ 1, ρ > 0. The quantile function is given by Q(x) = x1/ρ , x ≥ 1. Clearly F ∈ M(−ρ) and Q ∈ M(1/ρ). 3. In the same way, we can show that, for ρ > 0, − log F (·) ∈ M(ρ)

if and only if Q(exp(·)) ∈ M(1/ρ).

As an example consider a Weibull distribution with tail F (x) = e−αx , x, α, β > 0. Then − log F (x) = αxβ ∈ M(β) and Q(ex ) = (x/α)1/β ∈ M(1/β). Note that log Q (ex ) / log x → 1/β. β

x→∞

4. Assume that U = − log F ∈ M(−ρ), with F a d.f. and ρ > 0. Using (5), we have that βx−ρ L2 (x) ≤ − log F (x) ≤ αx−ρ L1 (x), ∀x ≥ x0 ,

(8)

where L1 , L2 are slowly varying. We define regularly varying sequences (an ) and (bn ) by the following relations: na−ρ n L1 (an ) → 1 n→∞

Note that an

→

n→∞

∞ and bn

→

n→∞

and

nb−ρ n L2 (bn ) → 1. n→∞

∞. Using the right hand side of (8), we can write, for an x ≥

−ρ x0 , −n log F (an x) ≤ αn(an x)−ρ L1 (an x) = αna−ρ n L1 (an )x

lim log F n (an x) ≥ −αx−ρ .

L1 (an x) , from which we deduce that L1 (an )

n→∞

In a similar way we have lim log F n (bn x) ≤ −βx−ρ . Since F n is the d.f. of the sequence of partial n→∞ maxima, we obtain that lim P (Mn ≤ an x) ≥ Gρ (α−ρ x)

n→∞

where Gρ (x) = exp (−x−ρ ) is a max-stable d.f.

and

lim P (Mn ≤ bn x) ≤ Gρ (β −ρ x),

n→∞

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Note that choosing α = β > 0, we have, as x → ∞, lim P (Mn ≤ bn x) ≤ Gρ (α−ρ x) ≤ lim P (Mn ≤ an x).

n→∞

n→∞

1.2. Karamata’s theorem Karamata’s theorem deals with integrals of regularly varying functions. Theorem 1.5 (Karamata’s theorem for RV (Karamata, 1930; de Haan, 1970)). Let U be a positive and measurable function with support R+ and Lebesgue summable on finite intervals. Let U ∈ RVρ . x xU (x) → ρ + 1. (i) If ρ > −1, then U (t)dt ∈ RV1+ρ and  x U (t)dt x→∞ 0 0

∞ (ii) If ρ < −1 then x

xU (x) → −ρ − 1. U (t)dt ∈ RVρ+1 and  ∞ U (t)dt x→∞ x

x

(iii) If ρ = −1, then

∞ U (t)dt ∈ SV . Moreover if

0

∞ U (t)dt < ∞, then

U (t)dt ∈ SV . x

0

The converse results are false in general. In the case (iii), this was one of the motivations for de Haan (1970) to introduce the class Π. 1.2.1. Karamata’s theorem for the class M In the next result we consider integrals for functions in the class M. Theorem 1.6 (Karamata’s theorem for M). Let U be a positive and measurable function with support R+ and Lebesgue summable on finite intervals. Assume that U ∈ M(ρ). x (i) If ρ > −1, then V (x) :=

U (t)dt ∈ M(1 + ρ). 0

∞

(ii) If ρ < −1, then W (x) :=

U (t)dt ∈ M(1 + ρ). x

∞

(iii) If ρ = −1, then V (x) ∈ M(0) and if

U (t)dt < ∞, then W (x) ∈ M(0). 0

Proof. Taking integrals in (5), we find that, ∀ > 0 and C > 0, there exists x0 = x0 (, C) such that, ∀x ≥ y ≥ x0 , x

x t L2 (t)dt ≤

C y

x U (t)dt ≤ 

ρ

y

tρ L1 (t)dt. y

If ρ > −1, we can use Karamata’s theorem for regularly varying functions to see that x tρ Li (t)dt y

∼

x→∞

1 xρ+1 Li (x), ρ+1

for i = 1, 2.

(9)

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y

x U (t)dt ≤

Note also that 0

U (t)dt = o(1)x1+ρ Li (x), for i = 1, 2. So we obtain 0

x

U (t)dt

0 lim 1+ρ L2 (x) x→∞ x

C ≥ ρ+1

x and

lim

x→∞

U (t)dt

0 x1+ρ L1 (x)

≤

ε ρ+1

x U (x)dx ∈ M(1 + ρ).

from which we deduce that 0

If ρ < −1, taking x = ∞ in (9) and using Karamata’s theorem provide the result. x If ρ = −1, combining (9) with Karamata’s theorem gives: C L4 (x) ≤ U (t)dt ≤ L3 (x), where L3 y

∞

U (t)dt < ∞ gives via Karamata’s theorem that

and L4 are slowly varying. Assuming additionally that b

∞ ˜ 4 (y) ≤ CL

˜ 3 (y) where L ˜ 3 and L ˜ 4 are SV. 2 U (t)dt ≤ L y

The converse result is false in general, as can be shown in the example below. Example. Consider the following function:  1 1/k

U (x) =

if if

  x ∈ (0, 1) or x ∈ k; k + 1 − 1/k2 , k = 1, 2, · · · x ∈ [k + 1 − 1/k2 ; k + 1), k = 1, 2, · · ·

log U (x) log U (x) = −1 and lim = 0. Now consider the integral V (x) = x→∞ log x log x x   U (t)dt. First assume that x ∈ k; k + 1 − 1/k2 . Then we have Clearly U ∈ / M since lim

x→∞

1

⎛ V (x) =

k−1  i=1

⎜ ⎝

2 i+1−1/i

i+1

U (t)dt + i

⎞ ⎟ U (t)dt⎠ +

i+1−1/i2

x U (t)dt = k

k−1 

1−

i=1

1 1 + 3 i2 i

 + x − k = x + O(1).

For x ∈ [k + 1 − 1/k 2 ; k + 1) we obtain in a similar way that V (x) = x + O(1). Therefore V ∈ RV1 ⊂ M(1). In the next lemma, we present partial converse results of Theorem 1.6. Lemma 1.1.

x U (t)dt ∈ M(1 + ρ). If U is nondecreasing, then U ∈ M(ρ).

(i) Suppose that V (x) = 0

∞ U (s)ds < ∞ and that W ∈ M(1 + ρ) with ρ < −1.

(ii) Suppose that W (x) = x

If U is nonincreasing, then U ∈ M(ρ).

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Proof. (i) Take x > 0 and t ≥ 1. Since U is nondecreasing, we have (t − 1)x U (x) ≤ V (xt) − V (x) ≤ (t − 1)x U (xt), and ρ ≥ 0 (since ∃x0 , ∀x ≥ x0 , U (x) ≥ C and so V (x) ≥ Cx + D, with C and D constants). Now choose  in such a way that 0 <  < ρ + 1. V (xt) ρ+1+ V (x) U (x) On one hand, we can write (t − 1) ρ+ ≤ t − ρ+1+ . x (xt)ρ+1+ x Since V ∈ M(ρ + 1), i.e. satisfies (2), it follows that lim U (x)/xρ+ = 0. x→∞

V (x) U (tx) ρ+1− t − 1 V (xt) . On the other hand, we have xρ+1− − ρ+1− ≤ x ρ+1− (xt) t (tx)ρ− t Since ρ + 1 −  > 0 and V ∈ M(ρ + 1), we obtain that lim U (tx)/(tx)ρ− = ∞. t→∞

We can conclude that U ∈ M(ρ). (ii) We proceed as in (i). Let x > 0 and t ≥ 1 to be defined later. Since U is nonincreasing, we have (t − 1)x U (xt) ≤ V (x) − V (xt) ≤ (t − 1)x U (x). Take  such that 0 <  < −ρ − 1. V (x) V (xt) ρ+1+ U (xt) On one hand we can write (t − 1)tρ+ ≤ ρ+1+ − t , from which we deduce, since (xt)ρ+ x (xt)ρ+1+ V (x) V (xt) V ∈ M(ρ+1), that lim U (xt)/(xt)ρ+ = 0. On the other hand we have − ≤ ρ+1− x→∞ (t − 1)x (t − 1)xρ+1− U (x) . xρ− Choosing t = x/2 , for x ≥ 1, we may observe that, for ρ + 1 +  < 0, V (x) V (x) x/2 → ∞ = (t − 1)xρ+1− xρ+1−/2 x/2 − 1 x→∞

and

V (xt) V (x3/2 ) x/2 → 0 = (t − 1)xρ+1− (x3/2 )ρ+1−(ρ+1+)/3 x/2 − 1 x→∞ hence the result. 2 1.2.2. Convolutions Now let us study convolutions of functions in the class M. Recall that the convolution of two functions x U and V is defined by U ∗ V (x) = U (x − y)V (y)dy. 0

Proposition 1.2 (Convolution). Assume that U ∈ M(ρ) and V ∈ M(θ). We can suppose w.l.o.g. that ρ ≤ θ. (i) If ρ ≤ θ < −1 or ρ < −1 ≤ θ, then the convolution U ∗ V ∈ M(λ) with λ = θ = ρ ∨ θ. (ii) If −1 < ρ ≤ θ, then U ∗ V ∈ M(λ) with λ = ρ + θ + 1. Proof. First observe that Theorem 1.6 (or (5)) tells us that for  > 0 and C > 0, we can find x0 = x0 (, C) such that for x ≥ x0 , we have C xρ L2 (x) ≤ U (x) ≤  xρ L1 (x) where Li , i = 1, 2, 3, 4 are SV.

and

C xθ L4 (x) ≤ V (x) ≤  xθ L3 (x).

(10)

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(i) First assume that ρ, θ < −1 so that both U and V are integrable on R+ . We have x/2 x/2 U ∗ V (x) = U (x − y)V (y)dy + V (x − y)U (y)dy 0

0

= I(x) + II(x). Consider the first integral I(x). Using (10) for U , we have, for all x ≥ 2x0 and 0 ≤ y ≤ x/2, (x − y)ρ L1 (x − y) U (x − y) z ρ L1 (xz) ≤  ≤  sup . ρ ρ x L1 (x) x L1 (x) 1/2≤z≤1 L1 (x) Hence I(x) z ρ L1 (xz) ≤  sup xρ L1 (x) 1/2≤z≤1 L1 (x)

x/2 V (y)dy,

(11)

0

I(x) ≤ A B, xρ L1 (x) ∞ z ρ L1 (xz) < ∞ and B := V (y)dy < ∞. where A := lim sup x→∞ 1/2≤z≤1 L1 (x) and since θ < −1, we obtain that lim

x→∞

0

I(x) ≥ CDB ρ L (x) x x→∞ 2

Using a similar approach and notations, we obtain that lim

z ρ L2 (xz) . It follows that the function I ∈ M(ρ). x→∞ 1/2≤z≤1 L2 (x) Consider now the second integral II(x). Using similar notations, we obtain where D := lim

inf

lim

x→∞

II ≤  A B  xθ L3 (x)

and

I ≥ C D B  , θ L (x) x x→∞ 4 lim

so that the function II ∈ M(θ). Proposition 1.1 (iii) allows one to conclude that U ∗ V ∈ M(λ) with λ = ρ ∨ θ. Now suppose that ρ < −1 ≤ θ. We need to reconsider I, but there is no change in the analysis of II (as ρ < −1), for which we obtain that II ∈ M(θ). Let us analyze I(x). Theorem 1.6 implies that x V (t)dt ∈ M(1 + θ), whereas using (10) for U gives 0

lim

x→∞

I(x) ≤  C , xθ+ρ+1 L1 (x)L3 (x)

for some constant C  . In a similar way we have a lim inf-result and obtain that I ∈ M(ρ + θ + 1). Then Proposition 1.1 (iii) provides that U ∗ V ∈ M(λ), where λ = θ ∨ (ρ + 1 + θ) = θ. (ii) Finally we consider the case where −1 ≤ ρ, θ. Using the previous approach, we obtain that I ∈ M(ρ + θ + 1) and II ∈ M(ρ + θ + 1). Then we can conclude via Proposition 1.1 (iii). 2 Remarks. 1. Bingham et al. [6] considered convolutions of regularly varying probability density functions f and g with conditions under which f ∗ g (x) ∼ f (x) + g(x). x→∞

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119

2. Let X and Y denote positive r.v. with d.f. FX and FY respectively, and let F X+Y denote the tail of the d.f. of X + Y . Using that (X > x) ∪ {Y > x) ⊆ (X + Y > x) ⊆ (X > x/2) ∪ (Y > x/2), we have A(x) ≤ F X+Y (x) ≤ A(x/2), where A(x) := F X (x) + F Y (x) − F X (x)F Y (x). If F X ∈ M(ρ) and F Y ∈ M(θ) with ρ, θ < 0, then A ∈ M(λ) where λ = ρ ∨ θ and it follows that F X+Y ∈ M(λ). The RV case was considered for example by Feller ([17], p. 278). Daley [13] also proved a similar result but using κ and inequalities of type xr + y r ≤ (x + y)r ≤ 2r−1 (xr + y r ) for r ≥ 1 and x, y > 0. 3. Let X1 , · · · , Xn be i.i.d. positive random variables with d.f. F . Consider the tail of the distribution of n the sum of the Xi ’s, F n (x) = P ( i=1 Xi > x). The previous remark shows that F ∈ M(ρ) implies that F n ∈ M(ρ), ∀n ≥ 2. 1.3. Karamata’s Tauberian theorem Suppose that U : R+ → R+ is measurable and Laplace Transform (LT) of U is defined by  (s) = s U

∞

∞ 0

e−sx |U (x)| dx < ∞ for all t > 0. Recall that the

e−sx U (x)dx, s > 0.

0

 (s) = If U is non-decreasing and U (0) = 0 we can write U

∞

e−xs dU (x), s > 0.

0

 is regularly varying at In 1931 Karamata proved that U is regularly varying at infinity if and only if U zero. Let us present his theorem as given in Bingham et al. (see [5], Theorem 1.7.1). Theorem 1.7 (Karamata (1931)). (see [5]) Suppose that U is a nondecreasing right continuous function with  (s) < ∞, ∀s > 0. Let L denote a slowly varying support R+ and satisfying U (0+) = 0. Assume that U ρ  (1/x) ∼ cΓ(1 + ρ)xρ L(x). function and let ρ ≥ 0, c ≥ 0. Then U (x) ∼ cx L(x) if and only if U x→∞

 (1/x) ∼ Γ(1 + ρ)U (x). If c > 0, both statements imply that U

x→∞

x→∞

We are going to prove a similar theorem for functions in the class M. Theorem 1.8 (Karamata’s Tauberian theorem for the class M). Assume that U is a nondecreasing right continuous function with support R+ , with U (0+) = 0, and such that x−η U (x) is a concave function for  (s) < ∞, ∀s > 0. Let 0 ≤ ρ ≤ ∞. Then U (x) ∈ M(ρ) if and some real number η ≥ 0. Suppose also that U  only if U (1/x) ∈ M(ρ). Proof. Theorem 1.8 results immediately from the following result: Lemma 1.2. Under the conditions of Theorem 1.8, there exist positive constants a, b, c, such that  ( 1 ) ≤ b U (cs), a U (s) ≤ U s

∀s > 0

(12)

and 1 1 1  c U ( ) ≤ U (s) ≤ U ( ) b s a s

(13)

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120

 (s Let us prove this lemma. First notice that U

−1

∞ )=

e−x U (sx)dx.

0

Since U is nondecreasing, we have  (1/s) ≥ U

∞

e−x U (sx) dx ≥ e−1 U (s).

(14)

1

On the other hand, since V (x) = x−η U (x), we can write  (s) = s U

∞

e−sx xη V (x) dx = s−η Γ(1 + η) E[V (Z)],

0

where Z is a random variable with density f (x) = 1 Γ(1 + η)

∞

xse−sx (sx)η dx =

1 s e−sx (sx)η and expectation E[Z] = Γ(1 + η)

Γ(η + 2) . Since V is concave, we can use Jensen’s inequality and obtain sΓ(1 + η)

0

 (s) ≤ s−η Γ(1 + η)V [E(Z)] = Γ(1 + η) U



Γ(η + 2) Γ(1 + η)

−η

 U

 (1/s) ≤ b U (cs) where b = Γ(1 + η) Replacing s by 1/s, it follows that U



Γ(η + 2) sΓ(1 + η)

Γ(η + 2) Γ(1 + η)

 .

−η and c =

Γ(η + 2) . Γ(1 + η)

Combining this latter inequality with (14) provides (12). Relation (13) follows from (12). 2  (1/x) ∈ M(ρ) can be proved without the concavity condition. The implication U (x) ∈ M(ρ) =⇒ U  (s) < ∞, ∀s > 0. Proposition 1.3. Suppose that U (.) is bounded on intervals [0, a] and that U  If U (·) ∈ M(ρ) with ρ ≥ 0, then U (1/·) ∈ M(ρ). Proof. Take  > 0, C > 0 and x0 = x0 (C, ) so that we have (5). Let y ≥ x0 . We can write  (s) = s U

y e

−sx

∞ U (x)dx + s

e−sx U (x)dx.

(15)

y

0

Using (5), we obtain ∞ Cs

e−sx xρ L2 (x)dx ≤ s

y

∞ y

e−sx U (x)dx ≤ s

∞

e−sx xρ L1 (x)dx.

(16)

y

Applying Theorem 1.7 gives, as s → 0, ∞ s

e−sx xρ Li (x)dx ∼ cΓ(1 + ρ) s−ρ Li (s−1 ), with c ≥ 0. s→0

0

(17)

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Now observe that K defined by K(x) = U (x) or K(x) = xρ Li (x), i = 1, 2, is bounded and satisfies y lim e−sx K(x)dx ≤ B, for some constant B. Using this bound and (17), we obtain s→0

0

y s 0 e−sx K(x)dx B ≤ lim = 0. lim s→0 cΓ(1 + ρ)s−ρ Li (s−1 ) s→0 cΓ(1 + ρ)s−1−ρ Li (s−1 )

(18)

Combining (15), (16), (17) and (18) provides lim

 (s) U

s→0 s−ρ L1 (s−1 )

≤  cΓ(1 + ρ)

and

lim

 (s) U

−ρ L (s−1 ) s→0 s 2

≥ C cΓ(1 + ρ) ,

hence the result. 2 As a consequence, we obtain the next result.  (x−1 ) = o(1)xρ L1 (x). If U is Corollary 1.1. Let U (x) = o(1)xρ L1 (x), with ρ ≥ 0 and L1 ∈ RV (0). Then U nondecreasing, then the converse holds. Proof. The first statement is straightforward from the previous proof. Conversely, if U (x) is nondecreas (x−1 ) = o(1)xρ L1 (x), then the inequality U (x) ≤ bU  (x−1 ) (see (12)) implies that U (x) = ing and U ρ o(1)x L1 (x). 2 2. Extensions and complements 2.1. The classes M(∞) and M(−∞) The class M (ρ), ρ ∈ R, is the class of positive and measurable functions U (x) such that (2) or (3) holds. This definition also makes sense when ρ = ∞ or −∞, and we will denote the corresponding classes of functions by M(∞), respectively M(−∞). Definition. Let U denote a positive and measurable function with support R+ and bounded on bounded intervals. Then U ∈ M(−∞) if and only if lim U (x)/xρ = 0, ∀ρ ∈ R, and U ∈ M(∞) if and only if lim U (x)/xρ = ∞, ∀ρ ∈ R.

x→∞

x→∞

In the next result, we list some properties of functions in M(±∞) (for more details, see [10] or [9]). Proposition 2.1. (i) We have U ∈ M(∞), respectively U ∈ M(−∞), if and only if U is of the form U (x) = eα(x) , respectively U (x) = e−α(x) , where α(x)/ log x → ∞ as x → ∞. (ii) Suppose that U is nondecreasing with inverse function V . Then U ∈ M(∞) if and only if V ∈ M(0). (iii) If U ∈ M(∞), then κU = −∞. If U ∈ M(−∞), then κU = ∞. (iv) We have U ∈ M(∞), respectively U ∈ M(−∞), if and only if lim log U (x)/ log x = ∞, respectively lim log U (x)/ log x = −∞.

x→∞

x→∞

(v) U ∈ M(−∞) if and only if U satisfies, for some function g ∈ M(∞), U (x) g(x) → 0. U ∈ M(∞) if and only if U satisfies, for some function h ∈ M(−∞), U (x) h(x) → ∞. x→∞

x→∞

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2.2. The class OM 2.2.1. Definition + In the class M(ρ), we require that lim log U (x)/ log x exists and is finite. The class OM is the class of x→∞ positive and measurable functions so that μ(U ) = lim

x→∞

log U (x) < ∞. log x

− The class OM is the class of positive and measurable functions so that

log U (x) > −∞. log x x→∞

ν(U ) = lim + − The class OM is defined by OM = OM ∩ OM .

Examples. 1. For U (x) = exp {sin(x) log(x)}, we have μ(U ) = 1 and ν(U ) = −1. 2. For U (x) = exp {x sin(x) log(x)}, we have μ(U ) = ∞ and ν(U ) = −∞. 3. Consider log(U (x)) = f (x) log(x), where f (x) = x(x −[x]). In this case we have lim log U (n)/ log n = 0, n→∞

log U (n + z) and for 0 < z < 1, lim = lim (n + z)z = ∞. n→∞ log(n + z) n→∞ 4. If U ∈ O − RV , then U ∈ OM . 2.2.2. Representation theorem If U ∈ OM , then log U (x) = a(x) log x, where a(x) is bounded away from ∞ and −∞. It can be rewritten as x log U (x) = α(x) +

β(t)t−1 dt

b

x where β(x) = a(x) and α(x) = a(x)

t

−1

x dt −

b

a(t)t−1 dt − a(x) log b.

b

Note that β(x) is bounded away from ±∞ and α(x) = O(1) log x. Hence we have the following result. Proposition 2.2 (Representation theorem for OM ). U ∈ OM

⎧ ⎨ if and only if

U (x) = exp

⎩

x α(x) +

⎫ ⎬

β(t) dt , x ≥ b, ⎭ t

b

where α(x) = O(1) log x and β(x) is bounded away from ±∞. 2.2.3. Karamata’s theorem Now suppose that U is integrable on finite intervals. If U ∈ OM , then there exist real numbers x0 , α and β with α < ν(U ) ≤ μ(U ) < β, such that α ≤ log U (x)/ log x ≤ β, ∀x ≥ x0 , or equivalently that xα ≤ U (x) ≤ xβ , ∀x ≥ x0 . Taking integrals, we find that

M. Cadena et al. / J. Math. Anal. Appl. 452 (2017) 109–125

xα+1 − y α+1 ≤ 1+α

x

123

xβ+1 − y β+1 , ∀x ≥ y ≥ x0 , 1+β

U (t)dt ≤ y

from which we deduce the following results. Proposition 2.3. With the notations introduced above, we have x (i) if α > −1, then

U (t)dt ∈ OM (with x ≥ b); b ∞

(ii) if β < −1, then

U (t)dt ∈ OM . x

2.2.4. Laplace transforms  (s) = s As in the previous section, we consider U

∞

e−sx U (x)dx.

0

Proposition 2.4. Assume that U ∈ OM and that U is bounded on bounded intervals.  (1/x) ∈ OM . (i) If α > 0, then U (ii) Under the conditions of Theorem 1.8, the converse holds true:  (1/x) ∈ OM ⇒ U ∈ OM . U Proof. (i) Using 0 < α ≤ β (defined above), we have, for all y ≥ x0 , ∞ s

e

∞

−sx α

x dx ≤ s

e

y

−sx

∞ U (x)dx ≤ s

y

e−sx xβ dx .

y

Clearly there exist constants c1 and c2 such that, as s → 0, ∞ s

e

−sx α

−α

x dx ∼ c1 s

∞ and s

0

e−sx xβ dx ∼ c2 s−β .

0

y If U is bounded on bounded intervals, then

e−sx U (x)dx = O(s) as s → 0. We also have

0

y s

e 0

−sx α

y

x dx + s

e−sx xβ dx = O(s).

0

 (s) ≤ c s−β , and We can conclude that, as s → 0, there exist constants c and d such that d s−α ≤ U  (1/s) ∈ OM . therefore that U (ii) The converse result follows from Lemma 1.2. 2

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3. Concluding remarks log U (x) = ρ. We may propose a direct extension as log x follows. Consider positive, measurable and nondecreasing functions U . For k = 1, 2, · · · , let log(1) f (x) = log f (x) and log(k+1) f (x) = log log(k) f (x). Then, for p, q = 1, 2, ..., we define the class Mp,q (ρ) of functions such that

1. For functions in the class M(ρ), we have lim

x→∞

logp U (x) = ρ. x→∞ logq (x) lim

Let us give some examples of functions belonging to Mp,q (ρ). ρ • Let U (x) = ex , ρ > 0. Then U ∈ M2,1 (ρ) since we have log U (x) = xρ and log(2) U (x) = ρ log x. x • Let U (x) = ee . We have log U (x) = ex and log(2) U (x) = x, hence U ∈ M3,1 (1). 2. We can also study (2) and (3) when considering sequences of positive numbers. We say that (an ) ∈ MS (ρ) if and only if it satisfies lim

n→∞

log an = ρ. log n

By considering the function U (x) = a[x] , we find that U ∈ M(ρ). We can then extend the properties we studied so far for functions of M, to sequences of MS . 3. Functions in the class M(ρ) being characterized by lim log U (x)/B(x) = ρ, where B(x) = log x, x→∞ considering other auxiliary functions B may also be quite interesting. Such an extension will be discussed in a forthcoming paper. 4. For functions in the class M(ρ), we have lim log(U (x)x−ρ )/ log x = 0. It may be useful to study x→∞ functions for which log(U (x)x−ρ ) = α, x→∞ b(x) lim

where b(x) = o(1) log x.

It will be the object of another paper. 5. It is quite challenging to find other Tauberian conditions than those proposed in Lemma 1.2 to have  (1/x) ∈ M =⇒ U (x) ∈ M. Further investigation would be required. the implication U 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 Fellowship) is also kindly acknowledged. References [1] S. Aljančić, D. Arandelović, O-regularly varying functions, Publ. Inst. Math. 22 (1977) 5–22. [2] V. Avakumović, On a O-inverse theorem, in: Rad Jugoslovenske Akademije Znanosti i Umjetnosti, t. 254, in: Razreda Matematičko-Prirodoslovnoga, vol. 79, 1936, pp. 167–186 (in Serbian). [3] N.H. Bingham, C.M. Goldie, Extensions of regular variation, I, II, Proc. Lond. Math. Soc. 44 (3) (1982) 473–496, 497–532. [4] N.H. Bingham, A.J. Ostaszewski, Beurling slow and regular variation, Trans. Lond. Math. Soc. 1 (2014) 29–56. [5] N. Bingham, C. Goldie, J. Teugels, Regular Variation, Cambridge University Press, 1989. [6] N.H. Bingham, C.M. Goldie, E. Omey, Regularly varying probability densities, Publ. Inst. Math. 80 (2006) 47–57. [7] M. Cadena, Revisiting extensions of the class of regularly varying functions, arXiv:1502.06488v2 [math.CA], 2015. [8] M. Cadena, A simple estimator for the M-index of functions in M, Hal-01142162, 2015. [9] M. Cadena, Contributions to the Study of Extreme Behavior and Applications, Doctoral Thesis, Université Pierre et Marie Curie, Paris, France, 2016.

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