Characterization of a general class of tail probability distributions

Statistics and Probability Letters 154 (2019) 108553

Contents lists available at ScienceDirect

Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro

Characterization of a general class of tail probability distributions ∗

Meitner Cadena a , Marie Kratz b , , Edward Omey c a

DECE, Universidad de las Fuerzas Armadas, Sangolqui, Ecuador ESSEC Business School, CREAR risk research center, France c KU Leuven @ Campus Brussels, Belgium b

article

info

Article history: Received 8 April 2019 Received in revised form 24 June 2019 Accepted 6 July 2019 Available online 18 July 2019 MSC: 26A12 45M05 60G70 26A42

a b s t r a c t Recently, new classes of positive and measurable functions, M(ρ ) and M(±∞), have been defined in terms of their asymptotic behavior at infinity, when normalized by a logarithm (Cadena et al., 2016–17). Looking for other suitable normalizing functions than logarithm seems quite natural. It is what is addressed here, studying general classes of log U(x) distribution functions of the type lim = ρ ≤ 0 for normalizing functions H x→∞ H(x) such that lim H(x) = ∞. x→∞

© 2019 Elsevier B.V. All rights reserved.

Keywords: Karamata functions Laplace transform Probability distribution Regularly varying function Semi-exponential tail distribution

1. Introduction Recently Cadena et al. (see Cadena (2016), Cadena and Kratz (2016), Cadena et al. (2017b)) introduced and studied the class M(ρ ) of positive and measurable functions with support R+ , bounded on finite intervals, such that lim

x→∞

log U(x) log x

=ρ

(1.1)

where ρ is a finite real number, called the order of the function U. Looking for other suitable normalizing functions than logarithm seems quite natural. It is what is proposed here, considering the general class of normalizing functions H such that lim H(x) = ∞. Examples include distributions with different tail distribution behaviors, for instance regularly varying x→∞

α

functions and semi-exponential tail distributions of the form A(x)e−x B(x) with slowly varying (noted RV0 ) functions A, B and 0 < α ≤ 1. The paper is organized as follows. In the next section, we define the new class and provide characterization theorems. Section 3 presents properties and relations with respect to other existing classes, as well as relations based on integrals and Laplace transforms. Some remarks conclude the paper. ∗ Corresponding author. E-mail address: [email protected] (M. Kratz). https://doi.org/10.1016/j.spl.2019.108553 0167-7152/© 2019 Elsevier B.V. All rights reserved.

2

M. Cadena, M. Kratz and E. Omey / Statistics and Probability Letters 154 (2019) 108553

2. The class M(H , ρ) 2.1. Definition and characterization theorems In our effort to generalize (1.1), we introduce the following class of functions. Let H be a real function satisfying lim H(x) = ∞. We define M(H , ρ ) as the class of positive and

Definition 2.1.

x→∞

+

measurable functions U with support R , bounded on finite intervals, such that

lim

x→∞

log U(x) H(x)

= ρ ≤ 0.

Note that the case ρ = 0 is understood as the limit case 0− , so that U tends to 0 in infinity. We can obtain a first characterization theorem for functions belonging to M(H , ρ ), inspired by Theorem 1.1 in Cadena (2016) or Cadena et al. (2017b): Theorem 2.2.

If lim H(x) = ∞, then we have the following equivalence: x→∞

U ∈ M(H , ρ ) ⇐⇒ ∀ϵ > 0, lim U(x)e−(ρ+ϵ )H(x) = 0 and lim U(x)e−(ρ−ϵ )H(x) = ∞. x→∞

x→∞

Remark 2.3. It is straightforward to deduce that if U ∈ M(H , ρ ), then for each ϵ > 0, there exists xϵ > 0 such that, for all x ≥ xϵ , exp{(ρ − ϵ )H(x)} ≤ U(x) ≤ exp{(ρ + ϵ )H(x)}.

(2.1)

Proof of Theorem 2.2. First assume that U ∈ M(H , ρ ) and let ϵ > 0. Via (2.1),

∀δ ∈ (0, ϵ ), ∃ xδ such that ∀x ≥ xδ , (ρ − δ )H(x) ≤ log U(x) ≤ (ρ + δ )H(x). It follows that U(x)e−(ρ+ϵ )H(x) ≤ e(δ−ϵ )H(x) → 0 x→∞

and, similarly, U(x)e−(ρ−ϵ )H(x) ≥ e(ϵ−δ )H(x) → ∞. x→∞

Now assume that the two limits on the RHS of the equivalence given in the theorem hold. For each δ > 0, ∃ xδ such that ∀x ≥ xδ , U(x) e−(ρ+ϵ )H(x) ≤ δ . Therefore we obtain log U(x) ≤ ρ + ϵ. log U(x) ≤ log δ + (ρ + ϵ )H(x) and lim sup H(x) x→∞ In a similar way, we have lim inf

log U(x)

≥ ρ − ϵ . Hence U ∈ M(H , ρ ). □

H(x)

x→∞

We can suggest another representation theorem for M(H , ρ ), as done for the Regularly Varying (RV) class: Theorem 2.4. U ∈ M(H , ρ ) ⇐⇒ log U(x) = α (x) +

x

∫

β (t)dH(t), a

with α (x) = o(H(x)) and lim β (x) = ρ . x→∞

Proof. If U ∈ M(H , ρ ), then we can deduce from Theorem 2.2 that log U(x) = (ρ + ϵ (x))

x

∫

with lim ϵ (x) = 0.

dH(t)

x→∞

a

Therefore we can write: log U(x) = α (x) +

α (x) := ϵ (x)

x

∫

x

∫

ϵ (t)dH(t) = o(1)

dH(t) − a

∫

x

β (t) dH(t), where β (x) := ρ + ϵ (x) and, as x → ∞, ∫ ax

a

dH(t). a

Conversely, applying L’Hopital’s rule gives lim

x→∞

log U(x) H(x)

= lim

x→∞

α (x) +

∫x a

β (t)dH(t)

H(x)

= lim β (x) = ρ. □ x→∞

2.2. Examples Let us consider the class of functions U of the type U(x) = eρ A(x) , where ρ < 0 and lim A(x) = ∞. Then U ∈ M(H , αρ ) for any H such that lim A(x)/H(x) = α > 0. x→∞

x→∞

M. Cadena, M. Kratz and E. Omey / Statistics and Probability Letters 154 (2019) 108553

3

A possible subclass of this class is that of semi-exponential tail distributions of the form F (x) = P[X > x] = A(x) exp{−B(x)xα } with A, B ∈ RV0 and 0 < α ≤ 1; see e.g. Borovkov (2002). Clearly, − log F (x) = − log A(x) + B(x)xα ∼

x→∞

B(x)xα , so F ∈ M(H , −β ) with H such that lim xα /H(x) = β > 0. Note that Gantert (see Gantert (1998)) uses this type x→∞

of functions to obtain asymptotic expressions for

log P[Sn /n ∈ A] nα B(n)

, as n → ∞, Sn denoting the nth partial sum.

3. Properties 3.1. First properties Let us state some relations between M(H , ρ ) and other existing classes. Some are straightforward, but worth mentioning, others less. Properties 3.1.

As x → ∞,

1. If H(x) / = o(log x), then M(H , ρ ) ⊂ M(0). 2. If 1 H(x) = o(log−α x) with α ≥ −1, then M(H , ρ ) ⊂ M(−∞) and M(ρ ) ⊂ M(H , 0). 3. If H = o(log x), then M(H , ρ ) ∩ RVα = ∅ for any ρ , α ≤ 0. Proof. 1. Let U ∈ M(H , ρ ) with H(x) = o(log x). The claim is trivial when noticing that lim

x→∞

log U log x

log U

= lim

×

H(x)

x→∞

H(x) log x

= ρ × 0 = 0.

2. Let U ∈ M(H , ρ ) with ρ < 0, then, since lim (log x)α H(x) = 0, we have

/

x→∞

lim

x→∞

log U(x) log x

= ρ lim

x→∞

H(x) (log x)α

× lim (log x)α−1 = −∞. x→∞

Assume now that U ∈ M(ρ ), then lim

x→∞

log U(x) H(x)

= ρ lim

(log x)α

x→∞

H(x)

× lim

x→∞

1 (log x)α−1

= 0.

3. Assume U ∈ M(H , ρ ) ∩ RVα (̸ = ∅) for some ρ ≤ 0 and α ≤ 0. The{∫ representation } theorem for RV functions x

β (t)t −1 dt , where lim c(x) = c ̸= 0

(see Bingham et al. (1989), Theorem 1.3.1) states that U(x) = c(x) exp

x→∞

a

and lim β (x) = α . Taking logarithms in this expression, applying l’Hopital’s rule, then using that H(x) = o(log x), x→∞

provide

lim

x→∞

log U(x) H(x)

= α × lim

x→∞

log x H(x)

= −∞,

which contradicts that U ∈ M(H , ρ ). □ Example 3.2. Some properties derived for the class M(H , ρ ) may help describe some tail behaviors. For instance, let X and Y be independent random variables, with distributions FX and FY , respectively. Assume F X = 1 − FX ∈ M(H , α ) with H(x) ∼ xκ , κ > 0, and FY has a density function fY ∈ M(Q , β ) with Q (x) ∼ ρ log x for some ρ > 0. Then, x→∞

x→∞

via Properties 3.1, (ii), we have FX × fY ∈ M(H , α ). It allows us to compute for instance lim P[XY > x], obtaining that

P[XY > x] x−ρβ tends to a positive constant as x → ∞.

x→∞

/

In the next result, we collect some algebraic results. Properties 3.3. Let U ∈ M(H , α ). (i) If V ∈ M(H , β ), then UV ∈ M(H , α + β ) and, if α ≤ β , U /V ∈ M(H , α − β ). (ii) If V ∈ M(G, β ) and lim G(x)/H(x) = 0, then UV ∈ M(H , α ). x→∞

(iii) If lim log U(x)/H(x) = γ ≥ 0, with lim H(x) = ∞, then 1/U ∈ M(H , −γ ). x→∞

x→∞

(iv) Consider a differentiable function V such that lim xV ′ (x)/V (x) = β > 0. Then U ◦ V ∈ M(W , αβ ), where H satisfies xH ′ (x) ∼ L(x) with L such that W (x) = x→∞

∫x b

x→∞

L(V (z))z −1 dz → ∞, for some b ∈ R.

4

M. Cadena, M. Kratz and E. Omey / Statistics and Probability Letters 154 (2019) 108553

Proof. The first three properties follow directly from Definition 2.1. Let us check (iv). The condition on V ensures that, log U(V (x)) for x large enough, V ′ (x) > 0, implying lim V (x) = ∞. Hence, since U ∈ M(H , α ), we obtain lim = α. x→∞ x→∞ H(V (x)) ∫ x log U(V (x)) log U(V (x)) H(V (x)) Now, writing = × , with W (x) := L(V (z))z −1 dz → ∞, b ∈ R, we obtain by using x→∞ W (x) H(V (x)) W (x) b l’Hopital’s rule, lim

log U(V (x)) W (x)

x→∞

= α × lim ∫ x x→∞

b

H(V (x)) L(V (z))z −1 dz

H ′ (V (x))V ′ (x)

= α × lim

L(V (x))x−1

x→∞

= α × lim

x→∞

xV ′ (x) V (x)

= α × β. □

Remark 3.4. In (iii), although 1/U cannot be a distribution tail, it may be extended to one, if nonincreasing. For instance, / it can be extended to a distribution tail on R+ when focusing on (b, ∞) for some b such that 1 U(x) ≤ 1. 3.2. Integrals and Laplace transforms Proposition 3.5. Let U ∈ M(H , ρ ) and eH ∈ RV0 . x

∫

t α U(t) dt ∈ M(H , ρ ).

(i) If α > −1, then x−1−α

∫a ∞

(ii) If α < −1, then x−1−α

t α U(t) dt ∈ M(H , ρ ).

x

Remark 3.6. Proving the converse result is an open problem. Moreover, it is not clear what occurs when α = −1. Proof. Let V = eH . (i) Use (2.1) with ϵ > 0. Multiplying by xα and taking integrals give xϵ

∫

t α U(t)dt +

∫

x

t α V ρ−ϵ (t)dt ≤

t α U(t)dt ≤

xϵ

∫

t α U(t)dt +

∫

x

t α V ρ+ϵ (t)dt.

xϵ

a

a

xϵ

a

x

∫

Since V ∈ RV0 , then V ρ±ϵ ∈ RV0 , so t α V ρ±ϵ (t) ∈ RVα . Moreover, via Karamata’s theorem (see Bingham et al. (1989), Proposition 1.5.8), x

∫

t α V ρ±ϵ (t)dt

xϵ

1

∼

1+α

x→∞

x1+α V ρ±ϵ (x) → ∞. x→∞

It follows that there exists x0 > xϵ such that, for all x ≥ x0 , 1−ϵ 1+α

V

ρ−ϵ

(x) ≤

x

∫

1 x1+α

t α U(t) dt ≤

a

1+ϵ 1+α

V ρ+ϵ (x).

Using lim log V (x) = ∞ and taking ϵ → 0 allow us to conclude. x→∞

(ii) The proof follows the same steps as in (i), using now (via Karamata) ∞

∫

t α V ρ±ϵ (t) dt

x

−1 ∼

x→∞

1+α

x1+α V ρ±ϵ (x). □

Proposition 3.7. Assume H twice differentiable and U differentiable such that U ′ (x) U(x) ∼ ρ H ′ (x), with ρ < 0. We have

/

x→∞

the following: (i) If lim −H ′′ (x) (H ′ (x))2 = α ≤ 0, then U ∈ M(H , ρ ),

/

→∞ ∫ x∞ x

∞

∫

U(t) dt ∈ M(H , ρ + α ) and

U(t) dt x

(ii) If lim −H ′′ (x) (H ′ (x))2 = −∞, then

/

→∞ ∫ x∞

U(t) dt ∈ M(H , −∞) and

x

∼

x→∞

−

U(x) (ρ + α )H ′ (x)

.

∞

∫

U(t) dt = o(1) U(x) H ′ (x).

/

x

Remark 3.8. Let U and H be positive functions with derivatives U ′ and positive H ′ , respectively. Assume that they satisfy U ′ (x)

→ α,

H ′ (x)U(x) x→∞

with

α ≤ 0.

M. Cadena, M. Kratz and E. Omey / Statistics and Probability Letters 154 (2019) 108553

5

Then, ∀ε > 0, ∃ xε such that, ∀x ≥ xε , (α − ε )H ′ (x) ≤

U ′ (x) U(x)

≤ (α + ε)H ′ (x),

and U ∈ M(H , α ). Proof of Proposition 3.7. Let W be defined by W (x) = 1/H ′ (x). (i) Clearly 1

U(x) ∼

W (x)U ′ (x) ⇔

ρ

x→∞

U ′ (x) U(x)

∼

x→∞

ρ/W (x).

It implies (see Remark 3.8, 1.) that U ∈ M(H , ρ ), lim U(x) = 0 and ∞

∫

U(t) dt x

∼

x→∞

x→∞

∞

∫

1

′

W (t)U (t) dt.

ρ

x

W (x)U(x) Now consider R(x) = ∫ ∞ . Applying l’Hopital’s rule provides: W (t)U ′ (t) dt x lim R(x) = lim

x→∞

W (x)U ′ (x) + W ′ (x)U(x)

−W (x)U ′ (x)

x→∞

= −1 − ρ −1 × lim W ′ (x).

(3.1)

x→∞

We deduce, under the assumption of (i), that lim R(x) = −1 − α/ρ . It follows that

U(t)dt x

∞

∫

x→∞

∞

∫

∼ −

W (x)U(x)

ρ+α

x→∞

∞

∫

. Applying once again Remark 3.8 for the function

U(t)dt, we obtain x

U(t) dt ∈ M(H , ρ + α ). x

(ii) If lim W ′ (x) = −∞, using (3.1) gives lim R(x) = −∞, as ρ < 0. x→∞

∫∞

So we have

x

x→∞

U(t) dt

W (x)U(x)

1

∼

x→∞

ρ R(x)

, which tends to 0 as x → ∞.

□

A related result follows directly from Theorem 4.12.10 in Bingham et al. (1989). Proposition 3.9. Assume U ∈ M(H , ρ ). If H ∈ RVα , with α > 0, then ∞

∫ − log

U −1/ρ (t) dt

x

∼ − log U −1/ρ (x) and

x→∞

∞

∫

U −1/ρ (t) dt ∈ M(H , −1).

x

/ ˆ Laplace Transforms. Let us turn to Laplace transforms. Recall that the Laplace transform of 1/U is given by 1 U(s) = ∫ ∞

e−sx U(x) dx. Let U be such that 1/U satisfies the conditions of Theorem 1.8 in Cadena et al. (2017b), namely: (i)

/

s 0

/ / ˆ U is/a nonincreasing right continuous function with support R+ and 1 U(0+) = 0, (ii) 1 U(s) < ∞, ∀s > 0, and (iii) x−η U(x) is a concave function for some real number η > 0. Then we can enunciate a Karamata Tauberian type of theorem. Theorem 3.10. Assume lim H(x) = ∞ and U satisfies the conditions (i) to (iii) above. Let ρ < 0. Then x→∞

U(x) ∈ M(H , ρ )

if and only if

( / ˆ

)−1 ∈ M(H , ρ ).

1 U(1/x)

Proof. First suppose that U ∈ M(H , ρ ). Using (2.1) with V = eH , we can say that there exist positive constants a, b, c, such that, ∀x > xϵ ,

/ ˆ a V −ρ−ϵ (x) ≤ 1 U(1/x) ≤ b V −ρ+ϵ (cx). Since V ∈ RV0 , it follows that 1

/ / (ˆ

)

1 U(1/x) ∈ M(H , ρ ). The converse implication can be proved in a similar way.

/ ˆ In fact, the implication U(x) ∈ M(H , ρ ) H⇒ 1 U(1/x) ∈ M(H , ρ ) can be proved without the concavity condition, as in Cadena et al. (2017b): / ˆ Proposition 3.11. Let U ∈ M(H , ρ ), with lim H(x) = ∞, U and eH bounded on bounded intervals, and 1 U(s) < ∞, )−1 ∀s > 0. Then 1 U(1/x) ∈ M(H , ρ ). ( / ˆ

x→∞

6

M. Cadena, M. Kratz and E. Omey / Statistics and Probability Letters 154 (2019) 108553

Remark 3.12. We could also consider O−versions of this class of functions. 4. Concluding remarks Note that many distribution functions F with density f satisfy a relation of the form lim

log(1 − F (x))

x→∞

be interesting to study the class of functions U satisfying the following relation: lim

x→∞

log U(x) log |U ′ (x)|

log f (x)

= 1. So it may

= 1. An example of such

distribution functions is the exponential one. We considered in this paper probability distribution functions (i.e. for ρ ≤ 0 in Definition 2.1), but we could turn to other types of functions (as e.g. density functions) as long as they satisfy Definition 2.1. Moreover, our results may be extended to the case ρ > 0 (see Cadena et al. (2017a)). References Bingham, N., Goldie, C., Teugels, J., 1989. Regular Variation. Cambridge University Press. Borovkov, A., 2002. On subexponential distributions and asymptotics of the distribution of the maximum of sequential sums. Sib. Math. J. 43 (6), 995–1022. Cadena, M., 2016. Contributions to the Study of Extreme Behavior and Applications (Doctoral Thesis). Université Pierre et Marie Curie, Paris, France. Cadena, M., Kratz, M., 2016. New results for tails of probability distributions according to their asymptotic decay. Stat. Probab. Lett. 109, 0–5. Cadena, M., Kratz, M., Omey, E., 2017a. New results on the order of functions at infinity, ESSEC Working Paper 1708. https://hal-essec.archivesouvertes.fr/hal-01558855. Cadena, M., Kratz, M., Omey, E., 2017b. On the order of functions at infinity. J. Math. Anal. Appl. 452 (1), 109–125. Gantert, N., 1998. Functional Erdös-Renyi laws for semi-exponential random variables. Ann. Probab. 26 (3), 1356–1369.