EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 13, No. 4, 2020, 739-757
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

Extremes, extremal index estimation, records, moment
problem for the Pseudo-Lindley distribution and

applications

Gane Samb LO1,2,3,∗, Modou Ngom4, Moumouni Diallo5

1 LERSTAD, Gaston Berger University, Saint-Louis, Sénégal
2 LSTA, Pierre and Marie Curie University, Paris VI, France.
3 AUST - African University of Sciences and Technology, Abuja, Nigeria
4 LERSTAD, Gaston Berger University, Saint-Louis, Sénégal Ministry of High School,
Sénégal
5 Faculté des Sciences Économiques et de Gestion (FSEG), Université des Sciences Sociale
et de Gestion de Bamako ( USSGB), Mali

Abstract. The pseudo-Lindley distribution which was introduced in Zeghdoudi and Nedjar (2016)
is studied with regards to it upper tail. In that regard, and when the underlying distribution
function follows the Pseudo-Lindley law, we investigate the behavior of its values, the asymptotic
normality of the Hill estimator and the double-indexed generalized Hill statistic process (Ngom
and Lo, 2016), the asymptotic normality of the records values and the moment problem.

2020 Mathematics Subject Classifications: 6oG70, 62G20,62H10,62H15
Key Words and Phrases: Lindley’s distribution, Pseudo-Lindley distribution, Extreme value
theory, record values, Hill’s estimator, asymptotic laws

1. Introduction

1. General facts.

The following probability distribution function (pdf ), named as the Pseudo-Lindley pdf,

f(x) = f(x, θ, β) =
θ(β − 1 + θx)e−θx

β
1(x≥0) (1)

with parameters θ > 0 and β > 1, has been introduced by [15] as a generalization of the
Lindley pdf :

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v13i4.3834

Email addresses: gane-samb.lo@ugb.edu.sn, gslo@aust.edu.ng, ganesamblo@ganesamblo.net (G.S.
LO), modou.ngom4@education.sn, ngom.modou1@ugb.edu.sn, ngomodoungom@gmail.com (M. Ngom),
moudiallo1@gmail.com (M. Diallo)

https://www.ejpam.com 739 c© 2020 EJPAM All rights reserved.



G.S. LO, M. Ngom, M.Diallo / Eur. J. Pure Appl. Math, 13 (4) (2020), 739-757 740

`(x) =
θ2(1 + x)e−θx

1 + θ
1(x≥0), (2)

in the sense that for β = 1 + θ, f(◦) is identical to `(◦).

Actually, f derives from ` by a mixture of a Lindley distributed random variable and
an independent Γ(2, θ) random variables with mixture coefficients r1 = (β − 1)/β and
r2 = 1/β, where 1 < r1, r2 < 1 and r1 + r2 = 1.

The cumulative distribution cdf function is given by

1− F (x) =
(
β−1(β + θx)e−θx

)
1(x≥0).

The Lindley original distribution is an important law that has been used and still is being
used in Reliability, in Survival analysis and other important disciplines. Because of its
original remarkable qualities, it kicked off a considerable number generalizations as pointed
out by [15]. The current generalization (1) has been tested on real data and simulated.
The results of those studies and simulations have shown a real interest of that model in
survival analysis. In ([15]) for example, that model has been tested on Guinean Ebola.
The paper of [6] focused on asymptotic tests of that law based on moments estimators of
the new law. The interest that distribution demonstrated in real data modeling motivated
us to give some asymptotic theories on it, in view of statistical tests. In this paper, we
deal with the properties of the upper tail, the extreme value distribution and the record
values. etc., each of them providing statistical tests.

Throughout the paper, X, X1, X2, · · · is a sequence of independent real-valued random
(rv), defined on the same probability space (Ω,A,P), with common cumulative distribution
function F , with the first asymptotic moment function and the generalized inverse function
defined by

R(x, F ) =
1

1− F (x)

∫ +∞

x
(1− F (y)) dy, x ∈]0,+∞[

and

F−1(u) = inf{x ∈ R, F (x) ≥ u} for u ∈]0, 1[ and F−1(0) = F−1(0+).

For each n ≥ 1, we denote the ordered statistics of the sample X1, · · · , Xn by

X1,n ≤ · · · ≤ Xn,n.



G.S. LO, M. Ngom, M.Diallo / Eur. J. Pure Appl. Math, 13 (4) (2020), 739-757 741

Usually, in extreme value theory, we focus on upper extreme and the hypothesis X > 0 and
the log-transform Y = logX is instrumental in all major results in that field. We denote
the cdf of Y by G(x) = F (ex), x ∈ R+. The Renyi representation is also of common use
in the following form. The sequence is replaced as follows

{{X1,n ≤ · · · ≤ Xn,n}, n ≥ 1} =d {{F−1(1− Un−j+1,n), 1 ≤ j ≤ n}, n ≥ 1}, (3)

where =d stands for the equality in distribution. Finally, the following Malmquist repre-
sentation (see [14], also [9], page 127) is also used : for each n ≥ 1, there exist a finite
sequence of standard independent exponential random variables E1,n, · · · , En,n such that{(

Ui+1,n

Ui,n

)i
, 1 ≤ i ≤ n

}
=d {Ei,n, 1 ≤ i ≤ n} . (4)

2. Extremes

We can directly see that F is the Gumbel distribution G0 by three different arguments.
First, by using the Von Mises’ argument (see [2] or [7], Proposition 24, page 184)

lim
x→+∞

f ′(x)(1− F (x))

f2(x)
= −1. (5)

A second argument comes from that Y = exp(X) has the distribution G(x) = F (log x) =
β−1(β + θ log x)x−θx. Since

∀λ > 0, lim
x→+∞

1−G(λx)

1−G(x)
= λ−θ, (6)

G ∈ D(G1/θ) and since F (x) = G(ex) for x ≥ 1, by Theorem [4] (Lemmas 9 and 10),
F ∈ D(G0).

A third argument is related to the development of the quantile function. In the appendix
(page 753), we give a number of expansions of that quantile that could be used for different
purposes. For example we have (see page 755),

∀λ > 0, F−1(1− u) = θ−1(log(1/u)− log log(1/u)) + θ−1K(u) (7)

with K(u) = O

(
(log 1/u)−2

)
. By using it, we get



G.S. LO, M. Ngom, M.Diallo / Eur. J. Pure Appl. Math, 13 (4) (2020), 739-757 742

F−1(1− λu)− F−1(1− u)

(1/θ)
→ − log λ as u→ 0.

By the π-variation criteria of [2] (See [9], Proposition 11, page 88), we have F ∈ D(G0)
and R(x, F ) → γ = 1/θ as x → +∞. Formula (7) is actually a second-order condition
for the quantile function (see [2]). We apply it right to get a rate of convergence of the
maximum observations. Put γ = 1/θ.

2. Expansion of the maximum values.

By the Renyi representation and by denoting Zn = − log(nU1,n), we have that log(1 +
Zn/(log n))→P 0 and since logU1,n = OP(log n)−1

Xn,n − F−1(1− 1/n) = γZn + γ log(1 + Zn/(log n))

+ O((log n)−2) +O((logU1,n)−2)

and hence

Xn,n − F−1(1− 1/n)

γ
= Zn +OP

(
(log n)−1

)
= Λ + oP(1). (8)

It is easy to see that Zn converges to Gumbel law Λ with cdf

G0(x) = exp(− exp(−x)), x ∈ R.

So we have that Xn,n converges to a Λ law. But we obtain the random rate of convergence
Zn/ log n, since

logZn
log n

(
Xn,n − F−1(1− 1/n)

γ
− Zn

)
= 1.

As well for k = k(n)→ +∞ such that k(n)/n→ 0, and by taking Tn = log(nUk,n/k) and
qn = n/k(n) which goes to +∞, we have

Xn−k,n − F−1(1− k/n)

γ
= Tn + log(1 + Tn/ log qn)) +OP((log qn)−2). (9)

3. Estimating the extreme value index γ = 1/θ.

The Hill’s estimator ([3], 1975)



G.S. LO, M. Ngom, M.Diallo / Eur. J. Pure Appl. Math, 13 (4) (2020), 739-757 743

Hn =
1

k(n)

k(n)∑
j=1

j (Xn−j+1,n −Xn−j,n) , (10)

is the most celebrated estimator of the extreme value index γ = 1/θ of Z = exp(X).
Among a significant number of generalizations of the Hill’s estimator, the Ngom-Lo gen-
eralization ([12], 2016), called the functional Double-indexed Hill estimator, is one the
sharpest one. It is defined as

Hn(f, s) =

k(n)∑
j=1

f(j) (Xn−j+1,n −Xn−j,n)s /an(f, s)

1/s

,

where f : N \ {0} → R+ \ {0} is a measurable mapping and s > 0, and

an(f, s) = Γ(s+ 1)

k(n)∑
j=1

f(j)j−s.

Let us define for s > 0 and f : N \ {0} → R+ \ {0} measurable,

C2(s) = Γ(2s+ 1)− Γ(s+ 1)2, s2n(f, s) = C2(s)

k(n)∑
j=1

f(j)2j−2s,

and

Bn(f, s) = max{f(j)j−s/sn(f, s), 1 ≤ j ≤ k(n)}.

We simply notice that the classical Hill’s estimator is Hn(Id, 1) where Id is the identity
function on N \ {0}. Let us give asymptotic normality for the functional Double-indexed
Hill estimator.

(a) Extreme Limit Theorem.

We begin with the simple Hill’s estimator.

Theorem 1. For ]0, n] 3 k(n)→ +∞ such that

k(n)3/4/ log n→ 0. (K1)

We have, as n→ +∞, √
k(n) (Hn − γ) N (0, γ2). (11)



G.S. LO, M. Ngom, M.Diallo / Eur. J. Pure Appl. Math, 13 (4) (2020), 739-757 744

We want to establish the random rate of convergence associated with the convergence 11
in the part (a) of the following corollary. In the part (b), we want to share that we do not
need any other condition on top of k(n)/n → 0 to have the central limit theorem if F−1

is reduced to

F−1∗ (1− u) = γ log u− C(γ) log log(1/u), u ∈]0, 1[, C(γ) ≥ 1. (12)

Corollary 1. We have the following results.

(a) Here again F is the cdf of the Pseudo-Lindley distribution with parameters θ > 0 and
β > 0 and the notation above. Let k(n)/ log n→ 0.

Let W (1) is a standard Gaussian random variable. Then we have

log n

γ
√
k(n)

(√
k(n)(Hn − γ)− γW (1)

)
→P 1,

(b) If F−1 were reduced as in Formula (12), we have the asymptotic normality√
k(n)(Hn − γ)→ N (0, γ2)

whenever k(n)/n→ 0 and

log n

(√
k(n)(Hn − γ)− γW (1)

)
= OP(1). ♦

Proof of Theorem 1. By the Malmquist representation (See [14] or [9], Proposition 32,
page 135), by Formula (38), we have for any 1 ≤ j ≤ k,

Xn−j+1,n −Xn−j,n = F−1(1− Uj,n)− F−1(1− Uj+1,n)

= γj−1Ej,n − γ
∫ Uj+1,n

Uj,n

du

u log(1/u)
+OP

(
(logn)−2

)
(13)

and next

j (Xn−j+1,n −Xn−j,n) = γEj,n − γj
∫ Uj+1,n

Uj,n

du

u log(1/u)
+OP

(
k (logn)−2

)
.

So for Zn = log nU1,n (which converges in law to Λ) and



G.S. LO, M. Ngom, M.Diallo / Eur. J. Pure Appl. Math, 13 (4) (2020), 739-757 745

∣∣∣∣∣
∫ Uj+1,n

Uj,n

du

u log(1/u)

∣∣∣∣∣ ≤ j−1Ej,n
| log n− Zn|

. (14)

Hence

1

k(n)

∣∣∣∣∣∣
k(n)∑
j=1

j

∫ Uj+1,n

Uj,n

du

u log(1/u)

∣∣∣∣∣∣ ≤
S∗k(n)

k
OP((log n)−1),

where S∗k(n) = Ej,n + · · ·+ Ek,n. We finally get

√
k(n) (Hn − γ) = γ

S∗k(n) − k√
k(n)

+OP

(
1

log n
,

k3/2

(log n)2

)

We conclude that, whenever (K1) holds, we have

√
k(n) (Hn − γ) = γ

S∗k(n) − k√
n

+ oP(1). �

Proof of the Corollary 1. The proof of Part (b) is the conclusion of the proof of
Theorem 1 up to the formula (14). If (12) holds, further steps are dismissed. And we
need only k(n)/n→ 0 to conclude. Let us set

Z∗n =
1√
k(n)

k(n)∑
j=1

j

∫ Uj+1,n

Uj,n

du

u log(1/u)
, n ≥ 1.

From the first part, we already know that Z∗n = OP(1/ log n). We denoted by W (1)
a standard Gaussian random variable. By the classical Kómlos-Màjor-Tusnàdy (KMT)
approximation, we have∣∣∣∣∣S∗k(n)− k(n)√

k(n)
− γW (1)

∣∣∣∣∣ = OP

(
log k(n)√
k(n)

)
.

Straightforward expansions using the different rates of convergence lead to√
k(n)(Hn − γ)− γW (1)

γZ∗n
→P 1,

whenever k(n)/n → 0. Now we apply Proposition in [9], page 22. Since the function
log(1/u) is slowly varying and that U1,n/Uk+1,n and Uk+1,n/U1,n are both asymptotically
bounded in probability, we have



G.S. LO, M. Ngom, M.Diallo / Eur. J. Pure Appl. Math, 13 (4) (2020), 739-757 746

tn = sup
1≤j≤k(n)

sup
s∈[Uj,n,Uj+1,n]

∣∣∣∣ log(1/s)

log n
− 1

∣∣∣∣→P 0.

It comes that

Z∗n =

√
k(n)

log n
(k−1(n)S

∗
k(n))(1 +O(tn)) =

√
k(n)

log n
(1 + o(1)),

which gives the desired result. �

We have the following convergence of the Double-indexed functional Hill statistics.

Theorem 2. We have the following two results.

(a) If the following conditions hold, as n→ +∞

sn(f, 1)/(sn(f, s) log n)→ 0 and Bn(f, s)→ 0,

then

Tn(f, s)− γsan(f, s)

sn(f, s)
 N

(
0, γ2s

)
.

(b) Furthermore, if an(f, s)/sn(f, s)→ +∞, then

an(f, s)

sn(f, s)

((
Tn(f, s)

an(f, s)

)1/s

− γ
)
 N (0, s−2γ2).

Proof. Let us exploit the proof of Theorem 1. We have for j ∈ {1, · · · , k(n)}, s ≥ 1,

Ai,n = f(j) (Xn−j+1,n −Xn−j,n)s

= f(j)

(
γj−1Ej,n − γ

∫ Uj+1,n

Uj,n

du

u log(1/u)
+OP

(
Fk(n) (logn)−2

))s
=: f(j)

(
γj−1Ej,n −Rj,n + Cj,n

)s
,

with

Cj,n = OP

(
(logn)−2

)
(uniformly in j),

∣∣∣∣∣γ
∫ Uj+1,n

Uj,n

du

u log(1/u)

∣∣∣∣∣ ≤ γj−1Ej,nb(n)

| log n− Zn|
.



G.S. LO, M. Ngom, M.Diallo / Eur. J. Pure Appl. Math, 13 (4) (2020), 739-757 747

We get, by the mean value theorem, j ∈ {1, · · · , k(n)}, s ≥ 1,

Ai,n − γsf(j)j−sEsj,n

≤ sf(j) |Rj,n + Cj,n|
(
γj−1Ej,n + |Rj,n|+ |Cj,n|

)s−1
≤
(
sγf(j)j−1Ej,n
| log n− Zn|

)(
γj−1Ej,n + |Rj,n|+ |Cj,n|

)s−1
.

In the lines below, we will bound the term with the power s − 1. If s = 1, there will
is nothing to bound. So formulas regarding that term are dismissed for s = 1 and are
used only for s > 2. For s ≥ 1, we will use the Cs−1 inequality ( for s ≤ 2, with
|a+ b|s−1 ≤ 2s−2|a|s−1 + |b|s−1 Cs−1 = 2s−2). For 0 < r < 1, it can be easily checked that,
for u > 0 fixed, the function g(v) = (u + v)r − ur − vr of v ≥ 0 takes the value g(v) = 0
and has a non-positive derivative function, so that g(v) ≤ g(0) = 0 for any v ≥ 0, which
is equivalent to (u + v)r ≤ ur + vr. We finally have that |a + b|s−1 ≤ Ds|a|s−1 + |b|s−1
with Ds = 1 for 1 < s < 2 and Ds = Cs−1 for s ≥ 2. Applying that inequality leads,
j ∈ {1, · · · , k(n)}, s ≥ 1, to

Ai,n − γsf(j)j−sEsj,n (A)

≤
(
sγf(j)j−1Ej,n
| log n− Zn|

)(
Dsγ

s−1js−1Es−1j,n +
D2
sγ

s−1js−1Es−1j,n

(| log n−Xn|s−1)
+OP

(
D2
s

(log n)2(s−1)

))
.

Let us denote

Sn(f, s) =

k(n)∑
j=1

f(j)j−sEsj,n

and

Tn(f, s) =

k(n)∑
j=1

f(j) (Xn−j+1,n −Xn−j,n)s .

By combining the results above, we arrive at

∣∣∣∣Tn(f, s)− γsSn(f, s)

∣∣∣∣ (B)

≤
(
sγSn(f, 1)

| log n− Zn|

)(
Dsγ

s−1Sn(Id, s− 1) +
D2
sγ

s−1Sn(Id, s− 1)

(| log n− Zn|s−1)
+OP

(
D2
s

(log n)2(s−1)

))
.



G.S. LO, M. Ngom, M.Diallo / Eur. J. Pure Appl. Math, 13 (4) (2020), 739-757 748

Let us study Sn(f, s). As a sequence of partial sums of real-value independent random
variables indexed by j ∈ {1, · · · , k(n)} with first and second moments

Γ(s+ 1)f(j)j−s and (Γ(2s+ 1)− Γ(s+ 1)2)f(j)2j−2s,

the asymptotic normality is given by the the theorem of Levy-Feller-Linderberg (See Theo-
rem 20 in [5]) we apply to the centered rrv ’s ξj = f(j)j−s(Esj,n−Γ(s+1)), after remarking
that {

Var(ξj)∑k(n)
j=1 Var(ξj)

, 1 ≤ j ≤ k(n)

}
= C(s)Bn(f, s).

So, as n→ +∞, 1

sn(f, s)

k(n)∑
j=1

(
f(j)j−s(Esj,n − Γ(s+ 1))

)
 N (0, 1)

 and Bn(f, s)→ 0

and the Lynderberg condition holds, that is, for any ε > 0,

g(n, ε) =
1

sn(f, s)

k(n)∑
j=1

∫
(|ξj |>εsn(f,s))

ξ2j dP→ 0.

But, for K2(s) = Γ(4s+ 1)− 4Γ(3s+ 1)Γ(s+ 1) + Γ(2s+ 1)Γ(s+ 1)2− 3Γ(3s+ 1)4,

Eξ4 = K(s)f(s)4j−4s

and, by the Cauchy-Schwarz inequality

∫
(|ξj |>εsn(f,s))

ξ2j dP

≤
(∫

ξ4j dP
)1/2(∫

1(|ξj |>εsn(f,s)) dP
)1/2

= Kf(j)2j2s
(∫

1(|ξj |>εsn(f,s)) dP
)1/2

= Kf(j)2j2sP (|ξj | > εsn(f, s)))1/2

≤ Kf(j)2j2s
(
K(s)2f(j)4j−4s

ε4s4n(f, s)

)1/2

= K(s)2
(
f(j)2j2s

)2
(s−2n (f, s)2



G.S. LO, M. Ngom, M.Diallo / Eur. J. Pure Appl. Math, 13 (4) (2020), 739-757 749

=
C(s)

K(s)
Bn(f, s)

Var(ξj)
s2 (f, s)

So

g(n, ε) =

(
K(s)

C(s)

)2

Bn(f, s)→ 0.

Our hypothesis Bn(f, s) → 0 makes the Lynderberg hold and the central limit theorem
holds for Sn(f, s), that is

Sn(f, s)− γsan(f, s)

sn(f, s)
 N (0, 1).

Now, let us return to the approximation (B) at page 747. We have that for s = 1, the
expression denoted as Cn between the pair of big parentheses should be equal to one as
explained before. If s > 1, we have σ2(s) =

∑
j≥1 j

−2(s−1) < +∞, we apply a theorem
of Kolmogorov (see [5], Proposition 25, page 233), Sn(Id, s − 1) weakly converges to the
random variable W (s) with variance σ2(s). Hence Cn = OP(1). We arrive at∣∣∣∣Tn(f, s)− an(f, s)

sn(f, s)
− γs(Sn(f, s)− an(f, s))

sn(f, s)

∣∣∣∣ ≤ OP

(
Sn(f, 1)

sn(f, s) log n

)
. (15)

The later bound goes to zero in probability if and only if Sn(f, 1)/(sn(f, s) log n) → 0.
Now, we have

an(f, s)

sn(f, s)

(
Tn(f, s)

an(f, s)
− γs

)
= Zn + oP(1).

If an(f, s)/sn(f, s)→ +∞, we can use the δ-method applied to g(t) = t1/s to get

an(f, s)

sn(f, s)

((
Tn(f, s)

an(f, s)

)1/s

− γ
)
 N (0, s−2γ2).�.

Remark. In [12], we gave a direct proof of the asymptotic normality of Sn(f, s) by using
the two hypotheses Bn(f, s) → 0 and sn(f, s) → +∞. Here, it seems that we only used
the first one. But that one could not hold if Sn(f, s) contains a sub-sequence converging
to a finite and positive number. That remark should be recalled in interpreting the results
in [12].



G.S. LO, M. Ngom, M.Diallo / Eur. J. Pure Appl. Math, 13 (4) (2020), 739-757 750

3. Upper records values

The main result is :

Theorem 3. If, for each n ≥ 1, X(n) stands for n-th record value, we have as n→ +∞,

X(n) − γn
γ
√
n

 N (0, 1).

Remark. We refer the reader to [8] for a simple introduction to records theory.

Proof. We already noticed that Z = exp(X) is the extremal domain of attraction of
Gγ(x) = exp(−(1 +γx)), for γx > −1. From Part (b) of Theorem 1 in [8], the n-th record
Z(n) = exp(X(n)) have the representation(

exp(X(n))

H−1(1− e−n

)1/
√
n

= exp(γS∗n) + oP(1) (16)

where S∗n has the same law as γ−1(Tn−n)/
√
n with Tn denoting a γ law with parameters

n and 1. Since H−1(1− u) = exp(F−1(1− u)), we have

X(n) − F−1 (1− e−n)

γ
√
n

= S∗n + oP(1) (17)

By the central limit theorem, it comes that

X(n) − F−1 (1− e−n)

γ
√
n

= N (0, 1) + oP(1). (18)

By using Formula (7), we get

X(n) − γn
γ
√
n

= S∗n + oP(1) (19)

X(n) − γn
γ
√
n

= N (0, 1) + oP(1). (20)

The proof is over. �



G.S. LO, M. Ngom, M.Diallo / Eur. J. Pure Appl. Math, 13 (4) (2020), 739-757 751

4. The moment problem

Typically, the moment problem on R(see [13] and more recently in [10]) is the following.
Given a sequences real numbers (mn)n≥1, can we find a distribution (not necessarily a
cdf ) F on R as the unique solution of the moments equations.

∀n ≥ 1, mn =

∫
xn dF (x).

This is a nice but a difficult mathematical question treated in [13] and more recently [10].
But in the context of probability theory on R, we may have a fixed cdf F of random
variable X having moments

∀n ≥ 1, EXn = mn finite.

The moment problem becomes : Is the sequence of moments (mn)n≥1 characterize the
probability law of X. In that regard, we have

Theorem 4. The moments of the pseudo-Lindely probability law are the following

∀n ≥ 1, mn =
n!(β + n)

θnβ
.

Any real-valued random variable have the moments (mn)n≥1 follows the pseudo-Limdley
law.

Proof. At the place of a simple proof, we proceed to slight round-up of the moment
problem and explain how to find a simple criteria based on Analysis. A possible tool is
the characteristic function which characterize its associated probability law. We have the
following expansion of any characteristic function of X (see [11] or [5], Lemma 5, page
255), we have

EeiuX = 1 +

n∑
k=1

(iu)kmk

k!
+ θ21−δµn+δ

|u|n+δ

(n+ 1)!
. (21)

By usual analysis tools, the series in Formula (21) converges in the ] − R,R[ where R is
found according the Cauchy rule

lim sup
n→+∞

(mn)1/n = R > 0.

The conclusion is that two random variables have the same moments of all orders have
characteristic functions coinciding on ] − R,R[. Finally, (see [11], page 225, Part B.; see
also [1]) two characteristic functions coinciding on an interval ]−R,R[ coincide everywhere



G.S. LO, M. Ngom, M.Diallo / Eur. J. Pure Appl. Math, 13 (4) (2020), 739-757 752

and thus, are associated to the same probability law.

Let us apply to the pseudo-Lindley law. In [15], the moments are given by

∀n ≥ 1, mn =
n!(β + n)

θnβ
.

Straightforward computation based on the Stirling formula leads to R = 1/θ. This is
enough to prove the claim of the theorem. (As remarked by the anonymous referee, We
might have use the Carleman criteria).�



G.S. LO, M. Ngom, M.Diallo / Eur. J. Pure Appl. Math, 13 (4) (2020), 739-757 753

Appendix . Let R = β/θ. In the computations below, u ∈ (0, 1) and x ≥ 0 are linked by
u = 1− F (x). So u→ 0 if and only if x→ +∞. Also, below, functions of x are functions
of u actually. We denote A(u) = log(1 +R/x). We have A(u)→ 0 as u→ 0. By writing

log(β + θx) = log(β + θx)− log θx+ log θx = log θx+A(u),

we see that u = 1− F (x) gives

θx = log(1/u) + logR+ log x+A(u). (22)

So, we have

F−1(1− u) = θ−1 log(1/u)(1 + o(1)). (23)

and

log x = log log(1/u)(1 + o(1)). (24)

Now, we wish to develop that asymptotic equivalence with rates of convergence. Let
B(u) = logR+ log x+A(u). From Formula 22, we have

x

θ−1 log(1/u)
− 1 =

B(u)

log(1/u)
. (25)

By Formula (25), we notice that

B(u) = logR+log x+(R/x)−(R/x)2/2+O(log(1/u)−3) = O(log x) = (log log u)(1+o(1)),
(26)

and hence, for D(u) = logR+A(u),

log(1/u)

log x

(
x

θ−1 log(1/u)
− 1

)
= 1 +

D(u)

log x
. (27)

Also

D(u)

log x
=

logR+ (R/x)− (R/x)2/2 +O(x−3

log x

Next, we have

log x

− logR

(
log(1/u)

log x

(
x

θ−1 log(1/u)
− 1

)
− 1

)
(28)



G.S. LO, M. Ngom, M.Diallo / Eur. J. Pure Appl. Math, 13 (4) (2020), 739-757 754

= 1 +
R

x logR
− R2

2x2 logR
+O(x−3)

and finally

x logR

R

(
log x

logR

(
log(1/u)

log x

(
x

θ−1 log(1/u)
− 1

)
− 1

)
− 1

)
(29)

= 1− R

2x
+O(x−2).

Now we want to do the same for log x. Hence, we get.

log(θx) = log log(1/u) + log(1 +B(u)/ log(1/u)) (30)

from which we get

log x− log log(1/u) = − log θ + (B(u)/ log(1/u)) +O
(
(B(u)/ log(1/u)2

)
. (31)

From Formula (27), we have

log(1/u)

log x

(
x

θ−1 log(1/u)
− 1

)
− log(1/u)

log log 1/u

(
x

θ−1 log(1/u)
− 1

)
=

(
x

θ−1 log(1/u)
− 1

)
−(log(1/u))(log x− log log 1/u)

(log x)(log log 1/u)

= (1 +D(u)/ log x)

(
1

(log x)(log log 1/u)

(
− log θ + (B(u)/ log(1/u)) +O(B(u)/ log(1/u)2)

))
= O((log log 1/u)2)

Formula (27) becomes

log(1/u)

log log 1/u

(
x

θ−1 log(1/u)
− 1

)
= 1 +

D(u)

log x
+O((log log 1/u)2). (32)

That formula will be used with Formula 31 and

B(u)

log 1/u
=

logR

log 1/u
+

log log 1/u

log 1/u
(1 + o(1)) (33)

+
(R/x)− (R/x)2/2

log 1/u
+O((log 1/u)−4).



G.S. LO, M. Ngom, M.Diallo / Eur. J. Pure Appl. Math, 13 (4) (2020), 739-757 755

From 22, and from the following formula we can check by using differentiation methods
to establish monotonicity

x− x2/2 ≤ log(1 + x) ≤ x

we have

(R/x)−R2/(2x2) + logR+ log x ≤ θx− log(1/u) ≤ (R/x) + logR+ log x. (34)

But we also have

x = log(1/u)

(
1 +

log β−1 + log x+A(u)

log(1/u)

)

which implies

log x = log log(1/u) + log

(
1 +

log β−1 + log x+A(u)

log(1/u)

)

By putting

H(u) =
log β−1 + log x+A(u)

log(1/u)
,

we finally get

H(u)−H(u)2/2 ≤ log x− log log(1/u) ≤ H(u). (35)

By combining Formulas (34) and (35), we get

|θx− log(1/u)− log(1/u)| ≤ 1

2

(
R2

x2
+H(u)2

)
. (36)

Since (R/x2) and H(u)2 are both O(log 1/u)−2), we have

F−1(1− u) = θ−1(log(1/u)− log log(1/u)) +O(log 1/u)−2). (37)

But since the derivative log log(1/u) is (−u log(1/u))−1, we have for d = − log log 2,

∀u ∈]0, 1[, log log(1/u)− =

∫ 1/2

u

1

u log(1/u
du,



REFERENCES 756

and finally

F−1(1− u) = d+ θ−1(log(1/u)−
∫ 1/2

u

1

u log(1/u)
du+O

(
(log 1/u)−2

)
. (38)

References

[1] P. Billingsley. Probability and Measure. Wiley, Third Edition, 1995.

[2] L. de Haan. On regular variation and its application to the weak convergence of sample
extremes. Mathematical Center Tracts,Amsterdam. (MR0286156), 1970.

[3] B. Hill. A simple general approach to the inference about the tail index of a distri-
bution. Ann. Statist., 3:1163–1174, 1975.

[4] G.S. Lo. Sur quelques estimateurs de l’Index d’une loi de Pareto : Estimateur de
Hill, de S.Csörgő-Deheuvels-Mason, de de Haan-Resnick et loi limites de sommes de
valeurs extrêmes pour une variable aléatoire dans le domaine d’attraction de Gumbel.
Université Pierre et Marie Curie, Sorbonne. Thèse de Doctorat, 1986.

[5] G.S. Lo. Mathematical foundation of probability theory. arXiv:1808.01713, doi :
10.16929/sbs/2016.0008, 2018.

[6] G.S. Lo, T. A. Kpanzou, and C.M. Haidara. Statistical tests for the pseudo-lindley
distribution and applications. Afrik. Statist., Doi :10.16929/as/2019.2127.151, 14
(4):2127–2139, 2019.

[7] G.S. Lo, T. A. Kpanzou, M. Ngom, and M. Diallo. Weak Convergence (IIA) -
Functional and Random Aspects of the Univariate Extreme Value Theory. Arxiv :
1810.01625, 2018.

[8] G.S. Lo and Ahnsanullah M. Asymptotic laws of the strong upper records values in
the extreme domain of attraction and beyond. Arxiv :1905.03380, 2019.

[9] G.S. Lo, M. Ngom, and T. A. Kpanzou. Weak Convergence (IA) - Sequences of
random vectors. arXiv:1810.01625, doi :10.16929/sbs/2016.0001, 2016.

[10] G.S. Lo, M. Touré, and A.B. Niang. Recent and Modern Approach to The Moment
Problem on R. In : Theory and Practice of Mathematics and Computer Science Vol.
1. [TPMCS-V1] (Ed. B. Chaouchi). ISBN: 978-93-90149-72-8 (print), 978-93-90149-
22-3 (ebook). Doi : DOI: 10.9734/bpi/tpmcs/v1. Book Publishing International,
Hooghly and London, 2020.

[11] M. Loève. Probability Theory I. Springer-Verlag, 4th Edition, 1997.

[12] M. Ngom and G.S. Lo. A double-indexed functional hill process and applications.
Journal of Mathematical Research (e-ISSN 1916-9809), Doi : 105539/jmr/v8n4p144,
8 (4):144–165, 2016.



REFERENCES 757

[13] J.A. Shohat and J.D. Tamarkin. The problem of moment, Mathematical Surveys and
monographs. American Society of Mathematics (Re-edited in 1950, 1963 and 1970),
1943.

[14] G.R. Shorack and J.A. Wellner. Empirical Processes with Applications to Statistics.
Wiley-Interscience, New-York, 1986.

[15] H. Zeghdoudi and S. Nedjar. A pseudo lindley-distribution and its application. Afrika
Statistika, DOI: http://dx.doi.org/10.16929/as/2016.923.83, 11 (1):923–932, 2016.