a generalization of karamata’s theorem on the … · 2018. 11. 16. · teor imovr. ta matem....

Teor�� Imov�r. ta Matem. Statist. Theor. Probability and Math. Statist.Vip. 81, 2010 No. 81, 2010, Pages 15–26

S 0094-9000(2010)00806-4Article electronically published on January 14, 2011

A GENERALIZATION OF KARAMATA’S THEOREM

ON THE ASYMPTOTIC BEHAVIOR OF INTEGRALSUDC 519.21

V. V. BULDYGIN AND V. V. PAVLENKOV

Abstract. A generalization of Karamata’s theorem on the asymptotic behavior ofintegrals of regularly varying functions with oscillating components is obtained in thepaper.

1. Introduction

Jovan Karamata [6, 7] introduced the notion of regularly varying (RV) functions andproved a number of fundamental theorems for them. The theorem on the asymptoticbehavior of integrals of RV functions is one of those results. This theorem found manyapplications in probability theory (see, for example, [5, 3]). Below we generalize thistheorem to a certain class of functions with a nondegenerate group of regular points;see [4].

Let R be the set of real numbers, R+ the set of positive numbers, Z the set of integers,and N the set of natural numbers.

We assume that A > 0 and let F+(A) be the set of positive and (Lebesgue) measurablefunctions f = (f(x), x ≥ A).

A function f ∈ F+(A) is called regularly varying (at infinity) in the Karamata senseif the limit

κf (λ) = limx→∞

f(λx)

f(x)

exists and is positive and finite for all λ > 0.We say that an RV function f is slowly varying (SV) if

κf (λ) = 1 for all λ > 0.

If f is an RV function, then there is a real number ρ such that

κf (λ) = λρ, λ > 0.

The number ρ is called the index of the function f . The index ρ = 0 characterizes SVfunctions.

Any RV function f of index ρ is represented in the following form:

f(x) = xρ�(x), x ≥ A,

where � is a corresponding SV function.A measurable real function ϕ(x), x ≥ A, is called locally integrable if it is (Lebesgue)

integrable on every interval [a, b] ⊂ [A, ∞).

2010 Mathematics Subject Classification. Primary 26A12, 26A48; Secondary 34C41.Key words and phrases. Regularly varying functions, Karamata’s theorem, asymptotic behavior of

integrals, oscillating functions.

c©2010 American Mathematical Society

15

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16 V. V. BULDYGIN AND V. V. PAVLENKOV

Recall that any of the following three relations: ϕ(x)∼x→∞ ψ(x), ϕ(x) ∼ ψ(x), andϕ ∼ ψ as x → ∞ means that

limx→∞

ϕ(x)

ψ(x)= 1.

We say that the functions ϕ and ψ are asymptotically equivalent (at infinity) in this case.There are two parts of the Karamata theorem, namely the forward and backward

Karamata theorems (see [6, 7] for continuous RV functions and [3, 5, 8] for locally inte-grable RV functions). In this paper, we consider the case of ρ > −1.

Forward theorem. Let f be a locally integrable regularly varying function of indexρ > −1. Then

(1)

∫ x

A

f(t) dt ∼x→∞

xf(x)

ρ+ 1.

Backward theorem. Let f be a locally integrable function belonging to F+(A). If thereis a number γ ∈ (0,∞) such that

(2)

∫ x

A

f(t) dt ∼x→∞

xf(x)

γ,

then f is an RV function of index ρ = γ − 1.

The aim of the paper is to generalize the above two theorems to the case of functionsf ∈ F+(A) such that

(3) f(x) = xρ�(x)H(lnx), x ≥ A,

where ρ ∈ R, � = (�(x), x ≥ A) is a slowly varying function and

H = (H(u), u ∈ R)

is a positive continuous periodic function.The class of functions introduced above is denoted by Φ.

2. Definitions and auxiliary results

Consider some properties of functions of the class Φ (that is, the functions representedin the form (3)). First of all, the functions of this class are positive and measurable. Thusall the factors in representation (3) are also positive and measurable functions.

The function H on the right hand side of (3) is positive, continuous, and periodic.This function is constant; that is, there exists a constant c > 0 such that H(x) = c forall x ∈ R if and only if f is a regularly varying function. In this case, H can be removedfrom representation (3) by adjoining this constant to the function �. Otherwise, H is nota slowly varying function and one cannot neglect it in representation (3).

In what follows we assume (if the opposite is not stated) that

(4) H(0) = 1.

Condition (4) does not restrict the class Φ, since the function f ∈ Φ can be representedas follows:

f(x) = xρ�0(x)H0(lnx), x ≥ A,

where H0(u) = H(u)/H(0) and �0(x) = H(0)�(x). It is clear that H0(0) = 1 and �0 is aslowly varying function.

The function

r(x) = xρ�(x), x ≥ A,


A GENERALIZATION OF KARAMATA’S THEOREM 17

is called the regular (RV) component of f ; the index ρ of the function r is called the indexof the function f (ρf = ρ). It is clear that

ρ = limx→∞

ln f(x)

lnx.

The function � is called the slowly varying (SV) component of f . Note that the function� is allowed to be discontinuous.

The function H is called the oscillating function or oscillating component of f .If ϕ = (ϕ(x), x ∈ R) is a continuous periodic function, then we denote by Sper(ϕ) the

set of periods of the function ϕ; respectively, by Spper(ϕ) we denote the set of positiveperiods of the function ϕ.

The number

T (ϕ) = inf Spper(ϕ)

is called the oscillating characteristic of the function ϕ. Since the function ϕ is contin-uous, either T (ϕ) = 0 (in which case the function ϕ is constant) or T (ϕ) > 0. In thelatter case, T (ϕ) is the least positive period of the function ϕ, that is,

Sper(ϕ) = {nT (ϕ), n ∈ Z} .If H is the oscillating component of a function f , then its characteristic T (H) is also

called the oscillating characteristic of the function f and is denoted by T (Hf ).The class of functions f ∈ Φ such that T (Hf ) > 0 is denoted by Φ+. Note that the

members of the class Φ+ are not regularly varying; they are so-called functions with anondegenerate group of regular points; see [4].

The class

Φ0 = Φ \ Φ+

contains functions f of Φ that are represented in the form (3) with functions H constant,whence, under condition (4),

H(x) ≡ 1.

Thus Φ0 coincides with the class of RV functions.If a function f belongs to the class Φ and is of index ρ > −1, then

(5)

∫ ∞

A

f(x) dx = ∞.

Indeed, ∫ ∞

A

r(x) dx = ∞

for an arbitrary RV function r(x), x ≥ A, of index ρ > −1 (see, for example, [3, 5, 8]).Thus (5) follows from the inequality∫ ∞

A

f(x) dx =

∫ ∞

A

xρ�(x)Hl(x) dx ≥ κ

∫ ∞

A

xρ�(x) dx = ∞,

where κ = infx∈R H(x) > 0.

3. Main results

We now state two theorems (forward and backward ones) generalizing Karamata’stheorem to functions belonging to the class Φ. The proofs will be given in furthersections where we use and develop some of the methods of the papers [6, 7] and [3, 5, 8].

We start with a generalization of the forward theorem; see (1).



Theorem 3.1 (Forward theorem). Let f ∈ Φ be a locally integrable function of indexρ > −1 whose oscillating component and characteristic are denoted by H and T (H) = T ,respectively. Then there is a positive continuous periodic function D = (D(x), x ∈ R)that depends on ρ and H and satisfies

(6)

∫ x

A

f(t) dt ∼x→∞

xD(lnx)f(x).

Moreover,

1) T (DH) = T (D) = T (H) = T , where T (D) is the oscillating characteristic of thefunction D and T (DH) is the oscillating characteristic of the function

DH = (D(x)H(x), x ∈ R).

2) If f is an RV function, that is, H(x) ≡ 1 and T = 0, then

(7) D(x) = DH,ρ(x) ≡1

ρ+ 1.

3) If f ∈ Φ+, that is, T > 0, then

(8) D(x) = DH,ρ(x) =

∫ T{x/T}0

e(ρ+1)yH(y) dy + C

H(x) exp (T (ρ+ 1){x/T}) , x ≥ 0,

where

C = CH,ρ =

∫ T

0e(ρ+1)yH(y)dy

eT (ρ+1) − 1

and {x/T} stands for the fractional part of the number x/T .

We now turn to a generalization of the backward theorem (see (2)).

Theorem 3.2 (Backward theorem). Let f be a locally integrable function of the classF+(A). If there is a positive continuous periodic function B = (B(x), x ∈ R) with oscil-lating characteristic T (B) such that

(9)

∫ x

A

f(t) dt ∼x→∞

xB(lnx)f(x),

then the function f belongs to the class Φ, its index ρ is > −1, and its oscillatingcharacteristic is H = (H(x), x ∈ R).

Moreover,

1) T (BH) = T (H) = T (B), where T (H) is the oscillating characteristic of thefunction H and T (BH) is the oscillating characteristic of the function

BH = (B(x)H(x), x ∈ R).

2) If T (B) = 0, that is, B(x) ≡ β > 0, then

(10) ρ =1

β− 1

and H(x) ≡ 1, that is, f is an RV function of index (10).3) If T (B) > 0, then f belongs to the class Φ+,

(11) ρ =1

T (B)

∫ T (B)

0

du

B(u)− 1,

and

H(x) =B(0)

B(x)exp

(∫ x

0

(1

B(t)−(B−1

)av

)dt

), x ≥ 0,



where (B−1

)av

=B(0)

T (B)

∫ T (B)

0

du

B(u).

Remark 3.1. Relation (9) implies that

lim infx→∞

xf(x) > 0

and ∫ ∞

A

f(t) dt = ∞.

Remark 3.2. Relations (6) and (9) uniquely determine the functions D and B, respec-tively. For example, if (6) holds for two positive continuous periodic functionsD1 andD2,then D1 ∼ D2, whence D1 = D2. Thus D = B in Theorems 3.1 and 3.2, and the relationsthat hold for the function B also hold for the function D, and vice versa.

Remark 3.3. Putting H(x) → 1, x ∈ R, and T → 0 in (8), we get

DH,ρ(x) →1

ρ+ 1, x ∈ R,

that is, equalities (7) and (8) are compatible. Similar compatibility holds for relations(10) and (11).

Remark 3.4. Let I(x) =∫ x

Af(t) dt, x > A. Theorem 3.1 implies that the function I

belongs to the class Φ if so does the function f and ρf > −1. Moreover ρI = ρf + 1,HI = HfB, and T (HI) = T (Hf ) in this case.

4. Lemma on the asymptotic behavior of integrals

We start with the following result for functions of the class Φ. With minor changes, itsproof follows the lines of that of the Karamata theorem for regularly varying functions(see, for example, [3, 5]. For the sake of convenience we set Hl(x) = H(lnx).

Lemma 4.1. Let f ∈ Φ be a locally integrable function of index ρ > −1 whose SVcomponent is denoted by � and whose oscillating component is denoted by H (see (3)).Then

(12)

∫ x

A

f(t) dt ∼x→∞

�(x)

∫ x

A

tρHl(t) dt.

Proof. Let

J(x) =

∫ 1

A/x

yρ(�(xy)

�(x)− 1

)Hl(xy) dy, x ≥ A,

and

Jε(x) =

∫ 1

ε

yρ(�(xy)

�(x)− 1

)Hl(xy) dy, x ≥ (A/ε),

for ε ∈ (0, 1).Since H is a positive continuous periodic function and ρ > −1, we conclude that, for

all ε ∈ (0, 1) and x ≥ (A/ε),

(13) |Jε(x)| ≤ K

∫ 1

ε

yρ∣∣∣∣�(xy)�(x)

− 1

∣∣∣∣ dy ≤ Ksupy∈[ε,1]

∣∣∣∣ l(xy)l(x)− 1

∣∣∣∣ ,where

K =supx∈R H(x)

ρ+ 1< ∞.



According to the uniform convergence theorem for slowly varying functions (see, forexample, [8]),

limx→∞

supy∈[ε,1]

∣∣∣∣ l(xy)l(x)− 1

∣∣∣∣ = 0

for all ε ∈ (0, 1).This together with (13) implies that

(14) limx→∞

Jε(x) = 0

for all ε ∈ (0, 1). Further,

|J(x)− Jε(x)| ≤∫ ε

A/x

yρ∣∣∣∣�(xy)�(x)

− 1

∣∣∣∣Hl(xy) dy ≤ K

(ερ+1 + (ρ+ 1)

∫ ε

A/x

yρ�(xy)

�(x)dy

)

for all ε ∈ (0, 1) and x ≥ (A/ε).Choosing δ ∈ (0, ρ+ 1), the Potter theorem (see, for example, [3]) implies that there

is a number x(δ) > 0 such that∫ ε

A/x

yρ�(xy)

�(x)dy ≤ 2

∫ ε

A/x

yρ−δ dy ≤ 2ερ+1−δ

ρ+ 1− δ

for x ≥ max{x(δ), (A/ε)}. Thus

lim supx→∞

|J(x)− Jε(x)| ≤ K

(ερ+1 +

2(ρ+ 1)ερ+1−δ

ρ+ 1− δ

)

for all ε ∈ (0, 1), whence

(15) lim supε→0

lim supx→∞

|J(x)− Jε(x)| = 0,

since ρ+ 1− δ > 0.Relations (14) and (15) imply that

limx→∞

J(x) = 0,

whence

(16)

∫ 1

A/x

yρ�(xy)Hl(xy) dy ∼x→∞

�(x)

∫ 1

A/x

yρHl(xy) dy

in view of

lim infx→∞

∫ 1

A/x

yρHl(xy) dy > 0.

To complete the proof of the lemma we note that∫ x

A

f(t) dt =

∫ x

A

tρ�(t)Hl(t) dt = xρ+1

∫ 1

A/x

yρ�(xy)Hl(xy) dy, x ≥ A,

according to representation (3). In view of asymptotics (16),∫ x

A

f(t) dt = xρ+1

∫ 1

A/x

yρ�(xy)Hl(xy) dy

∼ xρ+1�(x)

∫ 1

A/x

yρHl(xy) dy = �(x)

∫ x

A

tρHl(t) dt

as x → ∞ and relation (12) follows. �



Remark 4.1. Since ρ > −1 in Lemma 4.1, equality (5) implies that relation (12) isequivalent to the condition that, for all x1 ≥ A and x2 > 0, the integrals∫ x

x1

f(t) dt ∼x→∞

�(x)

∫ x

x2

tρHl(t) dt

are well defined.

Remark 4.2. Lemma 4.1 with H(x) ≡ 1 implies the forward Karamata theorem for RVfunctions (see (1)).

5. The first step of the proof of the forward theorem for functions

belonging to the class Φ+

Since Lemma 4.1 implies the forward Karamata theorem for RV functions, its gener-alization for the class Φ follows from that for the class Φ+. The following result is thefirst step of the corresponding proof; the rest of the proof of the forward theorem is givenin Section 7.

Lemma 5.1. Let f ∈ Φ+ be a locally integrable function of index ρ > −1 and let itsoscillating component and oscillating characteristic be H and T (H) = T , respectively.Then

(17)

∫ x

A

f(t) dt ∼x→∞

xD(lnx)f(x),

where (D(x), x ∈ R) is a positive continuous periodic function such that

D(x) = DH,ρ(x) =

∫ T{x/T}0

e(ρ+1)yH(y) dy + C

H(x) exp (T (ρ+ 1){x/T}) , x ≥ 0,

C = C(H, ρ) =

∫ T

0e(ρ+1)yH(y) dy

eT (ρ+1) − 1;

here {x/T} stands for the fractional part of the number x/T (H). Moreover,

(18) T (D) ≤ T (H),

where T (D) is the oscillating characteristic of the function D.

Proof. Lemma 4.1 together with relation (5) and Remark 4.1 implies that

(19)

∫ x

A

f(t) dt ∼x→∞

�(x)U1(x),

where � is the SV component of the function f (see representation (3)) and

U1(x) =

∫ x

1

tρH(ln t) dt, x ≥ 1.

It is clear that

U1(x) =

∫ lnx

0

e(ρ+1)uH(u) du = V (lnx),

where

V (x) =

∫ x

0

e(ρ+1)uH(u) du =

[x/T ]−1∑k=0

∫ (k+1)T

kT

e(ρ+1)uH(u) du+R(x),

R(x) =

∫ x

T [x/T ]

e(ρ+1)uH(u) du,

and [x/T ] is the integer part of the number x/T .



Since H is a periodic function with period T ,

∫ (k+1)T

kT

e(ρ+1)uH(u) du =

∫ T

0

e(ρ+1)(y+kT )H(y) dy = ekT (ρ+1)

∫ T

0

e(ρ+1)yH(y) dy

for all k ∈ N. Thus

V (x) = C(eT (ρ+1)[x/T ] − 1

)+R(x),

where

C =

∫ T

0e(ρ+1)yH(y) dy

eT (ρ+1) − 1> 0.

Moreover,

R(x) =

∫ x

T [x/T ])

e(ρ+1)uH(u) du = eT (ρ+1)[x/T ]

∫ T{x/T}

0

e(ρ+1)uH(u) du.

After simple algebra we obtain

(20) U1(x) = xρ+1Θ(lnx)− C, x ≥ 1,

with

Θ(x) =

∫ T{x/T}0

e(ρ+1)yH(y) dy + C

exp (T (ρ+ 1){x/T}) , x ≥ 0.

The function Θ is positive for x ≥ 0 and periodic with period T . It is straightforwardto check that Θ is continuous. Denote by Θ = (Θ(x), x ∈ R) the continuous periodicextension of this function to the whole axis R.

By the assumption of Lemma 5.1, ρ > −1, and thus equality (20) implies that thelimit limx→∞ U1(x) is infinite and

U1(x) ∼x→∞

xρ+1Θ(lnx).

Taking into account asymptotics (19), we prove relation (17) with

D(x) =Θ(x)

H(x).

It remains to note that (D(x), x ∈ R) is a positive continuous periodic function withperiod T , whence inequality (18) follows. �

6. The proof of the backward theorem

Theorem 6.1 below allows one to complete the proof of Theorem 3.2. It also containssome complements to that theorem. In the proof of Theorem 6.1 we use Lemma 5.1.

The proof of Theorem 3.2 will be given after the proof of Theorem 6.1.Note that the rest of the proof of Theorem 3.1 uses Theorem 3.2 (see Section 7).

Theorem 6.1. Let f be a locally integrable function belonging to F+(A). If there area number γ ∈ (0,∞) and a positive continuous periodic function (Γ(x), x ∈ R) whoseoscillating characteristic is T (Γ) = T with Γ(0) = 1 and

(21)

∫ x

A

f(t) dt ∼x→∞

xΓ(lnx)f(x)

γ,

then f belongs to the class Φ, its index ρ > is −1, and its oscillating component andcharacteristic are H = (H(x), x ∈ R) and T (H) = T , respectively.



Moreover,

1)

(22) T (ΓH) = T (H) = T,

where T (H) is the oscillating characteristic of the function H, and T (ΓH) is theoscillating characteristic of the function ΓH = (Γ(x)H(x), x ∈ R);

2) T = 0 if and only if H(x) ≡ 1 and

(23) ρ = γ − 1,

that is, if and only if f is a regularly varying function of index (23);3) T > 0 if and only if f ∈ Φ+, ρ = γ

(Γ−1

)av

− 1, and

H(x) =1

Γ(x)exp

(γ

∫ x

0

(1

Γ(t)−(Γ−1

)av

)dt

), x ≥ 0,

where (Γ−1

)av

=1

T

∫ T

0

du

Γ(u).

Proof. Put

(24) b(x) =xf(x)Γ(lnx)∫ x

Af(t) dt

, x > A.

According to the asymptotic formula (21),

(25) limx→∞

b(x) = γ.

Equality (24) implies thatb(x)

xΓ(lnx)=

f(x)∫ x

Af(t) dt

.

Integrating both sides, we see that∫ x

A+1

b(t) dt

tΓ(ln t)= ln

(∫ x

A

f(t) dt

)− ln

(∫ A+1

A

f(t) dt

)

for all x > A+ 1. Thus∫ x

A

f(t) dt = a · exp(∫ x

A+1

b(t) dt

tΓ(ln t)

), x > A+ 1,

where

a =

∫ A+1

A

f(t) dt.

This equality together with (24) implies the following representation:

f(x) =ab(x)

xΓ(lnx)exp

(∫ x

A+1

b(t) dt

tΓ(ln t)

), x ≥ A+ 1.

For convenience, we rewrite this representation as follows:

f(x) =ab(x)

xΓ(lnx)exp

(∫ x

A

b(t)− γ

tΓ(ln t)dt

)exp

(γ

∫ x

A

dt

tΓ(ln t)

).

For x > A+ 1, put

�1(x) = ab(x)c(x),

where

c(x) = exp

(∫ x

A

b(t)− γ

tΓ(ln t)dt

).



According to (25),

limt→∞

(b(t)− 1)

Γ(ln t)= 0,

whence we conclude that (c(x), x > A+1) is an SV function by the integral representationtheorem for slowly varying functions (see [3, 8]). The function (�1(x), x > A+ 1) also isan SV function as a product of SV functions.

Moreover, ∫ x

A+1

dt

tΓ(ln t)=

∫ ln x

x0

du

Γ(u)=

∫ ln x

0

du

Γ(u)−∫ x0

0

du

Γ(u)

for x ≥ A+ 1, where x0 = ln(A+ 1). Hence

(26) f(x) =�(x)

xΓ(lnx)M(lnx), x ≥ A+ 1,

where

M(x) = exp

(γ

∫ x

0

dt

Γ(t)

), x ≥ 0,

and � is a slowly varying function such that

�(x) = �1(x) exp

(−γ

∫ x0

0

dt

Γ(t)

), x > A+ 1.

Now we define a number(Γ−1

)av

and a positive continuous periodic function h =

(h(x), x ∈ R). Let (Γ−1

)av

=1

Γ(0)= 1

if T = 0, that is, if the function Γ is constant, and let

(Γ−1

)av

=1

T

∫ T

0

dt

Γ(t)

if T > 0. Moreover, let

h(x) ≡ 0

if T = 0, and let

h(x) =

∫ x

0

(1

Γ(t)−(Γ−1

)av

)dt, x ≥ 0,

if T > 0. Note that T is a period of h. Then∫ x

0

dt

Γ(t)= x

(Γ−1

)av

+

∫ x

0

(1

Γ(t)−(Γ−1

)av

)dt = x

(Γ−1

)av

+ h(x), x ≥ 0,

whence it follows that

M(x) = exp(xγ

(Γ−1

)av

+ γh(x)), x ≥ 0.

Taking into account equality (26), we represent the function f as follows:

(27) f(x) = xρ�(x)H(lnx), x > A+ 1,

whereρ = γ

(Γ−1

)av

− 1,

H(x) = exp(h(x)), x ∈ R,

and

(28) h(x) = γh(x)− ln Γ(x), x ∈ R.



Since H is a positive continuous periodic function with period T and H(0) = 1,equality (3) holds, which implies that the function f belongs to the class Φ and is ofindex

(29) ρ = γ(Γ−1

)av

− 1 > −1.

If T = 0, then Γ(x) ≡ 1. This implies that H(x) ≡ 1, that is, f is a regularly varyingfunction of index ρ = γ − 1.

Let H(x) ≡ 1. Then representation (28) implies that the positive continuous periodicfunction Γ is a continuously differentiable solution of the following first order lineardifferential equation with constant coefficients:

dΓ(x)

dx= γ

(Γ−1

)av

Γ(x)− γ, x ∈ R.

This equation holds if and only if Γ(x) ≡ 1, that is, T = 0.Therefore, the equality T = 0 holds if and only if f is a regularly varying function of

index ρ = γ − 1. This proves statement 2) of Theorem 6.1.In its turn, the inequality T > 0 holds if and only if the function H is not constant.

In the latter case, H is a positive continuous periodic function with period T . Thus

(30) 0 < T (H) ≤ T.

Hence T > 0 if and only if f ∈ Φ+. This together with (27)–(29) implies statement 3) ofTheorem 6.1.

To complete the proof of Theorem 6.1, one needs to prove equality (22). If f isa regularly varying function, then H(x) ≡ Γ(x) ≡ Γ(x)H(x) ≡ 1. This means thatT (H) = T (Γ) = T (ΓH) = 0 and thus (22) follows.

Now let f ∈ Φ+. Since H is the oscillating component of the function f , Lemma 5.1implies that if condition (29) holds, then there is a positive continuous periodic functionDfor which relation (17) holds and

(31) T (D) ≤ T (H).

Condition (21) implies that relation (17) also holds for the positive continuous periodicfunction

D1 =1

γΓ.

Therefore, two positive continuous periodic functionsD andD1 are asymptotically equiv-alent, which happens only if these functions coincide, whence we conclude that

T (D) = T (D1) = T.

Taking into account (30) and (31), we get

(32) T (H) = T.

Now consider the function ΓH = (Γ(x)H(x), x ∈ R). We derive from (28) that

Γ(x)H(x) = exp

(∫ x

0

(1

Γ(t)−(Γ−1

)av

)dt

), x ≥ 0.

Since f ∈ Φ+, that is, T > 0, the positive periodic function ΓH has period T . We showthat the function ΓH does not have a smaller period. This is the case if the function

Ψ(x) = ln(Γ(x)H(x)) =

∫ x

0

(1

Γ(t)−(Γ−1

)av

)dt, x ≥ 0,

does not have a period smaller than T . To the contrary, assume that there is a numberτ ∈ (0, T ) such that

Ψ(x+ τ ) = Ψ(x)



for all x > 0. The function Γ is positive and continuous, thus the latter equality impliesthat

Γ(x+ τ ) =

(dΨ(x+ τ )

dx

)−1

=

(dΨ(x)

dx

)−1

= Γ(x)

for all x > 0. This contradicts the assumption that T is the minimal positive period ofthe function Γ. Thus T (ΓH) = T and this together with (32) proves (22).

Theorem 6.1 is completely proved. �Proof of Theorem 3.2. Theorem 3.2 follows from Theorem 6.1 with

Γ(x) =B(x)

B(0)and γ =

1

B(0). �

7. The proof of the forward theorem

Now we are in a position to complete the proof of Theorem 3.1.

Proof. Statement 2) follows from Lemma 4.1, statement 1) follows from statement 1) ofTheorem 3.2, and statement 3) follows from Lemma 5.1. �

8. Concluding remarks

A generalization of Karamata’s theorem on the asymptotic behavior of integrals for RVfunctions of index ρ > −1 is obtained in the paper for the case of verying upper limit ofintegration. A generalization is also given for functions with oscillating components. Theresults of the paper allow one to study Abelian and Tauberian theorems for functions withoscillating components, which are important for some problems in probability theory.

Bibliography

1. S. Aljancic and D. Arandelovic, O-regularly varying functions, Publ. Inst. Math. (Beograd)(N.S.) 22(36) (1977), 5–22. MR0466438 (57:6317)

2. V. G. Avakumovic, Uber einen O-Inversionssatz, Bull. Int. Acad. Youg. Sci. 1936, no. 29-30,107–117.

3. N. H. Bingham, C. M. Goldie, and J. L. Teugels, Regular Variation, Cambridge UniversityPress, Cambridge, 1987. MR898871 (88i:26004)

4. V. V. Buldygin, O. I. Klesov, and J. G. Steinebach, On factorization representations for Avaku-movic–Karamata functions with nondegenerate groups of regular points, Anal. Math. 30 (2004),161–192. MR2093756 (2005f:26029)

5. L. de Haan, On Regular Variation and its Application to the Weak Convergence of SampleExtremes, Math. Centre Tracts, vol. 32, Amsterdam, 1975. MR0286156 (44:3370)

6. J. Karamata, Sur un mode de croissance reguliere des fonctions, Mathematica (Cluj) 4 (1930),38–53.

7. J. Karamata, Sur un mode de croissance reguliere. Theoremes fondamentaux, Bull. Soc. Math.France 61 (1933), 55–62. MR1504998

8. E. Seneta, Regularly Varying Functions, Springer, Berlin, 1976. MR0453936 (56:12189)

Department of Mathematical Analysis and Probability Theory, National Technical Uni-

versity of Ukraine (“KPI”), Peremogy Avenue 37, Kyiv 03056, Ukraine

E-mail address: [email protected]

Department of Mathematical Analysis and Probability Theory, National Technical Uni-

versity of Ukraine (“KPI”), Peremogy Avenue 37, Kyiv 03056, Ukraine

Received 3/NOV/2009

Translated by O. KLESOV


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