Volume 2008, Article ID 748929,35pages doi:10.1155/2008/748929
Research Article
New Limit Formulas for the Convolution of
a Function with a Measure and Their Applications
Istv ´an Gy ˝ori and L ´aszl ´o Horv ´ath
Department of Mathematics and Computing, University of Pannonia, 8200 Veszpr´em, Egyetem u. 10, Hungary
Correspondence should be addressed to Istv´an Gy˝ori,[email protected]
Received 16 July 2008; Accepted 23 September 2008
Recommended by Martin J. Bohner
Asymptotic behavior of a convolution of a function with a measure is investigated. Our results give conditions which ensure that the exact rate of the convolution function can be determined using a positive weight function related to the given function and measure. Many earlier related results are included and generalized. Our new limit formulas are applicable to subexponential functions, to tail equivalent distributions, and to polynomial-type convolutions, among others.
Copyrightq2008 I. Gy˝ori and L. Horv´ath. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction
This paper investigates the existence of the limit of the ratio of a convolution and a positive valued weight function. The limit is given by an explicit formula in terms of the elements in the convolution and of the weight function. Our results are formulated for the convolution of a function with a measure and also for the convolution of two functions.
Our work was inspired by two different applications. One of them is the asymptotic stability theory of differential and integral equations, where an important question is to determine the exact convergence rate to the steady state. The second one is related to the asymptotic representation of the distribution of the sum of independent random variables. In the above and several similar problems, the weighted limits of convolutions play important role with different types of weights.
Let μ be a given measure on the Borel sets of 0,∞ and let f : 0,∞→R be a measurable function. The convolutionf∗dμis defined by
f∗dμt:
t
0
ft−sdμs 1.1
The convolution of two locally Lebesgue integrable functions f, g : 0,∞→R is defined by
f∗gt:
t
0
ft−sgsds 1.2
fort∈0,∞for which the integral exists.
The motivation of our work came from the following three known results.
The first well-known result has been used frequently in the asymptotic theory of the solutions of differential and integral equationssee, e.g.,1.
Theorem 1.1. Letf, g:0,∞→Rbe locally integrable and assume that
f∞: lim
t→ ∞ft∈R,
∞
0
|gt|dt <∞. 1.3
Then
lim
t→ ∞f∗gt f∞
∞
0
gtdt. 1.4
The next well-known simple result plays a central role, for instance, in the asymptotic theory of fractional differential and integral equationssee, e.g.,2–5.
Theorem 1.2. Letα, β >0be given. Then
lim
t→ ∞ 1 tαβ−1
t
0
t−sα−1sβ−1dsBα, β, 1.5
whereBα, βis the well-known Beta function.
The third known result is formulated for continuous subexponential weight functions. A continuous functionγ:0,∞→0,∞is subexponential if
lim
t→ ∞
γ∗γt γt 2
∞
0
γtdt <∞, 1.6
lim
t→ ∞ γt−s
γt 1, for any fixeds >0. 1.7
The terminology is suggested by the fact that 1.7 implies that for every α > 0 limt→ ∞γtexpαt ∞.
Theorem 1.3. Let the weight functionγbe continuous and subexponential. Iff, g:0,∞→Rare continuous functions such that
Lγf: lim t→ ∞
ft
γt, Lγg:tlim→ ∞
gt
γt 1.8
are finite, then
Lγf∗g: lim t→ ∞
f∗gt
γt Lγf
∞
0
gLγg
∞
0
f. 1.9
Based on the above three known results, we conclude the next observations.
iAll of the above theorems give different limit formulas for the ratiof∗g/γat∞. In factγt 1, t≥0,inTheorem 1.1andft tα−1, gt tβ−1, γt tαβ−1, t≥0,
inTheorem 1.2.
iiThe weight functions in Theorems1.1and1.2satisfy condition1.7, but they do not satisfy condition1.6.
iiiThe condition forg in1.8is not necessarily true inTheorem 1.1. Instead of that limt→ ∞1/γttt−1g0 holds, whereγt 1, t≥0.
ivLγf Lγg 0 inTheorem 1.2and at the same time Lγf∗g Bα, βis not
zero.
Our first goal is to prove results which unify the above-mentioned theorems. Second, we want to extend the limit formulas for the convolution of a function with a measure. This makes possible the applications of our theorems to not only density but also distribution functions.
In fact we prove limit formulas which contain three terms, and the weight function does not satisfy condition1.6. The major idea in the proofs of the main results is borrowed from the theory of subexponential functions. Namely, for large enought, in factt ≥2T >0, the convolutionf∗dμtcan be split into three terms:
f∗dμt
0,Tft−sdμs
T,t−Tft−sdμs
t
t−T
ft−sdμs. 1.10
Under suitable assumptions and some time-tricky and technical treatments of the above three terms, we get the limit formula
Lγf∗dμ Lγfμ0,∞ lγf, μ Lγμ,1
∞
0
where the following limits are finite:
Lγf: lim t→ ∞
ft γt,
lγf, μ: lim T→ ∞
lim
t→ ∞
T,t−Tft−sdμs
,
Lγμ,1: lim t→ ∞
μt−1, t γt .
1.12
In the limit formula1.11, the termsLγfμ0,∞andLγμ,1
∞
0f are interpreted as zero wheneverLγf 0 andLγμ,1 0, respectively. So the values ofμ0,∞and
∞
0 f need not be finite in the applications.
The limit formula1.11can be reformulated for the convolution of two functionsf andg. Formally, it can be done if the measureμis such thatμB: Bg for every Borel set B⊂R.
In that caselγf, μ lγf, gandLγμ,1 Lγg,1, where
lγf, g: lim T→ ∞
lim
t→ ∞
t−T
T
ft−sgsds
,
Lγg,1: lim t→ ∞
1 γt
t
t−1gsds.
1.13
These indicate that our remarksi–ivare taking into account and the known Theorems1.1,
1.2, and1.3are unified in our results.
The organization of the paper is as follows. Section 2 contains notations and definitions. Section 3 lists and discusses the main results both for the convolution of a function with a measure and for the convolution of two functions. InSection 4we present the corollaries of our main results for subexponential and long-tailed distributions. InSection 5
we show that our results can be easily reformulated to an extended set of weight functions.
Section 6gives some corollaries of our main results for the case when the weight function
is of polynomial type. These results have possible applications in the asymptotic theory of fractional differential and integral equations. The proofs of the main results are given in
Section 8based on some preliminary statements stated and proved inSection 7.
2. The basic notations and definitions
First we introduce some notations . The set of real numbers is denoted byR, andRdenotes
the set of nonnegative numbers.
In our investigations we will make use of different sets of measures and functions given in the next definitions.
Definition 2.1. LetBbe theσ-algebra of the Borel sets ofR.Mdenotes the set of measuresμ
Let a, b ∈ R. In this paper we will write bafdμorbafsdμs for the μ-integral of f on the closed interval a, b. The μ-integral of f on the interval a, b is written as
a,bfdμor
a,bfsdμs.Whenμλ, instead of
b
afdλwe also write
b
af or
b
afsds. Definition 2.2. Ldenotes the set of functionsf : R→Rwhich are Lebesgue integrable on
any compact subset ofR. As usual,
L1
f∈ L:
∞
0
|f|<∞
. 2.1
Definition 2.3. Fb is the set of the Borel measurable functionsf :R→Rwhich are bounded
on any compact subset ofR.
Definition 2.4. A measureμfromMbelongs to the setMcif it is absolutely continuous with
respect toλ. In this caseμ gλ means thatg is a nonnegative function fromLsuch that μB Bgdλfor everyB∈ Bgis the Radon-Nikodym derivative ofμwith respect toλ.
It is not difficult to show that for anyf∈Fbandμ∈ Mthe convolution
f∗dμt:
t
0
ft−sdμs 2.2
of f and μ is well defined on R. It is known see, e.g., 9 that for any f, g ∈ L the convolution
f∗gt:
t
0
ft−sgsds 2.3
off and g is well defined for λ almost everyshortly a.e.t ∈ R. It follows that for any
μ∈ Mcandf∈ Lthe convolutionf∗dμis well defined for a.e.t∈R,and
f∗dμt f∗gt, a.e.t∈R, 2.4
whereμgλ.
In this paper our major goal is to give conditions—possibly sharp—which guarantee the existence of the finite limit of the ratio
1
γtf∗dμt 2.5
ast→ ∞. The weight functionγ : R→0,∞will belong to some special classes of the
Definition 2.5. LetΓbe the set of the functionsγ:R→0,∞such that
lim
t→∞
γt−s
γt 1 for any fixeds≥0. 2.6
The set of the functionsγ ∈Γfor which the above convergence is uniform on any compact interval0, Tis denoted byΓu.
It is clear that ifγ∈Γ, then
lim
t→∞ γts
γt tlim→∞
γts−s γts
−1
1 2.7
holds fors≥0,and hence it holds for anys∈R. Therefore for allβ >0,we have
lim
t→∞
γlnβt
γlnt tlim→∞
γlntlnβ
γlnt 1, 2.8
that is the function 1≤t→γlntis so called regularly varying at infinity. Thus applying the Karamata uniform convergence theorem see, e.g., 10it follows that the convergence in
2.8is uniform inβon any compact set of0,∞,assuming thatγ is Lebesgue measurable. From this we get that the convergence in2.6is uniform on any compact set ofRassuming
thatγis Lebesgue measurable. ThusΓucontains the Lebesgue measurable members ofΓ.On
the other hand from10we know that there exists a nonmeasurable functionγ∈Γsuch that γ/∈Γuand henceΓuis a proper subset ofΓ.
To give an explicit formula for the weighted limit of the convolutionf∗dμat∞,we should assume some limit relations betweenγandfand betweenγandμ.
Definition 2.6. Letγ∈Γ.
aFγ denotes the set of functionsf:R→Rsuch that the limit
Lγf: lim t→∞
ft
γt 2.9
is finite.
bMγdenotes the set of measuresμ∈ Msuch that for any fixedT >0 the limit
Lγμ, T: lim t→∞
μt−T, t
γt 2.10
is finite.
cLet
Fμ:
Fb, if μ∈ M \ Mc
L, if μ∈ Mc.
Definition 2.7. Letγ∈Γandμ∈ Mγ.Fγ,μdenotes the set of functionsf∈Fμfor which
lim
t→∞ 1 γt
t
t−T
ft−sdμs Lγμ,1
T
0
f 2.12
holds for any fixedT >0.
Remark 2.8. A measureμ∈ Mcbelongs toMγif and only if for any fixedT >0 the limit
Lγg, T: lim t→∞
1 γt
t
t−T
g 2.13
is finite, whereμ gλ. MoreoverLγμ, T Lγg, Tfor anyT > 0.It can be shownsee
Proposition 7.3that ifg∈Fγandγ∈ΓuthenLγμ, T Lγg, T LγgT, T >0.
We close this section with the following definition.
Definition 2.9. A functionf :R→Ris said to be oscillatory onRif there exist two sequences tn, tn≥0, n≥1, such thattn→∞andtn→∞asn→∞,moreoverftn<0< ftn, n≥1.
3. Main results
In this section we state our main results. Their proofs are relegated toSection 8. We use the following hypothesis.
Hγ∈Γu, μ∈ Mγ, f ∈Fγ∩Fγ,μ, and the improper integral
lim
T→ ∞Lγμ,1
T
0
f 3.1
is finite wheneverfis oscillatory.
Note that ifLγμ,1 0,thenHis satisfied for anyf∈Fγ∩Fγ,μ.
In the next result we give an explicit limit formula for the weighted limit of the convolution offandμat∞.
Theorem 3.1. Assume (H). Then the following results hold.
aThe following three statements are equivalent.
a1The limit
Lγf∗dμ: lim t→∞
1 γt
T
0
ft−sdμs 3.2
a2For someT >0,the limit
lim
t→∞ 1 γt
T,t−Tft−sdμs 3.3
is finite.
a3The values
lim inf
t→∞ 1 γt
T1,t−T1
ft−sdμs for a fixedT1 >0,
lim sup
t→∞ 1 γt
T2,t−T2
ft−sdμs for a fixedT2>0
3.4
are finite, moreover
lim
T→ ∞
lim inf
t→∞ 1 γt
T,t−Tft−sdμs
lim
T→ ∞
lim sup
t→∞ 1 γt
T,t−Tft−sdμs
.
3.5
bAssume that one of the statementsa1–a3is true. Then the limit3.3is finite for any T >0and
Lγf∗dμ Lγfμ0,∞ lγf, μ Lγμ,1
∞
0
f, 3.6
where
Lγfμ0,∞: lim
T→∞Lγfμ0, T, 3.7
lγf, μ: lim T→∞
lim
t→∞ 1 γt
T,t−Tft−sdμs
, 3.8
Lγμ,1
∞
0
f : lim
T→∞Lγμ,1
T
0
f 3.9
are finite.
Remark 3.2. Our theorem is applicable for the case when μ0,∞ ∞ and also when f/∈L1.Namely, ifL
γf 0, then3.7yields thatLγfμ0,∞is zero independently on
the value ofμ0,∞.Similarly ifLγμ,1 0, then from3.9it follows that Lγμ,1
∞
Now consider the caseμ∈ Mc,that is,μgλ. In this case we can applyTheorem 3.1
by using the hypothesis.
Hcγ ∈Γu, f∈Fγ∩ L, the functiong∈ Lis nonnegative such that
Lγg, T: lim t→∞
1 γt
t
t−T
g 3.10
is finite for everyT >0, and the improper integral limT→ ∞Lγg,1
T
0fis finite wheneverfis oscillatory.
Theorem 3.3. AssumeHc. Then the following results hold.
aThe following three statements are equivalent.
a1The limit
Lγf∗g: lim t→∞
1 γt
t
0
ft−sgsds 3.11
is finite.
a2For someT >0,the limit
lim
t→∞
1 γt
t−T
T
ft−sgsds 3.12
is finite.
a3The values
lim inf
t→∞
1 γt
t−T1
T1
ft−sgsds, for a fixedT1 >0,
lim sup
t→∞ 1 γt
t−T2
T2
ft−sgsds, for a fixedT2>0,
3.13
are finite, moreover
lim
T→∞
lim inf
t→∞ 1 γt
t−T
T
ft−sgsds
lim
T→∞
lim sup
t→∞
t−T
T
ft−sgsds
. 3.14
bAssume that one of the statementsa1–a3is true. Then the limit3.12is finite for any T >0, and
Lγf∗g Lγf
∞
0
glγf, g Lγg,1
∞
0
where
lγf, g: lim T→∞
lim
t→∞ 1 γt
t−T
T
ft−sgsds
, 3.16
Lγf
∞
0
g: lim
T→∞
Lγf
T
0 g
, Lγg,1
∞
0
f: lim
T→∞Lγg,1
T
0
f 3.17
are finite.
Wheng ∈ L ∩Fγ, thenLγg, T LγgT, T >0seeProposition 7.3, and we get the
following.
Theorem 3.4. Letγ∈Γu, f, g∈ L ∩Fγ, and assume that
ithe improper integral
Lγg
∞
0
f : lim
T→∞Lγg
T
0
f 3.18
is finite, wheneverfis oscillatory andgis not oscillatory;
iithe improper integral
Lγf
∞
0
g: lim
T→∞Lγf
T
0
g 3.19
is finite, wheneverfis not oscillatory andgis oscillatory.
Then the statements ofTheorem 3.3are valid and3.15can be written in the form
Lγf∗g Lγf
∞
0
glγf, g Lγg
∞
0
f. 3.20
Remark 3.5. aIfγ ∈Γu, f ∈ L1∩Fγ, andg∈ Lis nonnegative such thatLγg, Tdefined in
3.10is finite for everyT >0, then the conditions ofTheorem 3.3hold.
bIfγ∈Γu, andf, g∈ L1∩Fγ, thenTheorem 3.4is applicable.
Remark 3.6. The well-known result Theorem 1.1see, e.g., 1 is a straightforward conse-quence ofTheorem 3.3.
4. Applications of the main results to subexponential functions
In this section we concentrate on the so-called subexponential functions which are strongly related to the subexponential distributions. Such distributions play an important role, for instance, in modeling heavy-tailed data. Such appears in the situations where some extremely large values occur in a sample compared to the mean size of datasee, e.g., 11 and the references therein.
Definition 4.1. Assume that a functionγis called subexponential ifγ∈Γ∩ L1and
Lγγ∗γ: lim t→∞
1 γt
t
0
γt−sγsds2
∞
0
γ. 4.1
Remark 4.2. Letγ ∈Γ∩ Lsuch thatLγγ∗γis finite. Thenγ is measurable and henceγ ∈Γu.
Thus
lim
t→∞ 1 γt
t
0
γt−sγsds≥ lim
t→∞
T
0
γt−s
γt γsds
T
0
γsds, T >0, 4.2
which shows that
∞
0
γ≤Lγγ∗γ<∞. 4.3
Thereforeγ∈ L1and the normalized function 0≤t→γt∞ 0γ
−1
is a subexponential density function. This gives the meaning of the “density-type” subexponentiality.
From Theorems3.4and6.1, we get the following.
Theorem 4.3. Ifγis a subexponential function andf, g∈ L1∩F
γ, then
Lγf∗g Lγf
∞
0
gLγg
∞
0
f. 4.4
It is worth to note that formula4.4has been obtained by Appleby et al.7in the case when the functionsγ, f,andgare continuous onR. These types of limit formulas were
used effectively for studying the subexponential rate of decay of solutions of integral and integro-differential equationssee, e.g.,6,12.
Now we apply our main results to subexponential and long-tailed-type distribution functions.
Definition 4.4. Let H : R→R be a distribution function on R such that H0 0 and Ht<1 for allt >0. Then
aHis called subexponential if
lim
t→∞ 1 Ht
1−
t
0
Ht−sdHs
2 4.5
or equivalently
LHH∗dH: lim
t→∞ 1 Ht
t
0
whereHdenotes the tail ofH, that is,
Ht 1−Ht>0, t≥0. 4.7
bHis called long-tailed if
lim
t→∞
Ht−s
Ht 1 for anys≥0. 4.8
The definition of the subexponential distribution was introduced by Chistyakov13
in 1964 and there are a large number of papers in the literature dealing with them. For the major properties and also for applications, we refer to the nice introduction and review paper by Goldie and Kl ¨uppelberg11and the references in it.
Now we show the consequences of our main results for the above-defined class of distribution functions. The proofs will be explained inSection 8.
It is noted in14,15 see also11that the set of the subexponential distributions is a proper subset of the set of the long-tailed distributions.
In the first theorem, we give equivalent statements for subexponential distributions; and in the second one, we give a limit formula for the more general long-tailed distributions.
Theorem 4.5. LetH:R→Rbe a distribution function such thatH0 0andHt<1, t >0.
Then the following statements are equivalent.
aHis subexponential.
bHis long-tailed and there is aT >0such that the limit
lim
t→∞ 1 Ht
T,t−T
Ht−sdHs 4.9
is finite and
lim
T→∞
lim
t→∞ 1 Ht
T,t−THt−sdHs
0. 4.10
cHis long-tailed and there is aT >0such that
lim sup
t→∞ 1 Ht
T,t−THt−sdHs 4.11
is finite and
lim
T→∞
lim sup
t→∞ 1 Ht
T,t−T
Ht−sdHs
Theorem 4.6. LetF, G, H : R→R be distribution functions,F0 G0 0, andH is
long-tailed. If
LHF: lim
t→∞ Ft
Ht, LHG:tlim→∞ Gt
Ht 4.13
are finite, and
lim
T→∞
lim sup
t→∞ 1 Ht
T,t−TFt−sdGs
0, 4.14
then
lim
t→∞ 1 Ht
1−
t
0
Ft−sdGs
LHF LHG. 4.15
The above theorem can be easily applied for tail-equivalent distributions defined as followssee11.
Definition 4.7tail-equivalence. Two distributionsH, F:R→Rwith the conditionsH0
F0 0 andHt<1, Ft<1, t >0,are called tail-equivalent ifLHFis a positive number.
Corollary 4.8. LetF, G, H:R→Rbe distribution functions,F0 G0 0, Ft<1, Gt<
1, t >0, andHis long-tailed. If the conditions ofTheorem 4.6are satisfied, thenHandF∗dGare tail equivalent if and only ifLHF LHG>0, that is, at least one of the distribution functionsFand
Gis tail equivalent toH.
5. Further corollaries for an extended set of weight functions
First we consider the extension of the setΓ.
Definition 5.1. Letα∈R. ByΓαone denotes the set of the functionsγ:R→0,∞such that
lim
t→∞
γt−s γt e
−αs 5.1
for alls≥0. ByΓuαone denotes the set of the functionsγ∈Γαfor which the convergence
in5.1is uniform on 0≤s≤T, for anyT >0.
Remark 5.2. It is clear thatΓ Γ0, Γu Γu0, andγ ∈ Γα γ ∈ Γuα if and only if
Letγ∈Γuα, μ∈ M, andf ∈Fμ. Then
1
γtf∗dμt 1 γt
t
0
ft−sdμs
1
γte−αt
t
0
e−αt−sft−se−αsdμs
1
γ1t
t
0
f1t−sdμ1s 1
γ1tf1∗dμ1
t, t∈Df∗dμ,
5.2
whereγ1t:γte−αt, f1t:fte−αt, t≥0, andμ1B:
Be−αsdμsforB∈ B.
Thus our earlier results are applicable ifγ1∈Γuandμ1 ∈ Mγ1.But fromRemark 5.2we have thatγ1∈Γuif and only ifγ∈Γuα.Moreover,μ1∈ Mγ1if and only if the limit
Lγμ, α, T: lim t→∞
1 γt
t−T,t
eαt−sdμs L
γ1μ1, T 5.3
is finite for anyT >0.
Remark 5.3. μ1 ∈ Mc if and only if μ ∈ Mc. Namely, let μ ∈ Mc, μ gλ. Thusμ1B
Be−αsdμs
Bgse−αsdλs, B ∈ B, and henceμ1 ∈ Mc.Now letμ1 ∈ Mc. Thenμ1B
Be−αsdμs 0 for anyB ∈ Bsuch thatλB 0. Thereforee−αs 0 μ-almost everys ∈B,
and henceμB 0. From this it follows thatμis absolute continuous with respect toλ, that is,μ∈ Mc.
It can be seen thatf1∈Fμ1if and only iff ∈Fμ.
Remark 5.4. f1 ∈Fγ1,μ1if and only iff ∈Fγ,α,μ. HereFγ,α,μdenotes the set of functionsf ∈Fμ for which
lim
t→∞
1 γt
t
t−T
ft−sdμs Lγμ, α,1
T
0
f 5.4
for anyT >0.
The above remarks show that our main results, Theorems 3.1–3.4, can be easily reformulated for the classΓuα, assuming that we replace the hypothesesHandHcby
HαandHcα, respectively. In fact we use the following modified hypotheses.
Hα α∈R, γ ∈Γuα, μ∈ Mare such thatLγμ, α, Tdefined in5.3is finite for any
T >0, f∈Fγ∩Fγ,α,μand the improper integral
lim
T→ ∞Lγμ, α,1
T
0
fdλ 5.5
Hcα α∈R, γ∈Γuα, f∈Fγ∩ L, g ∈ Lis nonnegative such that
Lγg, α, T: lim t→∞
1 γt
t
t−T
eαt−sgsds 5.6
is finite for everyT >0, and the improper integral limT→ ∞Lγg, α,1
T
0fdλis finite, wheneverfis oscillatory.
The extended form ofTheorem 3.1is as follows.
Theorem 5.5. AssumeHα. Then the following results hold.
aThe following three statements are equivalent.
a1The limit
Lγf∗dμ: lim t→∞
1 γt
t
0
ft−sdμs 5.7
is finite.
a2For someT >0the limit
lim
t→∞ 1 γt
T,t−Tft−sdμs 5.8
is finite.
a3The values
lim inf
t→∞ 1 γt
T1,t−T1
ft−sdμs, for a fixedT1>0,
lim sup
t→∞ 1 γt
T2,t−T2
ft−sdμs, for a fixedT2 >0,
5.9
are finite, moreover
lim
T→∞
lim inf
t→∞ 1 γt
T,t−T
ft−sdμs
lim
T→∞
lim sup
t→∞
1 γt
T,t−Tft−sdμs
.
5.10
bAssume that one of the statementsa1–a3is true. Then the limit5.8is finite for any T >0and
Lγf∗dμ Lγf
∞
0
e−αsdμs l
γf, μ Lγμ, α,1
∞
0
where
Lγf
∞
0
e−αsdμs: lim
T→∞Lγf
T
0
e−αsdμs,
Lγμ, α,1
∞
0
f : lim
T→∞Lγμ, α,1
T
0 f,
5.12
andlγf, μ, defined in3.8, are finite.
The extensions of Theorems3.3and3.4are similar and are left to the reader.
6. Power-type weight function and the role of the middle term
The introduction of our middle term was motivated by two independent papers 2, 4. In both papers power-type estimations have been proved for the solutions of functional differential equations and of the wave equations with boundary condition, respectively. The joint idea was to transform the original problems into a convolution-type form. By treating the convolution form, power-type estimations were given without investigating any limit formula.
As a consequence ofTheorem 3.4, we prove the next result, and as a corollary of it we give a power-type limit formula.
Theorem 6.1. Letγ ∈Γu, and letp, q ∈ L ∩Fγbe positive such that the limitLγp∗qis finite. If
f∈ L ∩Fpandg∈ L ∩Fq, then the limitLγf∗gis finite and
Lγf∗g LγpLpf
∞
0
gLpfLqglγp, q LγqLqg
∞
0
f, 6.1
where
LγpLpf
∞
0
g : lim
T→∞LγpLpf
T
0 g,
LγqLqg
∞
0
f : lim
T→∞LγqLqg
T
0 f,
6.2
andlγp, q, defined in3.16, are finite.
The following corollary is a generalization ofTheorem 1.2and shows the importance of our middle term whenγis a power-type function.
Corollary 6.2. Letf, g∈ Land assume that the limits
a: lim
t→∞
ft
tα−1, b:t→lim∞
gt
are finite, whereα, β >0are given constants. Then
lim
t→∞ 1 tαβ−1
t
0
ft−sgsdsabBα, β, 6.4
whereB:0,∞×0,∞→Ris the Beta function, that is,Bα, β ΓαΓβΓαβ−1(Γis the well-known Gamma function).
In the above limit formula,γt tαβ−1, t >0, L
γf Lγg 0 and the middle term
lγf, g abBα, β/0,wheneverab /0.
7. Preliminary results
In this section we state and prove preliminary and auxiliary results. They will be used in the proofs of our main results in the next section.Ndenotes the set of the positive integers.
Proposition 7.1. Letγ ∈Γandμ∈ Msuch thatLγμ, Tis finite for anyT ∈0, T0with a fixed T0>0.Thenμ∈ Mγ.
Proof. LetT > T0andT1∈0, T0such thatn:T/T1∈N. Then
μt−T, t γt
1 γt
n
k1
μt−kT1, t−k−1T1
n
k1
μt−kT1, t−k−1T1 γt−k−1T1 ·
γt−k−1T1 γt ,
7.1
and this yieldsLγμ, T nLγμ, T1.
Proposition 7.2. Letγ ∈Γandμ∈ Mγ. Then the following hold.
aLγμ, T TLγμ,1for anyT ≥0.
blimt→∞μ{t}/γt 0,wheretis the only element of the set{t}.
Proof. aFirst we show thatLγμ,·is additive. In fact forT1, T2 ≥0 we have
μt−T1T2, t
γt
μt−T1, t γt
μt−T1−T2, t−T1 γt−T1 ·
γt−T1
γt 7.2
for t > T1 T2. This yieldsLγμ, T1 T2 Lγμ, T1 Lγμ, T2. Therefore Lγμ,· can be
extended in a unique way to Rsuch that it is additive. Now a follows since Lγμ,· is
nonnegative onR.
bFor anyT >0,we have
0≤ μ{t} γt ≤
μt, tT γtT
γtT
therefore
0≤lim sup
t→∞ μ{t}
γt ≤Lγμ, T TLγμ,1. 7.4
SinceT >0 is arbitrarily chosen, statementbis proved.
Proposition 7.3. Letγ ∈ Γuand assume thatμ ∈ Mc, that is,μ pλ. Ifp ∈Fγ, thenLγμ, T
LγpT, T ≥0.
Proof. γis a positive function, thereforeLγp≥0.Thus for anyε∈0,1there exists atε >0
such that
1−εLγp≤
pt
γt ≤εLγp, fort > tε. 7.5
From this it follows that
1−εLγp≤
pt−s
γt−s ≤εLγp, s∈0, T, t > tεT. 7.6
On the other hand there is atε> tεT such that
1−ε < γt−s
γt <1ε, s∈0, T, t > tε, 7.7
where we used thatγ∈Γu.
Thus
1−ε2Lγp≤
pt−s γt
pt−s
γt−s
γt−s γt
≤1εεLγp, s∈0, T, t > tε,
7.8
therefore
1−ε2LγpT≤ 1
γt
T
0
pt−sds
1
γt
t
t−T
pdλ
μt−T, t
γt ≤1εεLγpT, t > tε.
From this it follows
1−ε2LγpT≤lim inf t→∞
μt−T, t γt
≤lim sup
t→∞
μt−T, t γt
≤1εεLγpT
7.10
for any fixedε∈0,1. This completes the proof asε→0.
Definition 7.4. For anyx≥0 andB∈ B, letεxBbe defined by
εxB:
1, if x∈B,
0, if x/∈B. 7.11
It is clear that for any fixedx≥0, εxis a measure onBthe unit mass atx, andεx∈ M.
Proposition 7.5. Lettn∞n1be a given sequence inRsuch that
δ: inf
n∈Ntn1−tn>0, 7.12
and letαn∞n1be a sequence of nonnegative numbers. Supposeγ∈Γ.
aIf the measure μ : n∞1αnεtn belongs to Mγ, then limn→∞αn/γtn 0 and
Lγμ, T 0, T ≥0.
bIfγ∈Γuandlimn→∞αn/γtn 0,thenμ∈ Mγ.
Proof. It is clear thatμ∈ M.
aLetT0∈0, δbe fixed. Sinceμtn−T0, tn 0 for everyn2,3, . . . ,we have
lim
n→∞
μtn−T0, tn
γtn
0, 7.13
and henceLγμ, T0 0.This and statementaofProposition 7.2imply thatLγμ, T 0, T ≥
0.Therefore
0Lγμ, T0 nlim→∞
μtn, tnT0 γtnT0
lim
n→∞
αn
γtnT0
lim
n→∞
αn
γtn
γtn
γtnT0
.
7.14
bLetε >0 and 0< T < δbe fixed. Then
μt−T, t γt
⎧ ⎨ ⎩
αn
γt, ift∈tn, tnT, 0, ift∈tnT, tn1
7.15
forn≥2.Thus
0≤ μt−T, t γt
≤
⎧ ⎪ ⎨ ⎪ ⎩
αn
γt αn
γtn
γtn
γt αn
γtn
γt−δnt
γt , t∈tn, tnT, δnt∈0, δ
0, t∈tnT, tn1
7.16
forn≥2.Butγ∈Γu, limn→∞αn/γtn 0 andtn→ ∞,therefore
Lγμ, T lim t→∞
μt−T, t
γt 0. 7.17
The proof is complete.
Definition 7.6. LetT >0 be fixed.
aBTdenotes theσ-algebra of the Borel sets of0, T.
bMT,edenotes the set of the finite measures onBT.
c A topology defined on MT,e is said to be the weak topology onMT,e if it is the
weakest one which makes the mapping
ν−→
T
0
fdν, ν∈ MT,e 7.18
continuous for all continuousf:0, T→R.
Definition 7.7. For a fixedT >0 andt≥T, define the shift operator
ST,t:t−T, t−→0, T, ST,ts:t−s. 7.19
Letμ∈ M. ForB∈ BT
ST,tμB:μ
S−1T,tB. 7.20
Proposition 7.8. Letγ ∈Γ, μ∈ Mγ, andT >0be fixed. Then
lim
t→∞ 1
γtST,tμ λγ,μ, 7.21
Proof. We should prove that for any fixed continuous functionf:0, T→R, we have
lim
t→∞
1 γt
T
0
f dST,tμ
T
0
f dλγ,μ. 7.22
For anyA⊂0, T,the functionχA :0, T→Rdenotes the characteristic function ofA.
Let
p:0, T−→R, pc1χ0,t1
k
i2
ciχti−1,ti, 7.23
wherek∈N, 0t0<· · ·< tkT,andci∈Ri1, . . . , k.
Then from the statementbofProposition 7.2, it follows
lim
t→∞
1 γt
T
0
pdST,tμ lim t→∞
1 γt
c1·ST,tμ0, t1
k
i2
ciST,tμti−1, ti
lim
t→∞ 1 γt
c1μt−t1, t
k
i2
ciμt−ti, t−ti−1
lim
t→∞
c1μ{t} γt
k
i1 ci
μt−ti, t
γt −
μt−ti−1, t
γt
k
i1
ciλγ,μti−1, ti
T
0 pdλγ,μ.
7.24
It is known that for a fixed continuous functionf:0, T→R, there exists a sequence of step functionspnsuch that it converges tofuniformly on0, T.Thus for arbitrarily fixedε >0,
there is an indexn0∈Nsuch that
|pn0t−ft|< ε, t∈0, T. 7.25
In that case
γ1tT
0
fdST,tμ−
T 0 fdλγ,μ ≤ 1 γt
T
0
|f−pn0|dST,tμ γ1tT
0
pn0dST,tμ− T
0
pn0dλγ,μ
T
0
|pn0−f|dλγ,μ
≤ εST,tμ0, T
γt
γ1tT
0
pn0dST,tμ− T
0
pn0dλγ,μ
ελγ,μ0, T
≤ελγ,μ0, T ε εελγ,μ0, T
for alltlarge enough. Here we used the conclusion of the first part of our proof and statement
bofProposition 7.2. Sinceε >0 is fixed but arbitrary, the proof is complete.
Corollary 7.9. Letγ ∈Γandμ∈ Mγ.
aIff :R→Ris Borel measurable and Riemann integrable on any interval0, T, T >0,
thenf∈Fγ,μ.
bIfλγ,μ 0andf ∈Fb, thenf∈Fγ,μ.
cLetμ∈ Mc, μpλ. Ifγ ∈Γu, p∈Fγandf∈ L, thenf ∈Fγ,μ.
Proof. FromProposition 7.8, it followssee, e.g., 12that iff ∈ Fb isλγ,μ-a.e. continuous,
thenf ∈Fγ,μ.From this we get statementsaandb.
cLetT >0 andε >0 be fixed. Sincep∈Fγ, there is at0>0 such that
pγtt−Lγp
< ε, t > t0, 7.27
and hence
γptt−−ss−Lγp
< ε, s∈0, T, t > Tt0. 7.28
Sinceγ ∈Γu,there is at1> t0T such that
γγt−ts −1< ε, s∈0, T, t > t1. 7.29
Thus
fs
pt−s
γt −Lγp
≤ |fs|pt−s
γt−s−Lγp
γγt−ts|fs|Lγp
γγt−ts −1 ≤ |fs|ε1ε εLγp, s∈0, T, t > t1.
7.30
From the general transformation theorem for integrals see, e.g., 12 and from the translation invariance of the Lebesgue measureλ, we get: for anyB∈ BT andt≥T,
ST,tμB μ
S−1T,tB
S−1T,tBpdλ
B
p◦S−1T,tdST,tλ
B
ButProposition 7.3shows thatμ∈ Mγandλγ,μLγpλ.So
γ1tT
0
fdST,tμ−
T
0 fdλγ,μ
≤
T
0
|fs|pt−s
γt −Lγp
ds
≤ε1ε εLγp
T
0 |f|dλ.
7.32
Sinceε >0 is arbitrary, this completes the proof.
Proposition 7.10. Letγ :R→0,∞, μ ∈ M, and assume thatf ∈Fμ is not oscillatory onR.
Then the following mappings have limits inRe:R∪ {−∞,∞}asT→∞:
0≤T −→lim sup
t→∞ 1 γt
T,t−Tft−sdμs∈Re,
0≤T −→lim inf
t→∞ 1 γt
T,t−Tft−sdμs∈Re.
7.33
Proof. Letf be nonnegative ont0,∞, wheret0 is large enough. Then fort0 ≤ T1 < T2and t∈2T2,∞∩Df∗μ, we get
1 γt
T1,t−T1
ft−sdμs 1 γt
T1,T2
ft−sdμs
1
γt
T2,t−T2
ft−sdμs 1 γt
T2,t−T2
ft−sdμs
1
γt
t−T2,t−T1
ft−sdμs≥ 1 γt
T2,t−T2
ft−sdμs.
7.34
Thus the above-defined mappings are decreasing, and hence their limits exist inRe asT →
∞. Whenf is eventually nonpositive, then the above procedure can be applied for−f.The proof is complete.
In the next two results, we give explicit formulas for the limit inferior and limit superior of the weighted convolution offandμat∞.
Theorem 7.11. Assume (H). Then the following results hold.
aThe following two statements are equivalent.
a1The limit inferior
Lγf∗dμ:lim inf
t→∞ 1 γt
t
0
ft−sdμs 7.35
a2For someT >0, the limit inferior
lim inf
t→∞ 1 γt
T,t−Tft−sdμs 7.36
is finite.
bIf the limit inferior7.36is finite for a fixedT >0, then it is finite for anyT >0and
Lγf∗dμ Lγfμ0,∞ lγf, μ Lγμ,1
∞
0
f, 7.37
where
lγf, μ: lim
T→∞
lim inf
t→∞
1 γt
T,t−Tft−sdμs
, 7.38
Lγfμ0,∞andLγμ,1
∞
0 f(they are defined in3.7and3.9, resp.) are finite.
Proof. LetT >0 be fixed. Then for anyt >2T,andt∈Df∗μ,we get
1 γt
t
0
ft−sdμs
1
γt
0,T
ft−sdμs 1 γt
T,t−T
ft−sdμs 1 γt
t
t−T
ft−sdμs.
7.39
First we show that
lim
t→∞
1 γt
0,Tft−sdμs Lγfμ0, T. 7.40
In fact fort≥T ands∈0, T,we have
fγt−ts −Lγf
≤ft−s
γt−s −Lγf
γγt−tsγt−s
γt −1
|Lγf|. 7.41
Butγ∈Γu, f ∈Fγ,therefore forε >0 there existst0>0 such that
γγt−ts −1< ε, s∈0, T, t > t0,
fγtt−Lγf
< ε, t > t0.
Thus7.41yields
fγt−ts−Lγf
< ε1ε ε|Lγf|, s∈0, T, t > t0T, 7.43
and hence
γ1t
0,Tft−sdμs−Lγf
0,T1dμ
≤
0,T
fγt−ts −Lγf
dμs≤ε1ε|Lγf|μ0, T, t > t0T, t∈Df∗μ,
7.44
which implies7.40.
Sincef ∈Fγ,μ,we have
lim
t→∞ 1 γt
t
t−T
ft−sdμs
T
0
fdλγ,μ. 7.45
Assume thata2holds. Then7.39,7.40, and7.45implya1. On the other hand, from
7.39,7.40, and7.45we get thata1yieldsa2, and hence statementais proved. This also verifies the first part of statementb.
Now we prove the second part of statementb. Assume that7.36is finite for any T >0. Then7.39yields
lim inf
t→ ∞ 1 γt
t
0
ft−sdμs
Lγfμ0, T lim inf t→ ∞
T,t−T
ft−sdμs
T
0
fdλγ,μ, T >0.
7.46
Now assume thatf is not oscillatory. Then there existst0 > 0 such that eitherft ≥ 0 for everyt≥t0orft≤0 for everyt≥t0. We consider the case whenft≥0 fort≥t0, the other case can be handled similarly.
All the three terms on the right-hand side of7.46have limit asT→ ∞inRe : R∪
{−∞,∞}.In fact
lim
T→∞Lγfμ0, T Lγfμ0,∞∈0,∞, 7.47
and byProposition 7.10, we get
lim
T→∞
lim inf
t→ ∞ 1 γt
T,t−T
ft−sdμs
moreover
lim
T→∞
T
0
fdλγ,μ
t0
0
fdλγ,μ lim T→∞
T
t0
fdλγ,μ∈−∞,∞. 7.49
Now the second part of b is proved, since the left-hand side of 7.46 is finite and independent onT.
Now assume thatfis oscillatory onRand as we assumed limT→∞T0fdλγ,μis finite.
In that caseLγf 0,and hence
lim
T→∞Lγfμ0, T 0. 7.50
Thus by using similar arguments to those we used above, statementbis proved again.
Theorem 7.12. Assume (H). Then the following results hold.
aThe following two statements are equivalent.
a1The limit superior
Lγf∗dμ:lim sup t→∞
1 γt
t
0
ft−sdμs 7.51
is finite.
a2For someT >0the limit superior
lim sup
t→∞
1 γt
t
T,t−Tft−sdμs 7.52
is finite.
bIf the limit superior7.52is finite for a fixedT >0,then it is finite for anyT >0and
Lγf∗dμ Lγfμ0,∞ lγf, μ Lγμ,1
∞
0
f, 7.53
where
lγf, μ: lim T→∞
lim sup
t→∞ 1 γt
T,t−Tft−sdμs
, 7.54
Lγfμ0,∞andLγμ,1
∞
0 f(they are defined in3.7and3.9, resp.) are finite.
Theorem 7.13. Letγ∈Γu, f, g∈ L ∩Fγ, and assume that
ithe improper integral
Lγg
∞
0
f : lim
T→∞Lγg
T
0
f 7.55
is finite, wheneverfis oscillatory andgis not oscillatory,
iithe improper integral
Lγf
∞
0
g: lim
T→∞Lγf
T
0
g 7.56
is finite, wheneverfis not oscillatory andgis oscillatory.
Then the following results hold.
aThe following two statements are equivalent.
a1The limit inferior
Lγf∗g:lim inf
t→ ∞ 1 γt
T
0
ft−sgsds 7.57
is finite.
a2For someT >0the limit inferior
lim inf
t→ ∞ 1 γt
t−T
0
ft−sgsds 7.58
is finite.
bIf the limit inferior7.58is finite for a fixedT >0, then it is finite for anyT >0and
Lγf∗g Lγf
∞
0
glγf, g Lγg
∞
0
f, 7.59
where
lγf, g: lim
T→∞
lim inf
t→∞ 1 γt
t−T
T
ft−sgsds
, 7.60
Lγf
∞
0gandLγg
∞
Proof. LetT >0 be fixed. Then for eacht >2T andt∈Df∗g, we have
1 γt
t
0
ft−sgsds
1
γt
T
0
ft−sgsds 1 γt
t−T
T
ft−sgsds 1 γt
t
t−T
ft−sgsds.
7.61
The proof of7.40can easily be modified to show that
lim
t→ ∞
1 γt
T
0
ft−sgsdsLγf
T
0
g. 7.62
Since
t
t−T
ft−sgsds
T
0
fsgt−sds, t >2T, t∈Df∗g, 7.63
it follows from7.62that
lim
t→ ∞ 1 γt
t
t−T
ft−sgsdsLγg
T
0
f. 7.64
By using7.61,7.62, and7.64instead of7.39,7.40, and7.45, the argument employed in the proof ofTheorem 7.11aand the first part ofbextends to giveaand the first part ofb.
Consider now the proof of7.59. Suppose that7.58is finite for everyT >0. By7.61,
Lγf∗g Lγf
T
0
glim inf
t→ ∞ 1 γt
t−T
T
ft−sgsdsLγg
T
0
f 7.65
for eachT >0. We separate the proof into four steps.
hSuppose first thatf andg are oscillatory. ThenLγf Lγg 0, hence7.65
implies that
Lγf∗g lim inf
t→ ∞ 1 γt
t−T
T
ft−sgsds, T >0. 7.66
It now follows thatlγf, gis finite and
jSuppose next that exactly one of the functionsfandgis oscillatory. Without loss of generality, we can assume thatfis oscillatory andgis not oscillatory. ThenLγf 0, hence
7.65shows that
Lγf∗g lim inf
t→ ∞ 1 γt
t−T
T
ft−sgsdsLγg
T
0
f, T >0. 7.68
Byi,lγf, gis finite and
Lγf∗g lγf, g Lγg
∞
0
f. 7.69
kSuppose that there isT0>0 such thatft≥0 andgt≥0 for everyt≥T0. Then it follows fromLγf≥0 and
Lγf
T
0
gLγf
T0
0
gLγf
T
T0
g, T > T0, 7.70
that
lim
T→ ∞Lγf
T
0
g∈−∞,∞. 7.71
A similar argument gives that
lim
T→ ∞Lγg
T
0
f ∈−∞,∞. 7.72
Since
lim inf
t→ ∞ 1 γt
t−T
T
ft−sgsds≥0, T > T0, 7.73
we have
lim inf
T→ ∞
lim inf
t→ ∞ 1 γt
t−T
T
ft−sgsds
≥0. 7.74
Using7.65, we deduce from7.71,7.72, and7.74that the limits in7.71and7.72are finite, and thereforelγf, gexists and is finite. This gives7.59. Ifft ≤0 andgt ≤0 for everyt≥T0, then a similar proof can be applied.
Theorem 7.14. Under the hypotheses ofTheorem 7.13the following results hold.
aThe following two statements are equivalent.
a1The limit superior
Lγf∗g:lim sup t→ ∞
1 γt
t
0
ft−sgsds 7.75
is finite.
a2For someT >0, the limit superior
lim sup
t→ ∞ 1 γt
t−T
T
ft−sgsds 7.76
is finite.
bIf the limit superior7.76is finite for a fixedT >0, then it is finite for anyT >0and
Lγf∗g Lγf
∞
0
glγf, g Lγg
∞
0
f, 7.77
where
lγf, g: lim T→∞
lim sup
t→∞
1 γt
t−T
T
ft−sgsds
, 7.78
Lγf
∞
0gandLγg
∞
0f(they are defined inTheorem 3.3) are finite.
Proof. The proof is similar to the proof of the previous theorem, therefore it is omitted.
8. The proofs of the main results
In this section, we give the proofs of the results stated in Sections3–6.
Proof ofTheorem 3.1. A similar argument employed in the proof ofTheorem 7.11 gives the equivalence ofa1anda2, and partb. It is clear froma2andbthata2impliesa3. Ifa3holds, then by Theorems7.11and7.12, the values ofLγf∗dμandLγf∗dμare finite.
Sincelγf, μ lγf, μ, it follows from7.37 and 7.53 thatLγf∗dμ Lγf∗dμ. This
shows thata3yieldsa1.
Proof ofTheorem 3.3. This is an immediate consequence ofTheorem 3.1.
Proof ofTheorem 3.4. The argument ofTheorem 7.13 can easily be generalized to prove the equivalence ofa1anda2, and partb.a1andbimplya3. Ifa3holds, then we can apply Theorems7.13and7.14. It now follows from7.59and7.77thatLγf∗g Lγf∗g,
Proof ofTheorem 1.1. First we suppose thatgis nonnegative. It follows fromg ∈ L1that
lim
t→ ∞
t
t−T
g0 8.1
for everyT ≥0. We can see that the hypothesisHcis satisfied withγ:R→0,∞, γt 1,
andLγg,1 0. This shows thatTheorem 3.3can be applied. Consider now the proof that
a3holds, andlγf, g 0. There existst0>0 such that
|ft|<|f∞|1, t > t0. 8.2
This implies that
t−T
T
|ft−s|gsds≤|f∞|1
t−T
T
g, T > t0, t >2T, t∈Df∗g, 8.3
hence
lim sup
t→ ∞
t−T
T
|ft−s|gsds≤|f∞|1
∞
T
g, T > t0, 8.4
and therefore, byg ∈ L1,
lim
T→ ∞
lim sup
t→ ∞
t−T
T
|ft−s|gsds
0. 8.5
Since
−lim sup
t→ ∞
t−T
T
|ft−s|gsds≤lim inf
t→ ∞
t−T
T
ft−sgsds
≤lim sup
t→ ∞
t−T
T
ft−sgsds
≤lim sup
t→ ∞
t−T
T
|ft−s|gsds,
8.6
it follows from8.5thata3is true, andlγf, g 0. Now3.15gives the result.
In the general case, the preceding can be applied to bothgandg−.
Proof ofTheorem 4.3. γis a subexponential function, and thereforeγ∈Γuand
Lγγ∗γ 2
∞
0
γ. 8.7
Theorem 3.4may now be applied withfg:γ, andlγγ, γ 0 is obtained. Thus the result
Proof ofTheorem 4.5. Suppose that H is subexponential. Then H is long-tailed as is well knowndetails can be found in11, and thusH∈Γu. The distribution functionHgenerates
a distributionμH∈ M. Since
μHt−T, t
Ht
Ht−Ht−T
Ht −1
Ht−T
Ht , T >0, t≥T, 8.8
we can see thatLHμH, T 0 for everyT ≥ 0, and thereforeμ ∈ MH. It follows that the
hypothesisHis satisfied withγ f : Handμ : μH. This shows thatTheorem 3.1can
be applied. Since H is subexponential, the equivalence ofa1and a2implies4.9, and
3.6gives4.10. We have proved thatbcomes froma. On the other hand,cobviously follows fromb.
By the equivalence ofa3anda1inTheorem 3.1,cimpliesa.
Proof ofTheorem 4.6. His long-tailed, henceH∈Γu. The distribution functionGcharacterizes
a distributionμG∈ M. Then
μGt−T, t
Ht
Gt−Gt−T Ht
−Gt Ht
Gt−T Ht−T
Ht−T
Ht , T >0, t≥T,
8.9
giving
lim
t→ ∞
μGt−T, t
Ht −LHG LHG 0, T≥0, 8.10
and thereforeμG∈ MHandLHμG, T 0, T ≥0.
It follows that the conditionHis satisfied withγ:H, μ:μGandf :F.
According toTheorem 3.1a3and3.6, we now have
lim
t→ ∞ 1 Ht
t
0
Ft−sdGs LHF. 8.11
Applying this and taking into accountProposition 7.2b, the result follows, since
1 Ht
1−
t
0
Ft−sdGs
1−Gt
Ht 1 Ht
t
0
Ft−sdGs
Gt
Ht−
μG{t}
Ht 1 Ht
t
0
Ft−sdGs, t >0.
8.12
Proof ofTheorem 5.5. By the correspondence between hypotheses H and Hα,
