# New Limit Formulas for the Convolution of a Function with a Measure and Their Applications

- István Győri
^{1}Email author and - László Horváth
^{1}

**2008**:748929

https://doi.org/10.1155/2008/748929

© I. Győri and L. Horváth. 2008

**Received: **16 July 2008

**Accepted: **23 September 2008

**Published: **15 October 2008

## Abstract

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.

## Keywords

## 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.

for all for which the integral exists.

for 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 equations (see, e.g., [1]).

Theorem 1.1.

The next well-known simple result plays a central role, for instance, in the asymptotic theory of fractional differential and integral equations (see, e.g., [2–5])

Theorem 1.2.

where is the well-known Beta function.

The terminology is suggested by the fact that (1.7) implies that for every .

The next result has been proved in [6] and it plays a central role to get exact rates of subexponential decay of solutions of Volterra integral and integro-differential equations (see, e.g., [6–8]).

Theorem 1.3.

Based on the above three known results, we conclude the next observations.

(i) All of the above theorems give different limit formulas for the ratio at . In fact in Theorem 1.1 and , in Theorem 1.2.

(ii) The weight functions in Theorems 1.1 and 1.2 satisfy condition (1.7), but they do not satisfy condition (1.6).

(iii)The condition for in (1.8) is not necessarily true in Theorem 1.1. Instead of that holds, where .

(iv) in Theorem 1.2 and at the same time is 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 the limit formula (1.11), the terms and are interpreted as zero whenever and , respectively. So the values of and need not be finite in the applications.

The limit formula (1.11) can be reformulated for the convolution of two functions and . Formally, it can be done if the measure is such that for every Borel set .

These indicate that our remarks (i)–(iv) are taking into account and the known Theorems 1.1, 1.2, and 1.3 are 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. In Section 4 we present the corollaries of our main results for subexponential and long-tailed distributions. In Section 5 we show that our results can be easily reformulated to an extended set of weight functions. Section 6 gives 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 8 based on some preliminary statements stated and proved in Section 7.

## 2. The Basic Notations and Definitions

First we introduce some notations. The set of real numbers is denoted by , and denotes 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.

Let be the -algebra of the Borel sets of . denotes the set of measures defined on such that the -measure of any compact subset of is a nonnegative number.

Note that the classical Lebesgue measure defined on , denoted by , is an element of .

Let . In this paper we will write for the -integral of on the closed interval . The -integral of on the interval is written as When , instead of we also write

Definition 2.2.

Definition 2.3.

Definition 2.4.

A measure from belongs to the set if it is absolutely continuous with respect to . In this case means that is a nonnegative function from such that for every ( is the Radon-Nikodym derivative of with respect to ).

as . The weight function will belong to some special classes of the functions given in the following definitions.

Definition 2.5.

The set of the functions for which the above convergence is uniform on any compact interval is denoted by .

that is the function is so called regularly varying at infinity. Thus applying the Karamata uniform convergence theorem (see, e.g., [10]) it follows that the convergence in (2.8) is uniform in on any compact set of assuming that is Lebesgue measurable. From this we get that the convergence in (2.6) is uniform on any compact set of assuming that is Lebesgue measurable. Thus contains the Lebesgue measurable members of On the other hand from [10] we know that there exists a nonmeasurable function such that and hence is a proper subset of .

To give an explicit formula for the weighted limit of the convolution at we should assume some limit relations between and and between and .

Definition 2.6.

is finite.

Definition 2.7.

Remark 2.8.

is finite, where . Moreover for any It can be shown (see Proposition 7.3) that if and then

We close this section with the following definition.

Definition 2.9.

A function is said to be oscillatory on if there exist two sequences , such that and as moreover .

## 3. Main Results

In this section we state our main results. Their proofs are relegated to Section 8.

We use the following hypothesis.

is finite whenever is oscillatory.

Note that if then (H) is satisfied for any

In the next result we give an explicit limit formula for the weighted limit of the convolution of and at +

Theorem 3.1.

- (a)
The following three statements are equivalent.

is finite.

is finite.

are finite.

Remark 3.2.

Our theorem is applicable for the case when and also when Namely, if , then (3.7) yields that is zero independently on the value of Similarly if , then from (3.9) it follows that is independently zero on . We will see that this character of our theorem is important for getting limit formulas for polynomial-type convolutions (see Corollary 6.2 in Section 6).

Now consider the case that is, . In this case we can apply Theorem 3.1 by using the hypothesis.

is finite for every , and the improper integral is finite whenever is oscillatory

Theorem 3.3.

is finite.

is finite.

are finite.

When , then (see Proposition 7.3), and we get the following.

Theorem 3.4.

is finite, whenever is oscillatory and is not oscillatory;

is finite, whenever is not oscillatory and is oscillatory.

Remark 3.6.

The well-known result Theorem 1.1 (see, e.g., [1]) is a straightforward consequence of Theorem 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 data (see, e.g., [11] and the references therein).

First we consider the "density-type" subexponential functions.

Definition 4.1.

Remark 4.2.

Therefore and the normalized function is a subexponential density function. This gives the meaning of the "density-type" subexponentiality.

From Theorems 3.4 and 6.1, we get the following.

Theorem 4.3.

It is worth to note that formula (4.4) has been obtained by Appleby et al. [7] in the case when the functions and are continuous on . These types of limit formulas were used effectively for studying the subexponential rate of decay of solutions of integral and integro-differential equations (see, e.g., [6, 12]).

Now we apply our main results to subexponential and long-tailed-type distribution functions.

Definition 4.4.

Let be a distribution function on such that and for all . Then

The definition of the subexponential distribution was introduced by Chistyakov [13] 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 Klüppelberg [11] and 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 in Section 8.

It is noted in [14, 15] (see also [11]) that 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.

Let be a distribution function such that and . Then the following statements are equivalent.

Theorem 4.6.

The above theorem can be easily applied for tail-equivalent distributions defined as follows (see [11]).

Definition 4.7 (tail-equivalence).

Two distributions with the conditions and are called tail-equivalent if is a positive number.

Corollary 4.8.

Let be distribution functions, , and is long-tailed. If the conditions of Theorem 4.6 are satisfied, then and are tail equivalent if and only if , that is, at least one of the distribution functions and is tail equivalent to .

## 5. Further Corollaries for an Extended Set of Weight Functions

First we consider the extension of the set

Definition 5.1.

for all . By one denotes the set of the functions for which the convergence in (5.1) is uniform on , for any .

Remark 5.2.

It is clear that , and if and only if , where is defined by .

Remark 5.3.

It can be seen that if and only if .

Remark 5.4.

The above remarks show that our main results, Theorems 3.1–3.4, can be easily reformulated for the class , assuming that we replace the hypotheses (H) and by and , respectively. In fact we use the following modified hypotheses.

is finite, whenever is oscillatory.

is finite for every , and the improper integral is finite, whenever is oscillatory.

The extended form of Theorem 3.1 is as follows.

Theorem 5.5.

is finite.

is finite.

and , defined in (3.8), are finite.

The extensions of Theorems 3.3 and 3.4 are 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 of Theorem 3.4, we prove the next result, and as a corollary of it we give a power-type limit formula.

Theorem 6.1.

and , defined in (3.16), are finite.

The following corollary is a generalization of Theorem 1.2 and shows the importance of our middle term when is a power-type function.

Corollary 6.2.

where is the Beta function, that is, ( is the well-known Gamma function).

## 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. denotes the set of the positive integers.

Proposition 7.1.

Let and such that is finite for any with a fixed Then .

Proof.

Proposition 7.2.

Let and . Then the following hold.

(b) where is the only element of the set .

- (b)

Since is arbitrarily chosen, statement (b) is proved.

Proposition 7.3.

Let and assume that , that is, . If then

Proof.

for any fixed . This completes the proof as .

Definition 7.4.

It is clear that for any fixed is a measure on (the unit mass at ), and .

Proposition 7.5.

and let be a sequence of nonnegative numbers. Suppose .

Proof.

The proof is complete.

Definition 7.6.

continuous for all continuous .

Definition 7.7.

Proposition 7.8.

where the convergence is in the weak topology of .

Proof.

For any the function denotes the characteristic function of

for all large enough. Here we used the conclusion of the first part of our proof and statement (b) of Proposition 7.2. Since is fixed but arbitrary, the proof is complete.

Corollary 7.9.

(a)If is Borel measurable and Riemann integrable on any interval , then .

Proof.

- (c)

Since is arbitrary, this completes the proof.

Proposition 7.10.

Proof.

Thus the above-defined mappings are decreasing, and hence their limits exist in as . When is eventually nonpositive, then the above procedure can be applied for The proof is complete.

In the next two results, we give explicit formulas for the limit inferior and limit superior of the weighted convolution of and at

Theorem 7.11.

- (a)
The following two statements are equivalent.

is finite.

is finite.

Proof.

which implies (7.40).

Assume that holds. Then (7.39), (7.40), and (7.45) imply . On the other hand, from (7.39), (7.40), and (7.45) we get that yields , and hence statement (a) is proved. This also verifies the first part of statement (b).

Now assume that is not oscillatory. Then there exists such that either for every or for every . We consider the case when for , the other case can be handled similarly.

Now the second part of (b) is proved, since the left-hand side of (7.46) is finite and independent on .

Thus by using similar arguments to those we used above, statement (b) is proved again.

Theorem 7.12.

- (a)
The following two statements are equivalent.

is finite.

Proof.

Its proof is similar to the proof of Theorem 7.11, therefore it is omitted.

Theorem 7.13.

is finite, whenever is not oscillatory and is oscillatory.

- (a)
The following two statements are equivalent.

is finite.

is finite.

Proof.

By using (7.61), (7.62), and (7.64) instead of (7.39), (7.40), and (7.45), the argument employed in the proof of Theorem 7.11(a) and the first part of (b) extends to give (a) and the first part of (b).

- (j)

Theorem 7.14.

- (a)
The following two statements are equivalent.

is finite.

is 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 Sections 3–6.

Proof of Theorem 3.1.

A similar argument employed in the proof of Theorem 7.11 gives the equivalence of and , and part (b). It is clear from and (b) that implies . If holds, then by Theorems 7.11 and 7.12, the values of and are finite. Since , it follows from (7.37) and (7.53) that . This shows that yields .

Proof of Theorem 3.3.

This is an immediate consequence of Theorem 3.1.

Proof of Theorem 3.4.

The argument of Theorem 7.13 can easily be generalized to prove the equivalence of and , and part (b). and (b) imply . If holds, then we can apply Theorems 7.13 and 7.14. It now follows from (7.59) and (7.77) that , and therefore is satisfied.

Proof of Theorem 1.1.

it follows from (8.5) that is true, and . Now (3.15) gives the result.

In the general case, the preceding can be applied to both and .

Proof of Theorem 4.3.

Theorem 3.4 may now be applied with , and is obtained. Thus the result follows from Theorem 6.1 with .

Proof of Theorem 4.5.

we can see that for every , and therefore . It follows that the hypothesis (H) is satisfied with and . This shows that Theorem 3.1 can be applied. Since is subexponential, the equivalence of and implies (4.9), and (3.6) gives (4.10). We have proved that (b) comes from (a). On the other hand, (c) obviously follows from (b).

By the equivalence of and in Theorem 3.1, (c) implies (a).

Proof of Theorem 4.6.

It follows that the condition (H) is satisfied with and .

Proof of Theorem 5.5.

By the correspondence between hypotheses (H) and (H( )), Theorem 3.1 implies the result.

Proof of Theorem 6.1.

Proof of Corollary 6.2.

which comes from the definition of the Beta function.

## Declarations

### Acknowledgment

This work is supported by Hungarian National Foundation for Scientific Research Grant no. K73274.

## Authors’ Affiliations

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