New Limit Formulas for the Convolution of a Function with a Measure and Their Applications
© I. Győri and L. Horváth. 2008
Received: 16 July 2008
Accepted: 23 September 2008
Published: 15 October 2008
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.
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.
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., ).
Based on the above three known results, we conclude the next observations.
(ii) The weight functions in Theorems 1.1 and 1.2 satisfy condition (1.7), but they do not satisfy condition (1.6).
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.
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
In our investigations we will make use of different sets of measures and functions given in the next definitions.
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 ).
that is the function is so called regularly varying at infinity. Thus applying the Karamata uniform convergence theorem (see, e.g., ) 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  we know that there exists a nonmeasurable function such that and hence is a proper subset of .
We close this section with the following definition.
3. Main Results
In this section we state our main results. Their proofs are relegated to Section 8.
We use the following hypothesis.
The following three statements are equivalent.
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).
The well-known result Theorem 1.1 (see, e.g., ) 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.,  and the references therein).
First we consider the "density-type" subexponential functions.
From Theorems 3.4 and 6.1, we get the following.
It is worth to note that formula (4.4) has been obtained by Appleby et al.  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.
The definition of the subexponential distribution was introduced by Chistyakov  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  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.
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.
The above theorem can be easily applied for tail-equivalent distributions defined as follows (see ).
Definition 4.7 (tail-equivalence).
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
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.
The extended form of Theorem 3.1 is as follows.
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.
7. Preliminary Results
The proof is complete.
The following two statements are equivalent.
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).
Thus by using similar arguments to those we used above, statement (b) is proved again.
The following two statements are equivalent.
Its proof is similar to the proof of Theorem 7.11, therefore it is omitted.
The following two statements are equivalent.
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).
The following two statements are equivalent.
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.
Proof of Theorem 4.3.
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).
Proof of Theorem 4.6.
Proof of Theorem 5.5.
Proof of Theorem 6.1.
Proof of Corollary 6.2.
which comes from the definition of the Beta function.
This work is supported by Hungarian National Foundation for Scientific Research Grant no. K73274.
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