On Convergence Rates of Some Limits

: In 2019 Seneta has provided a characterization of slowly varying functions L in the Zygmund sense by using the condition, for each y > 0, x (cid:16) L ( x + y ) L ( x ) − 1 (cid:17) → 0 as x → ∞ . Very recently, we have extended this result by considering a wider class of functions U related to the following more general condition. For each y > 0, r ( x ) (cid:16) U ( x + yg ( x )) U ( x ) − 1 (cid:17) → 0 as x → ∞ , for some functions r and g . In this paper, we examine this last result by considering a much more general convergence condition. A wider class related to this new condition is presented. Further, a representation theorem for this wider class is provided.


Introduction
The notion of ultimately monotony introduced by Zygmund says that a function U ≥ 0 is slowly varying if for each > 0 the function x U(x) is ultimately increasing and x − U(x) is ultimately decreasing ( [1], p. 186). A different kind of slowly varying functions was defined by Karamata [2] known as simply the class of slowly varying functions (KSV). It is known that any ZSV function is a KSV function (see [1], p. 186 and, e.g., [3], p. 49).
Recently, Seneta [4] found that the slowly varying functions L in the sense of Zygmund are characterized by the relation: L(x + y) L(x) − 1 = 0, ∀y.
More recently, Omey and Cadena's [5] functions extended the results of Seneta, and they considered functions for which the following relation holds: L(x + yg(x)) L(x) − 1 = 0, ∀y.
Here, the function g(x) is self-neglecting (notation: g ∈ SN) and r is in the class Γ 0 (g) with r(x) → ∞. The class Γ 0 (g) is deeply studied in [6]. Recall that g ∈ SN if it satisfies lim x→∞ g(x + yg(x)) g(x) = 1, locally uniformly in y. In addition, recall that, for g ∈ SN, we have f ∈ Γ α (g) if f satisfies lim x→∞ f (x + yg(x)) f (x) = e αy , ∀y. Now, we study more general relations of the form lim x→∞ r(x) U(x + yg(x)) U(x) − e αy = θ(y), ∀y, where we assume that the convergence is l.u. in y. As before, we assume that r ∈ Γ 0 (g), r(x) → ∞ and that g ∈ SN.
Throughout this paper, we use the notation f (x) ∼ g(x) for representing f (x) g(x) → 1 as x → ∞.
We study in detail the two cases: α = 0 and α = 0. The case α = 0 can be considered as the class SN with a rate of convergence in the definition. This case is presented in the following section. The case where α = 0 can be considered as the class Γ α (g) with a rate of convergence in the definition. This case is presented in Section 3. For each case, characterizations of the involved functions are provided. Concluding remarks are presented in the last section.

The Limit Function
Suppose that U, g, r > 0 are measurable functions and suppose that the following relation holds: and we assume that Equation (1) holds locally uniformly in y. As before, we assume that r(x) → ∞, r ∈ Γ 0 (g) and that g ∈ SN.
We conclude that and (since θ is measurable) hence also that θ(y) = θy for some constant θ. Conversely, we have the following (cf. [6]): if then this relation holds l.u. in y.
To conclude, we have the following theorem.

Representation
Three different ways to represent the functions satisfying Equation (1) follow.

First Form
For further use, let A(x) = x a 1/g(t)dt. Clearly, we have is an increasing function of y for which f x (y) → y as x → ∞. As a consequence, the inverse function also satisfies f −1 x (y) → y. To calculate the inverse, we set and We conclude that so that (replacing A(x) by x and t by y) l.u. in y. Now, let K(x) := W(A −1 (x)). We have (using l.u. convergence in the last step): It follows that l.u. in y. Taking the integral 1 y=0 (.)dy in Equation (3) we have We see that K(x) is of the form Using W(x) = K(A(x)), we find that We prove the following result: Theorem 2. Assume that g ∈ SN and that r ∈ Γ 0 (g), r(x) → ∞.
Proof. The proof of (a) is given above. To prove (b), we have Clearly, we have for some β ∈ (0, y). It follows that For the second term, we have The result follows.

Remark 1.
1. In the special case where g(x) = 1, we have The previous representation result shows that

Second Form
In Equation (3), we find that r( . From de Haan's theorem ( [7], Theorem 3.7.3), we find that K(log x) can be written as

Third Form
In [5], we found that relations of the form in Equation (1) hold with limit function θ(x) = 0. In that case, we have As usual, we assume that g ∈ SN, r ∈ Γ 0 (g) and r(x) → ∞. From Theorem 3 in [5], we get the following representation:

Sufficient Conditions
In the next result, we assume that the kth derivative of U exists and we assume that We find that We conclude that (d) In general, we get a result of the type As a special case, we can take g(

More Results
Proposition 1. Suppose that F(x) = x −α L(x) where L(·) is a normalized slowly varying (SV) function (that is, xL (x) L(x) → 0). Assume that g(x) and r(x) satisfy g(x) x → 0 and r(x)g(x) x → δ > 0. Then, Proof. We have F(x) = L(x)x −α and then It follows that For I(a), we have because L is SV and g(x) x → 0. We also have x , so that For the second term, we have x .
We conclude that Combining these results, we obtain the desired result.

Remark 2.
The condition on L(x) in the previous theorem is equivalent to the requirement that where f (x) = F (x) is the density of F.

Example 1
Assume that U(x) = exp x β with β > 1. We have Using g(x) = x −γ , we find The results of this section show that
Taking g ∈ SN and r(x) = x g(x) (→ ∞) we find

Example 4
Proposition 1 can be extended for some stable distributions. For instance, consider the density of an asymmetric stable distribution. The representation of such a stable density in the form of a convergent series is, for 0 < α < 1 and for any x > 0 (see, e.g., [8]), Let g(x) and r(x) be positive functions satisfying g(x) x → 0 and r(x)g(x)/x → δ > 0. Note that, for each n > 1 and for x large enough, we have, making use of z − 1 ∼ log z as z → 1, x .
Then, we have for x large enough Hence, we have

Special Case
We assume that W is differentiable and that g(x)W (x) → α.
In this case, we have Now, suppose in addition that r(x) g(x)W (x) − α → δ and that We have For the first integral, by assumption, we have

Representation Theorem
x a 1/g(t)dt as before. We prove above that then we also have Using Q(x) = W(x) − αA(x), we see that Hence, using Equation (3), l.u. in y. As in the previous subsection, we conclude that for some real number λ. The first representation of the previous subsection gives where C(x) → C and r(x)g(x)T (x) → λ.
Theorem 4. We have Equation (3) if and only if W(x) is of the form where C(x) → C and r(x)g(x)T (x) → λ.

More Results
In our next result, we consider the function h(x) = f (x) F(x), where f is the density of F. We make the following assumptions about h: Recall that r ∈ Γ 0 (g) means that r(x + yg(x)) r(x) → 1 as x → ∞.

Lemma 2. If (a) and (b) hold, then
It follows that (recall g(x) = 1/h(x)) and using Lemma 1, it follows that This proves the result. Now, we arrive at the main result here.
Theorem 5. If (a) and (b) hold, then Proof. Using Lemma 2, we have Using log z ∼ z − 1, it follows that The previous theorem can be useful in extreme value theory as follows.
We assume that (a) and (b) hold and that F is strictly increasing. We define a n by the equality nF(a n ) = 1. It is clear that a n ↑ ∞. In the result of Theorem 5, we replace x by a n to see that r(a n ) nF(a n + yg(a n )) − e −y → β y 2 2 e −y . Now, we use log(z) + (1 − z) = O(1)(1 − z) 2 and write nF(a n + yg(a n )) = nF(a n + yg(a n )) + n log F(a n + yg(a n )) − n log F(a n + yg(a n )) = O(1)nF 2 (a n + yg(a n )) − log F n (a n + yg(a n )).
If r(a n ) n → 0, we obtain that r(a n ) log F n (a n + yg(a n )) + e −y → −β y 2 2 e −y , and hence also that r(a n ) log e exp −y F n (a n + yg(a n )) → −β y 2 2 e −y , and r(a n ) e exp −y F n (a n + yg(a n )) − 1 → −β y 2 2 e −y , or r(a n ) F n (a n + yg(a n )) − exp −e −y → −β y 2 2 e −y exp −e −y .
It means that, if X i are independent and identically distributed random variables with distribution function F, then r(a n ) P M n − a n g(a n ) ≤ y − Λ(y) → Φ(y), where M n = max(X 1 , X 2 , ..., X n ), Λ(y) = exp −e −y and Φ(y) = −β y 2 2 e −y exp −e −y . It means that F is in the max-domain of attraction of the double exponential and the convergence rate is determined by r(a n ).
As for g(x), we have g(x) x → 0 and g(x + yg(x)) g(x) x .

Concluding Remarks
In this paper, new results on the condition, for some functions r and g, lim x→∞ r(x) U(x + yg(x)) U(x) − e αy = θ(y), ∀y, where we assume that the convergence is l.u. in y, are presented. This limit generalizes the ones analyzed by Seneta [4] and Omey and Cadena [5], both of them being related to the monotony of functions in the Zygmund sense. Under this analysis, properties of θ(y) are described. Representations of the functions U involved in this limit are provided.