The purpose of this paper is to investigate the asymptotic behavior of the difference
as , where X and Y are fat-tailed random variables (loss variables) and is a family of risk measures. The case where is an a-percentile value-at-risk (VaR), has been treated in  , where it was shown that the asymptotic behavior of drastically changes according to the relative magnitudes of the thicknesses of the tails of X and Y (the definition of the VaR is given in (2.1) in the next section). In this paper, we study a progressive case in which is given as a parameterized spectral risk measure, and we obtain similar results as in  . In particular, we find that if X and Y are independent and if the tail of X is sufficiently fatter than that of Y, then converges to the expected value as whenever are spectral risk measures converging to a risk measure of the worst case scenario. That is, whenever
for each loss random variable Z in some sense. Our result does not require any specific form for , implying that this property is robust. Furthermore, assuming some technical conditions for the probability density functions of X and Y, we study the asymptotic behavior of the Euler contribution, defined as
(see Remark 17.1 in  ), and show that is asymptotically equivalent to as . Here, is a constant determined according to the relative magnitudes of the thicknesses of the tails of X and Y.
We now briefly review the financial background for this study. In quantitative financial risk management, it is important to capture tail loss events by using adequate risk measures. One of the most standard risk measures is the VaR. The Basel Accords, which provide a set of recommendations for regulations in the banking industry, essentially recommend using VaR as a measure of risk capital for banks. VaRs are indeed simple, useful, and their values are easy to interpret. For instance, a yearly 99.9% VaR calculated as means that the probability of a risk event with a realized loss larger than is 0.1%. In other words, an amount of risk capital is sufficient to prevent a default with 99.9% probability. The meaning of the amount is therefore easy to understand. However, VaRs are often criticized for their lack of subadditivity (see, for instance,    and  ). VaRs do not reflect the risk diversification effect.
The expected shortfall (ES) has been proposed as an alternative risk measure that is coherent (in particular, subadditive) and tractable, with the risk amount at least that of the corresponding VaR. Note that there are various versions of ES, such as the conditional value-at-risk (CVaR), the average value-at-risk (AVaR), the tail conditional expectation (TCE), and the worst conditional expectation (WCE). These are all equivalent under some natural assumptions (see    , and  ). It should be noted that the Basel Accords have also considered recently the adoption of ESs as a minimal capital requirement, in order to better capture market tail risks (see for instance  and  ).
A spectral risk measure (SRM) has been proposed as a generalization of ESs, in  . SRMs are characterized by a weight function that represents the significance of each confidence level for the risk manager. SRMs are equivalent to comonotonic law-invariant coherent risk measures (see Remark 1 in the next section).
VaRs and ESs as risk measures depend on a confidence level parameter. We let (resp.,) denote the VaR (resp., ES) with confidence level a. When a is close to 1, the values of and are increasing without bound as in (1.2). The parameter a corresponds to the risk aversion level of the risk manager. Higher values of a indicate that the risk manager is more risk-averse and evaluates the tail risk as more severe.
In this paper, we consider a family of SRMs parameterized by the confidence level a. We make a mathematical assumption that intuitively implies situation (1.2) and investigate the asymptotic behaviors of (1.1) and (1.3) as, when the tail probability function of X (resp., Y) is regularly varying with index (resp.,). Our main theorem asserts that the asymptotic behaviors of (1.1) and (1.3) strongly depend on the relative magnitudes of and. Note that our results include the case, the inclusion of which was discussed as a future task in  .
The rest of this paper is organized as follows. In Section 2, we prepare the basic settings and introduce the definitions for SRMs based on confidence level. In Section 3, we give our main results. We numerically verify our results in Section 4. Finally, Section 5 summarizes our studies. Throughout the main part of this paper, we assume that X and Y are independent. The more general case where X and Y are not independent is studied in Appendix 1. All proofs are given in Appendix 2.
Let be a standard probability space and let denote a set of non-negative random variables defined on. For each, we denote by the distribution function of Z and by its tail probability function; that is, and. Moreover, for each, we define
Note that is exactly the left-continuous version of the generalized inverse function of.
We now introduce the definition of SRMs.
1) A Borel measurable function is called an admissible spectrum if is right-continuous, non-decreasing, and satisfies
2) A risk measure is called an SRM if there is an admissible spectrum such that, where
Remark 1 SRMs are law-invariant, comonotonic, and coherent risk measures. However, as shown in   , and  , if is atomless, then for any law-invariant comonotonic convex risk measure, there is a probability measure on such that
for each. This is due to the generalized Kusuoka representation theorem (Theorem 4.93 in  ), where is the a-percentile expected shortfall of Z:
Moreover, such a is always coherent and satisfies the Fatou property  . Furthermore, representation (2.3) can also be rewritten as, where
Here, it is easy to see that is non-negative, non-decreasing, right-continuous, and satisfies
meaning that is an admissible spectrum (see  ). Therefore, any law-invariant comonotonic convex (or coherent) risk measure is completely characterized as an SRM. Arguments similar to those above, replacing with, where, can be found in  and  .
Next, we introduce a family of SRMs parameterized by the confidence level a.
Definition 2 Let be a family of admissible spectra and let. Then is called a set of confidence-level-based spectral risk measures (CLBSRMs) if
where is a probability measure on defined by and is the Dirac measure with unit mass at 1.
Condition (2.5) formally implies (1.2). Indeed, if is a bounded random variable with a distribution function that is continuous and strictly increasing on, where, then the function is bounded and continuous, so that (2.5) gives
where we recognize. Moreover, we see that
Lemma 1 Relation (2.5) is equivalent to
We now give some examples of CLBSRMs.
Example 1. Expected Shortfalls
defined by (2.4) is a typical example of a CLBSRM. The corresponding admissible spectra are given as
It is easy to see that (2.5) does hold. Indeed, for any bounded continuous function f defined on, we see that
due to the bounded convergence theorem. Equivalently, we can also check that satisfies (2.6).
is characterized as the smallest law-invariant coherent risk measures that are greater than or equal to  . Note that if the distribution function of the target random variable Z is continuous, then coincides with, where
(see  for details).
Example 2. Exponential/Power SRMs
An admissible spectrum f corresponding to an SRM represents the preferences of a risk manager for each quantile of the loss distribution. Therefore, the form taken by f corresponds to the manager’s risk aversion, which is also described in terms of utility functions in classical decision theory. Recently, the relation between expected utility functions and SRMs has been studied, though it has not been entirely resolved. Here we introduce some examples of SRMs based on specific utility functions.
The exponential utility function is a typical example of tractable utility functions
where p denotes the profit-and-loss (indicating profit) and g characterizes the degree of risk preference. We focus on the case so that describes a risk-averse utility function. We transform the parameter g into the confidence level using. Note that the original parameter g can be recovered using the inverse. The exponential utility of the loss l with confidence level a is then given as. Cotter and Dowd  have proposed an SRM based on the exponential utility by constructing an admissible spectrum for some, so that satisfies (2.2). Then, must be set as, giving
Note that the theoretical validity of the above method is still unclear. Other methods to adequately construct SRMs from exponential utility functions have been discussed in   , and  , but no definite answer has been reached. In particular, it is pointed out in  that there exists no general consistency between expected utility theory and SRM-decision making. In any case, we can easily verify that as defined above satisfies (2.5)-(2.6), which implies that is actually a CLBSRM.
Similarly to the above, an SRM based on the power utility function has been studied in  . After changing the risk aversion parameter to the confidence level as above, is given as
We can also verify that is a CLBSRM.
We now introduce some notations and definitions used in asymptotic analysis and extreme value theory.
Let f and g be positive functions defined on, where and. We say that f and g are asymptotically equivalent (denoted as) as if. When, we say that f is regularly varying with index if it holds that for each. Moreover, we say that f is ultimately decreasing if f is non-increasing on for some. For more details, we refer the reader to  and  .
3. Main Results
Our main purpose is to investigate the property of (1.1) for a CLBSRM and random variables whose distributions are fat-tailed. To consider this case, we assume that and are regularly varying functions with indices and, respectively. That is, for each and
In  , we study the asymptotic property of (1.1) as when. The results display the following five patterns: (i), (ii), (iii), (iv), and (v). In cases (iv) and (v), we consider the difference instead of, and the results are restated consequences of cases (i) and (ii). Hence, we assume here that and focus on cases (i)-(iii) only. We further assume that. This assumption guarantees the integrability of X and Y (see, for instance, Proposition A3.8 in  ).
Let be a CLBSRM with a family of admissible spectra. Here we assume that
for each. Then, Lemma A.23 in  implies that
for each. This immediately implies that. Furthermore, by (17.9b) and Proposition 17.2 in  , we see that
where is given by (1.3) if is continuously differentiable in h. Note that inequality (3.3) holds for each whenever is coherent.
Our main purpose in this section is to investigate in detail the asymptotic behavior of, as well as if it is defined, as. To clearly state our main results, we establish the following conditions, which are assumed to hold in Section 4 of  .
[C1] X and Y are independent.
[C2] There is some such that has a positive, non-increasing
density function on; that is,.
[C3] The function converges to some real number k as.
Let us adopt the notation
for. Note that is finite for each fixed (see Corollary 1 in Appendix 2). Our main results are the two following theorems.
Theorem 1 Assuming [C1]-[C3], as.
Formally, assertions (i)-(iii) of Theorem 4.1 in  are the same as the assumptions of Theorem 1, by setting. That is, we have as, where
Theorem 1 justifies the following relation:
Note that condition [C3] is not required for Theorem 1 when. Moreover, when, Theorem 1 implies that converges to as. The limit does not depend on the forms of, so this result is robust. The second main result is as follows.
Theorem 2 Assume [C1] and [C3]. Moreover, assume that
[C4] X and Y have positive, continuous, and ultimately decreasing density functions and, respectively, on.
Under these assumptions, as, where is a positive constant given by
Theorems 1 and 2 together imply that if X and Y are independent, and if and have adequate density functions, then
Note that is always smaller than or equal to 1, so that (3.7) is consistent with inequality (3.3). In particular, if, then the asymptotic equivalence between the marginal risk contribution and the Euler contribution is justified (see (17.10) in  for the definition of marginal risk contributions).
Note that is always larger than or equal to so long as the random vector satisfies a suitable technical condition, such as Assumption (S) in  . (Here, we modify some conditions of the original version of Assumption (S) to facilitate focusing on non-negative random variables.) Indeed, because is a convex risk measure, the function is convex. Thus, we get
where the last equality in the above relation is obtained from (5.12) in  ,
due to the dominated convergence theorem. Therefore, if, then
In Section 4, we numerically verify the above relation. Note that we can also verify a version of Assumption (S) under [C4].
1) If is continuous, then has a uniform distribution on (see, for instance, Lemma A.21 in  ). Therefore, with is rewritten as
where denotes the expectation operator with respect to the probability measure defined as
Note that we have, and so represents the risk scenario that attains the maximum in the following robust representation of:
where is a set of probability measures on. Also note that if, then is given by
Until the end of Remark 2, we assume that and are continuous.
2) We can relax the independence condition [C1] so that X may weakly depend on Y within the negligible joint tail condition (see Remark A.1 in  ). In this case, under some additional assumptions such as [A5] and [A6] in  , we can make the same assertion as in Theorem 1, where the value in the definition (3.4) of is replaced by. In particular, if, then
Indeed, our proof in Appendix 2 also works by applying Theorem A.1 in  instead of Theorem 4.1. Note that we need some additional condition to have that
(see Proposition 3 in Appendix 2).
3) As mentioned in Appendix A.1 of  , we can get another version of Theorem A.1 by switching the roles of and X and by imposing modified (though somewhat artificial) mathematical conditions such as [A5’] and [A6’] in  . In particular, if, we see that
and then (by the same proof as Theorem 1 with (3.13))
under some assumptions. Here, is a probability measure defined by (3.10) with replacing X by. If X and Y are independent (with natural assumptions on the density functions), then (3.7) implies that (3.14) is also true. Here, note that the last equality of (3.14) is obtained by (1.3), (3.9), and the dominated convergence theorem. Indeed, we have
because is uniformly distributed on. In Appendix 1, we will show that under some technical conditions that are more natural than both [A5]-[A6] and [A5’]-[A6’] in  , relations (3.11) and (3.14) simultaneously hold in the case, even if X and Y are dependent.
Note that if, then
which is known as the component CVaR (also known as the CVaR contribution) and widely used, particularly in the practice of credit portfolio risk management (see for instance   , and  ).
4. Numerical Analysis
In this section, we numerically investigate the behavior of. Throughout this section, we assume that the distributions of X and Y are given as and, respectively, with and, where denotes the generalized Pareto distribution whose distribution function is given by,. Then, and satisfy (3.1) with and. Note that condition [C3] is satisfied with
(see (5.2) in  ). Also note that and are analytically solved as
We numerically compute, and, where we let for brevity. In all calculations, we fix and. For and, we examine several patterns to study each of the following three cases: 1), 2), and 3).
We set and. Hence, and, so that holds. Figure 1 shows the graphs of, and. These values are always larger than whenever, and they converge to for both and. Indeed,
holds because, for each. The limit as is a consequence of Theorem 1. Moreover, the forms of these graphs are unimodal. That is, the function increases on and decreases on for some. Intuitively, the values of seem to become large as a increases because a larger a implies a greater risk sensitivity. However, our result implies that the impact of adding loss variable Y into the prior risk profile X is maximized at some.
Figure 2 shows the relation between and. We see that takes a maximum at, where is a solution to
Indeed, we have the following result.
Proposition 1 If there is a unique solution to (4.2), then
Note that unlike the case of SRMs, takes a value smaller than if a is small. This is because VaR is not a convex risk measure, so the relation (3.8) is not guaranteed for. In particular, we observe that
Figure 1. Graphs of (blue), (orange), (green) and (black, dashed) with and. The red solid line shows. The horizontal axis corresponds to a.
Figure 2. Graphs of (blue) and (brown, dashed) with and. The red solid line shows. The horizontal axis corresponds to a.
Figure 3 shows the approximation errors, defined as
with () and (). We see that is close to 0 as for each case of. Moreover, we numerically verify the assertion of Theorem 2 for in Figure 4. We observe that converges to as.
By contrast, the convergence speed of as decreases if the tails of X and Y are less fat-tailed. Figure 5 shows with () and (). We find that decreases as a tends to 1, but the gap between and 0 is still large, even in the case.
Similarly to Case 2), the convergence speed of decreases as the tails of X and Y become thinner. Figure 8 shows the graph of with. The approximation error tends to zero as, but remains smaller than −20% even when.
5. Concluding Remarks
In this paper, we have studied the asymptotic behavior of the difference between and as when is a parameterized SRM satisfying (1.2). We have shown that is asymptotically equivalent to given by (3.4), whose form changes according to the relative magnitudes
Figure 3. Approximation errors defined by (4.4) with and. Blue line:. Orange line:. Green line:. The horizontal axis corresponds to a.
Figure 4. (blue) and (red). We set and. The horizontal axis corresponds to a.
Figure 5. Approximation errors defined by (4.4) with and. Blue line:. Orange line:. Green line:. The horizontal axis corresponds to a.
Figure 6. Approximation errors defined by (4.4) with. Blue line:. Orange line:. Green line:. The horizontal axis corresponds to a.
Figure 7. (blue) and (red). We set. The horizontal axis corresponds to a.
Figure 8. Approximation errors defined by (4.4) with. Blue line:. Orange line:. Green line:. The horizontal axis corresponds to a.
of the thicknesses of the tails of X and Y. In particular, for, we found the convergence for general CLBSRMs. Moreover, we also found that as for a constant given by (3.6). This clarifies the asymptotic relation between the marginal risk contribution and the Euler contribution.
Our numerical results in the case showed that is not increasing but is unimodal with respect to a, which implies that the impact of Y in the portfolio does not always increase with a. Interestingly, this phenomenon is inconsistent with intuition.
Our results essentially depend on the assumption that X and Y are independent. However, the dependence structure of the loss variables X and Y plays an essential role in financial risk management. The case of dependent X and Y for has already been studied in Section A.1 of  . As mentioned in Remark 2, we have now generalized this result to the case of CLBSRMs. However, we require the somewhat strong assumption that X and Y are not strongly dependent on each other. With the additional analysis in Appendix 1, we will see that our main results still hold for a general dependence structure if, but that they are easily violated if. In future work, we will continue to study the asymptotic behavior of as, without the independence condition.
Appendix 1. A Short Consideration of the Dependent Case
Here, we briefly investigate the asymptotic behavior of as when X and Y are not independent. Throughout this section, we assume that, , and are continuous. With this, (3.8) is rewritten as. Combining this result with (3.3), we have
Note that (A.1) holds for general SRM whenever (3.9) holds.
1.1. Comonotonic Case
We consider the case where X and Y are comonotone. In other words, they are perfectly positively dependent (see Definition 4.82 of  and Definition 5.15 in  ). In this case, the following proposition is straightforwardly shown.
Proposition 2 If X and Y are comonotone, then
This proposition implies that when, the asymptotic relations (3.11) and (3.14) still hold, even if X and Y are strongly correlated, but that the assertions of Theorems 1 and 2 do not necessarily hold when.
1.2. Additional Numerical Analysis
Similarly to Section 4, we assume that and with,. To describe the dependence between X and Y, we introduce a copula. By Sklar’s theorem, we see that the joint distribution function of the random vector is represented by
for a copula, which is a distribution function with uniform marginals. Here, we examine the following three copulas:
1) The Gaussian copula, ,
2) The Gumbel copula, ,
3) The countermonotonic copula,
where is the distribution function of the standard
normal distribution (for more details on the copulas, see, for instance, Chapter 5 of  ). The parameters in (a) and in (b) describe the strength of the dependence between X and Y. We always set and in this section. If, then X and Y are perfectly negatively dependent. In particular, in that case, X and Y are represented as and, where U is a random variable with uniform distribution on.
Figure A1 summarizes the results with and. We compare the values of (with) and. We find that all these values converge to the same value, which is not equal to, by letting. Note that when X and Y are countermonotonic, they converge to zero as, so (3.12) does not hold in this case.
Figure A2 shows the graphs of the relative errors defined by (4.4) with when we set and. We find that does not converge to zero as. Similar phenomena are observed in Figure A3 with the settings. Therefore, the assertion of Theorem 1 does not hold when if X and Y are correlated.
Figure A1. Graphs of (blue), (orange), (green) and (black, dashed) with and. The red solid line shows. The horizontal axis corresponds to a. Top: with. Center: with. Bottom:.
Figure A2. Approximation errors defined by (4.4) with and. Blue line:. Orange line:. Green line:. The horizontal axis corresponds to a. Top: with. Center: with. Bottom:.
Note that the above findings are consistent with the comonotonic case (Proposition 2).
1.3. Theoretical Result in the Case
We describe the following conditions.
[C5] For each, has a positive, non-increasing density function on, where is the conditional distribution function of X given. Moreover, is continuous in and y.
[C6] There is a such that is uniformly regularly varying with index in the following sense:
Figure A3. Approximation errors defined by (4.4) with. Blue line:. Orange line:. Green line:. The horizontal axis corresponds to a. Top: with. Center: with. Bottom:.
for each. Moreover, is ultimately decreasing.
[C7] It holds that
Conditions [C5]-[C7] strongly correspond to conditions [A5]-[A6] in  . It should be noted that the index parameter is assumed to be equal to in condition [A6] in  , but that this equality is not required to obtain our results. Note also that may be different from. Indeed, we can verify, at least numerically, that for each, the function is regularly varying with index (resp.,) if we adopt (resp.,) as a copula for the random vector whose marginal distributions are given by the generalized Pareto distribution.
Using a similar argument as in the proof of the uniform convergence theorem (Theorem 1.2.1 in  ), together with the continuity of in y, we get from (A.3) that
for each compact set.
We now introduce the following result.
Theorem 3 Assume [C5]-[C7] and (3.12). If, it holds that
This theorem claims that both (3.11) and (3.14) are true under some conditions, even when X and Y are dependent.
Appendix 2. Proofs
\Proof of Lemma 1. Assume (2.5). Fix any. Then, (2.5) implies that
Because is non-decreasing and non-negative, we see that
Combining (B.1) with (B.2), we have.
Conversely, if we assume (2.6), then Prokhorov’s theorem implies that for
each increasing sequence with there is a further subsequence and a probability measure on such that
weakly converges to as. Then, for each, we see that
This immediately leads us to, hence. We therefore arrive at (2.5).
Proof of Proposition 1. Let. We observe that
where. By (4.1), (4.3), and Theorem 1, we see that g is continuous on, and. Moreover, by the assumption, it holds that and for all. Together, these imply that g is positive on and negative on, and that has the same pattern. Therefore, takes a maximum at.
Proof of Proposition 2. Because is comonotonic, we obviously have
Here, we see that and for some random variable U with uniform distribution on (see Lemmas 4.89-4.90 in  and their proofs). Then we have
Similarly, because, we have
and, which completes the proof.
2.1. Proof of Theorem 1
We first state some propositions and prove them. For this, let be given as (3.5). Note again that defined in (3.4) satisfies
Proof. If, we see that because Y is non-negative and is positive. If, we observe
where is a real number satisfying. The existence of such an can be proven using Propositions 1.5.1 and 1.5.15 in  . Similarly, if, we have
Proof. If, the assertion is obvious from the assumption. If, we see that
because of. If, we have
Proof. This follows from (3.2) and Proposition 4.
Proof of Theorem 1. Let,. Note that
by virtue of Theorem 4.1(i)-(iii) in  . Moreover, (B.3) immediately implies
Furthermore, it holds that
hence is integrable. The integrability of is guaranteed by Proposition 4.
Temporarily fix any. From (2.6) and (B.5), we easily see that
Similarly, we have
Additionally, we have
where. Using (B.7) and Proposition 3, we obtain
By (B.8) and (B.9), we have
Combining this with (B.6) and Proposition 3, we arrive at
Because is arbitrary, we obtain the desired assertion by (B.4).
2.2. Proof of Theorem 2
Let for brevity. We see that Z has a density function
Lemma 2 is positive and continuous on. Moreover, is regularly varying with index and it holds that
Proof. Continuity and positivity are obvious. By [C4] and Theorem 1.1 in  , we see that, and that is regularly varying with index. The last assertion is obtained by Proposition 1.5.10 in  .
Let be the conditional distribution function of Y given. Then we have
Proposition 5 It holds that
Proof. For each, a straightforward calculation gives
which implies our assertion.
Note that (B.11) and Proposition 5 lead to
Proposition 6 If, then
Then, we see that
Therefore, we need to show that
First, we show that
Using (B.10), Lemmas A.1 and A.3 in  , and Proposition A3.8 in  , we obtain
Furthermore, we observe that
and that the function is regulary varying with index. Thus, we obtain
Now, (B.18) is obvious.
Next, we observe that
Because and are convergent (as), they are bounded. Thus, we have
for some. By (B.18) and (B.20), we can apply the dominated convergence theorem to obtain (B.17).
Proposition 7 If, then
Proof. Let, , and be the same as in (B.13)-(B.15). First, we have, by the same argument as in the proof of Proposition 6. Next, for each, we see that
due to [C3], (B.10), Proposition A3.8 in  , Proposition 3.1(i) in  , and Lemmas A.1 and A.3 in  . Moreover, we have (B.19), and the right-hand side of this inequality converges to as, and so it is bounded. Therefore, we apply the dominated convergence theorem to obtain as. We complete the proof by combining these with (B.16).
Proposition 8 If, we have
Proof. Let, and be set as earlier. Similarly to the proof of Propositions 6 and 7, we get,. This implies that, , where. Therefore, it suffices to show that as, which is easy to see by using similar calculations as in the proof of Proposition 7 and by using Proposition 3.1(i) in  .
Proposition 9 If, then
Proof. Similarly to the proof of Proposition 8, we need to show only that
where. Note that Lemmas A.1 and A.2 in  imply, and,. Therefore, for each, we observe
by [C3], (B.10), Proposition A3.8 in  , and Lemma A.3 in  . Moreover, we have
and thus we obtain (B.21) by applying the dominated convergence theorem.
Proof of Theorem 2. We can verify that the random vector satisfies (a version of) Assumption (S) in  by using a standard argument. Therefore, (3.9) is true from (5.13) in  . Additionally, using Propositions 6-9, we see that for each, there is an such that
where we denote. Moreover, it is easy to see that and are bounded on. Therefore, combining (3.9), (3.15), and (B.22), we get
where, which is positive due to Proposition 3. Because
is arbitrary, we obtain the desired assertion.
2.3. Proof of Theorem 3
First, note that condition [C5] immediately implies [C2] with and
Second, note that by [C6], Proposition 3.1(i) in  (see also Remark 3.2 therein) and Proposition A3.8 in  , we have (B.10) and
To prove Theorem 3, we give the following three propositions.
Proposition 10 is continuously differentiable in and it holds that
Proposition 10 is obtained by an argument similar to the proof of Lemma 5.3 in  , using the implicit function theorem.
Proposition 11 The function is regularly varying with index.
Proof. Fix any. We observe that
and therefore, using [C5], we arrive at
Proof. Fix any. Then we have
where we denote and
By [C7] and the Chebyshev inequality, we get
for some. Because Proposition 11 tells us that is regularly varying with index, the right-hand side of (B.24) converges to zero as (see Proposition 1.5.1 in  ).
Moreover, we see that
Here, we observe that
for each. Note that if (resp.,), we have (resp.,). Moreover, by (B.23), converges to 1 as, and so it is bounded. Therefore, we get
Now we arrive at
by using (A.5) and (B.23). Because is arbitrary, we obtain the desired assertion.
Proof of Theorem 3. First, note that Proposition 10 guarantees that
Then, fix any. By Propositions 11-12 and Lemma A.3 in  , we see that
Thus, there is an such that
Therefore, we have
by virtue of (3.12). Because is arbitrary, we get that,. Combining this result with (A.1), we obtain the desired assertion.