Portfolio credit risk with Archimedean copulas: asymptotic analysis and efficient simulation

Abstract

In this paper, we study large losses arising from defaults of a credit portfolio. We assume that the portfolio dependence structure is modelled by the Archimedean copula family as opposed to the widely used Gaussian copula. The resulting model is new, and it has the capability of capturing extremal dependence among obligors. We first derive sharp asymptotics for the tail probability of portfolio losses and the expected shortfall. Then we demonstrate how to utilize these asymptotic results to produce two variance reduction algorithms that significantly enhance the classical Monte Carlo methods. Moreover, we show that the estimator based on the proposed two-step importance sampling method is logarithmically efficient while the estimator based on the conditional Monte Carlo method has bounded relative error as the number of obligors tends to infinity. Extensive simulation studies are conducted to highlight the efficiency of our proposed algorithms for estimating portfolio credit risk. In particular, the variance reduction achieved by the proposed conditional Monte Carlo method, relative to the crude Monte Carlo method, is in the order of millions.

Appendix: Proofs

To simplify the notation, for any two positive functions g and h, we write $g\lesssim h$ or $h > rsim g$ if $\lim \sup g/h\le 1$.

1.1 A.1 Proofs for LT-Archimedean copulas

We first list a series of lemmas that will be useful for proving Theorem 4.1 and Theorem 4.2. The following is a restatement of Theorem 2 of Hoeffding (1963).

Lemma A.1

If $X_{1},X_{2},\ldots ,X_{n}$ are independent and $a_{i}\le X_{i}\le b_{i}$ for $i=1,\ldots ,n$, then for $\varepsilon >0$

$$\begin{aligned} \mathbb {P}\left( \left| \bar{X}_{n}-\mathbb {E}\left[ \bar{X}_{n}\right] \right| \ge \varepsilon \right) \le 2\exp \left( -\frac{2n^{2} \varepsilon ^{2}}{\sum _{i=1}^{n}(b_{i}-a_{i})^{2}}\right) , \end{aligned}$$

with $\bar{X}_{n}=\left( X_{1}+X_{2}+\ldots +X_{n}\right) /n$.

Applying Lemma A.1, we obtain the following inequality:

Lemma A.2

For any $\varepsilon >0$ and any large M, there exists a constant $\beta >0$ such that

$$\begin{aligned} \mathbb {P}_{v}\left( \left| \frac{1}{n}\sum _{i=1}^{n} c_{i} 1_{\{U_{i}>1-l_{i}f_{n}\}}-r(v)\right| \ge \varepsilon \right) \le \exp (-n\beta ), \end{aligned}$$

uniformly for all $0<v\le M$ and for all sufficiently large n, where $\mathbb {P}_{v}$ denotes the original probability measure conditioned on $V=\frac{v}{\phi (1-f_{n})}$.

Proof

Note that $U_{i}$ are conditionally independent on V. Then by Lemma A.1, for every n,

$$\begin{aligned} \mathbb {P}_{v}\left( \left| \frac{1}{n}\sum _{i=1}^{n} c_{i} 1_{\{U_{i}>1-l_{i}f_{n}\}}-\frac{1}{n}\sum _{i=1}^{n}c_{i}p(v,i)\right| \ge 2\varepsilon \right) \le 2\exp \left( -\frac{8n^{2}\varepsilon ^{2}}{\sum _{i=1}^{n} c_{i}^{2}}\right) \le \exp (-n\beta ), \end{aligned}$$

(A.1)

where $\beta$ is some unimportant constant not depending on n and v.

Using (A.1), to obtain the desired result, it suffices to show the existence of N, such for all $n\ge N$,

$$\begin{aligned} \left| \frac{1}{n}\sum _{i=1}^{n}c_{i}p(v,i)-r(v)\right| \le \varepsilon \end{aligned}$$

(A.2)

holds uniformly for all $v\le M$. Recall that $r(v)=\sum _{j\le |\mathcal {W} |}c_{j}w_{j}\tilde{p}(v,j)$. Note that $n_{j}$ denotes the number of obligors in sub-portfolio j. Then

$$\begin{aligned} \left| \frac{1}{n}\sum _{i=1}^{n}c_{i}p(v,i)-r(v)\right|&=\left| \sum _{j\le |\mathcal {W}|}c_{j}\left( p(v,j)\frac{n_{j}}{n} -\tilde{p}(v,j)w_{j}\right) \right| \nonumber \\&\le \sum _{j\le |\mathcal {W}|}c_{j}p(v,j)\left| \frac{n_{j}}{n} -w_{j}\right| \nonumber \\&\quad +\, \sum _{j\le |\mathcal {W}|}c_{j}w_{j}\left| p(v,j)-\tilde{p} (v,j)\right| \nonumber \\&\le \sum _{j\le |\mathcal {W}|}c_{j}\left| \frac{n_{j}}{n}-w_{j} \right| +\bar{c}\max \limits _{j\le |\mathcal {W}|}\left| p(v,j)-\tilde{p}(v,j)\right| \end{aligned}$$

(A.3)

where $\bar{c}=\sum _{j\le |\mathcal {W}|}c_{j}w_{j}$. By Assumption 2.1, there exists $N_{1}$ satisfying $\sum _{j\le |\mathcal {W}|} c_{j}\left| \frac{n_{j}}{n}-w_{j}\right| \le \frac{\varepsilon }{2}$ for all $n\ge N_{1}$. For the second part of (A.3), by noting that $e^{x}\ge 1+x$ for all $x\in \mathbb {R}$, we have

$$\begin{aligned} \left| p(v,j)-\tilde{p}(v,j)\right|&=\exp \left( -v\left( \frac{\phi (1-l_{j}f_{n})}{\phi (1-f_{n})}\wedge l_{j}^{\alpha }\right) \right) \left( 1-\exp \left( -v\left| \frac{\phi (1-l_{j}f_{n})}{\phi (1-f_{n} )}-l_{j}^{\alpha }\right| \right) \right) \\&\le v\left| \frac{\phi (1-l_{j}f_{n})}{\phi (1-f_{n})}-l_{j}^{\alpha }\right| \\&\le M\left| \frac{\phi (1-l_{j}f_{n})}{\phi (1-f_{n})}-l_{j}^{\alpha }\right| . \end{aligned}$$

Since $\phi \in \mathrm {RV}_{\alpha }(1)$, there exists $N_{2}$ such that for all $n\ge N_{2}$, $\bar{c}\max \limits _{j\le |\mathcal {W}|,v\in A}\left| p(v,j)-\tilde{p}(v,j)\right| \le \frac{\varepsilon }{2}$.

Combining the upper bound for both parts in (A.3) and letting $N=\max \{N_{1},N_{2}\}$, (A.2) holds uniformly for all $v\le M$. The proof is then completed. $\square$

The following proof of Theorem 4.1 is motivated by the proof of Theorem 1 in Bassamboo et al. (2008).

Proof of Theorem 4.1

Let $v_{\delta }^{*}$ denote the unique solution to the equation $r(v)=b-\delta$. By using continuity and monotonicity of r(v) in v, we have

$$\begin{aligned} v_{\delta }^{*}\rightarrow v^{*} \end{aligned}$$

as $\delta \rightarrow 0$.

Fix $\delta >0$. We decompose the probability of the event $\{L_{n}>nb\}$ into two terms as

$$\begin{aligned} \mathbb {P}\left( L_{n}>nb\right)&=\mathbb {P}\left( L_{n}>nb,V\le \frac{v_{\delta }^{*}}{\phi (1-f_{n})}\right) +\mathbb {P}\left( L_{n}>nb,V>\frac{v_{\delta }^{*}}{\phi (1-f_{n})}\right) \\&=I_{1}+I_{2}. \end{aligned}$$

The remaining part of proof will be divided into three steps. We first show that $I_{1}$ is asymptotically negligible. Then we develop upper and lower bounds for $I_{2}$ with the second and third step.

Step 1. We show $I_{1}=o(f_{n})$. Note that for any $v\le v_{\delta }^{*}$, $r(v)\le b-\delta$. Thus, by Lemma A.2, for all sufficiently large n, there exists a constant $\beta >0$ such that

$$\begin{aligned} \mathbb {P}_{v}\left( L_{n}>nb\right) \le \mathbb {P}_{v}\left( \frac{1}{n}\sum _{i=1}^{n}c_{i}1_{\{U_{i}>1-l_{i}f_{n}\}}-r(v)>\delta \right) \le \exp (-n\beta ) \end{aligned}$$

uniformly for all $v\le v_{\delta }^{*}$. So the same upper bound holds for $I_{1}$. Due to the condition on $f_{n}$, $I_{1}=o(f_{n})$.

Step 2. We now develop an asymptotic upper bound for $I_{2}$. Note that

$$\begin{aligned} I_{2}\le \mathbb {P}\left( V>\frac{v_{\delta }^{*}}{\phi (1-f_{n})}\right) =\overline{F}_{V}\left( \frac{v_{\delta }^{*}}{\phi (1-f_{n})}\right) . \end{aligned}$$

Recall that $\phi ^{-1}$ is the LS transform for random variable V. Then by $\phi (1-\frac{1}{\cdot })\in \mathrm {RV}_{-\alpha }$ and Karamata’s Tauberian theorem, we obtain

$$\begin{aligned} I_{2}&\le \overline{F}_{V}\left( \frac{v_{\delta }^{*}}{\phi (1-f_{n} )}\right) \\&\sim \frac{{1-\phi ^{-1}\left( \frac{\phi (1-f_{n})}{v_{\delta }^{*} }\right) }}{{\Gamma (1-1/\alpha )}}\\&\sim f_{n}\frac{(v_{\delta }^{*})^{-1/\alpha }}{\Gamma (1-1/\alpha )}, \end{aligned}$$

where in the first step we used $\overline{F}_{V}\in \mathrm {RV}_{-1/\alpha }$ and the second step is due to $1-\phi ^{-1}(\frac{1}{\cdot })\in \mathrm {RV} _{1/\alpha }$. Letting $\delta \downarrow 0$, we obtain

$$\begin{aligned} I_{2}\lesssim f_{n}\frac{(v^{*})^{-1/\alpha }}{\Gamma (1-1/\alpha )}. \end{aligned}$$

(A.4)

Step 3. We now develop an asymptotic lower bound for $I_{2}$. Denote $v_{\widehat{\delta }}^{*}$ as the unique solution to the equation $r(v)=b+\delta$. Similarly, we have $v_{\widehat{\delta }}^{*}\rightarrow v^{*}$ as $\delta \rightarrow 0$. It also follows from the monotonicity of r(v) that $v_{\widehat{\delta }}^{*}\ge v_{\delta }^{*}$. Thus,

$$\begin{aligned} I_{2}\ge \mathbb {P}\left( L_{n}>nb,V>\frac{v_{\widehat{\delta }}^{*}}{\phi (1-f_{n})}\right) . \end{aligned}$$

Note that for any large $M>0$, applying Lemma A.2, it holds uniformly for $v\in \left[ v_{\hat{\delta }}^{*},M\right]$ that $r(v)\ge b+\delta$ and then as $n\rightarrow \infty$, by Lemma A.2

$$\begin{aligned} \mathbb {P}_{v}\left( L_{n}>nb\right)&\ge \mathbb {P}_{v}\left( \frac{1}{n}\sum _{i=1}^{n}c_{i}1_{\{U_{i}>1-l_{i}f_{n}\}}-r(v)>-\delta \right) \\&=1-\mathbb {P}_{v}\left( \frac{1}{n}\sum _{i=1}^{n}c_{i}1_{\{U_{i} >1-l_{i}f_{n}\}}-r(v)\le -\delta \right) \rightarrow 1. \end{aligned}$$

Hence,

$$\begin{aligned} I_{2}& > rsim \overline{F}_{V}\left( \frac{v_{\hat{\delta }}^{*}}{\phi (1-f_{n})}\right) -\overline{F}_{V}\left( \frac{M}{\phi (1-f_{n} )}\right) \\&\sim f_{n}\frac{(v_{\hat{\delta }}^{*})^{-1/\alpha }}{\Gamma (1-1/\alpha )}-f_{n}\frac{M^{-1/\alpha }}{\Gamma (1-1/\alpha )}. \end{aligned}$$

Taking $M\rightarrow \infty$ followed by $\delta \rightarrow 0$, we get

$$\begin{aligned} I_{2} > rsim f_{n}\frac{(v^{*})^{-1/\alpha }}{\Gamma (1-1/\alpha )}. \end{aligned}$$

(A.5)

Combining (A.4), (A.5) with Step 1 completes the proof of the theorem. $\square$

Proof of Theorem 4.2

We first note that the expected shortfall can be rewritten as in (4.6). Using Theorem 4.1, in order to get the desired result, it suffices to show that

$$\begin{aligned} \int _{b}^{\infty }\mathbb {P}\left( L_{n}>nx\right) \mathrm {d}x\sim f_{n} \frac{\int _{v^{*}}^{\infty }r^{\prime }(v)v^{-1/\alpha }\mathrm {d}v}{\Gamma (1-1/\alpha )}. \end{aligned}$$

(A.6)

We decompose the left-hand side of (A.6) into the following two terms

$$\begin{aligned} \int _{b}^{\infty }\mathbb {P}\left( L_{n}>nx\right) \mathrm {d}x&=\int _{b}^{\bar{c}}\mathbb {P}\left( L_{n}>nx\right) \mathrm {d}x+\int _{\bar{c} }^{\infty }\mathbb {P}\left( L_{n}>nx\right) \mathrm {d}x\\&:=J_{1}+J_{2}, \end{aligned}$$

where $\bar{c}=\sum _{j\le |\mathcal {W}|}c_{j}w_{j}$. The remaining part of proof will be divided into three steps. We first show $\mathbb {P}\left( L_{n}>n\bar{c}\right)$ and $J_{2}$ are asymptotically negligible in the first two steps. Then we develop the asymptotic for $J_{1}$ in the last step. For simplicity, we denote the unique solution of the equation $r(v)=s$ for $0\le s\le \bar{c}$ by $r^{\leftarrow }(s)$.

Step 1. In this step, we show

$$\begin{aligned} \mathbb {P}\left( L_{n}>n\bar{c}\right) =o(f_{n}). \end{aligned}$$

(A.7)

Fix an arbitrarily small $\delta >0$. Proceeding in the same way as in step 1 in the proof of Theorem 4.1, for all sufficiently large n, there exists a constant $\beta >0$ such that

$$\begin{aligned} \mathbb {P}\left( L_{n}>n\bar{c},V\le \frac{r^{\leftarrow }(\bar{c}-\delta )}{\phi (1-f_{n})}\right) \le \exp (-n\beta ). \end{aligned}$$

Due to the condition on $f_{n}$ and letting $\delta \downarrow 0$, we have the desired result in (A.7).

Step 2. In this step, we show $J_{2}=o(f_{n}).$ Note that $J_{2}$ can be rewritten as follows,

$$\begin{aligned} J_{2}&=\mathbb {E}\left[ \left( \frac{L_{n}}{n}-\bar{c}\right) _{+}\right] \\&=\mathbb {E}\left[ \left( \frac{L_{n}}{n}-\bar{c}\right) 1_{\left\{ L_{n}>n\bar{c}\right\} }\right] . \end{aligned}$$

Since $\frac{L_{n}}{n}<\max \limits _{j\le \vert \mathcal {W}\vert }c_{j}$, we have

$$\begin{aligned} J_{2}\le \left( \max \limits _{j\le \vert \mathcal {W}\vert }c_{j}-\bar{c}\right) \mathbb {P}\left( L_{n}>n\bar{c}\right) . \end{aligned}$$

It follows from (A.7) that $J_{2}=o(f_{n})$.

Step 3. To this end, we show

$$\begin{aligned} \lim _{n\rightarrow \infty }\int _{b}^{\bar{c}}\frac{\Gamma (1-1/\alpha )}{f_{n} }\mathbb {P}\left( L_{n}>nx\right) \mathrm {d}x=\int _{v^{*}}^{\infty }r^{\prime }(v)v^{-1/\alpha }\mathrm {d}v. \end{aligned}$$

First note that, for any $x\in [b,\bar{c}]$, by Theorem 4.1 we have

$$\begin{aligned} \lim _{n\rightarrow \infty }\frac{\Gamma (1-1/\alpha )}{f_{n}}\mathbb {P}\left( L_{n}>nx\right) =(r^{\leftarrow }(x))^{-1/\alpha }. \end{aligned}$$

Further, the following inequality holds any $x\in [b,\bar{c}]$

$$\begin{aligned} \frac{\Gamma (1-1/\alpha )}{f_{n}}\mathbb {P}\left( L_{n}>nx\right) \le \frac{\Gamma (1-1/\alpha )}{f_{n}}\mathbb {P}\left( L_{n}>nb\right) . \end{aligned}$$

Applying the dominated convergence theorem, we obtain

$$\begin{aligned} \lim _{n\rightarrow \infty }\int _{b}^{\bar{c}}\frac{\Gamma (1-1/\alpha )}{f_{n} }\mathbb {P}\left( L_{n}>nx\right) \mathrm {d}x&=\int _{b}^{\bar{c}}\left( \lim _{n\rightarrow \infty }\frac{\Gamma (1-1/\alpha )}{f_{n}}\mathbb {P}\left( L_{n}>nx\right) \right) \mathrm {d}x\\&=\int _{b}^{\bar{c}}(r^{\leftarrow }(x))^{-1/\alpha }\mathrm {d}x\\&=\int _{v^{*}}^{\infty }r^{\prime }(v)v^{-1/\alpha }\mathrm {d}v. \end{aligned}$$

The last equality is by changing the variable and let $v=r^{\leftarrow }(x)$.

Combing Step 2 and Step 3 completes the proof of the theorem. $\square$

1.2 A.2 Proofs for algorithm efficiency

Lemma A.3 and A.4 will be used in proving Lemma 5.1.

Lemma A.3

For sufficiently large n, there exists a constant C such that

$$\begin{aligned} \frac{f_{V}(x)}{f_{V}^{*}(x)}\le C\left( -\log \phi (1-f_{n})\right) \end{aligned}$$

(A.8)

for all x, where $f_{V}^{*}(x)$ is defined in (5.6).

Proof

By definition of $f_{V}^{*}(x)$, the ratio $\frac{f_{V}(x)}{f_{V}^{*}(x)}$ equals 1 for $x<x_{0}$. Hence, to show (A.8), it suffices to show the existence of a constant C for all $x\ge x_{0}$.

Note that when $x\ge x_{0}$,

$$\begin{aligned} \frac{f_{V}(x)}{f_{V}^{*}(x)}=\frac{f_{V}(x)}{\overline{F}_{V}(x_{0})} x_{0}^{1/\log \phi (1-f_{n})}\left( -\log \phi (1-f_{n})\right) x^{1-\frac{1}{\log \phi (1-f_{n})}}. \end{aligned}$$

By Assumption 4.1 that V has a eventually monotone density function, we have $f_{V}\in \mathrm {RV}_{-1/\alpha -1}$. Then by Potter’s bounds [see e.g. Theorem B.1.9 (5) of de Haan and Ferreira (2007)], for any small $\varepsilon >0$, there exists $x_{0}>0$ and a constant $C_{0}>0$ such that for all $x\ge x_{0}$

$$\begin{aligned} f_{V}(x)\le C_{0}x^{-\frac{1}{\alpha }-1+\varepsilon }. \end{aligned}$$

Thus,

$$\begin{aligned} \frac{f_{V}(x)}{f_{V}^{*}(x)}&\le \frac{C_{0}}{\overline{F}_{V}(x_{0} )}x_{0}^{1/\log \phi (1-f_{n})}\left( -\log \phi (1-f_{n})\right) x^{-1/\alpha -\frac{1}{\log \phi (1-f_{n})}+\varepsilon }\nonumber \\&\le C\left( -\log \phi (1-f_{n})\right) , \end{aligned}$$

(A.9)

which yields our desired result by noting the fact that $x\ge x_{0}$ and $-1/\alpha -\frac{1}{\log \phi (1-f_{n})}+\varepsilon <0$. $\square$

Lemma A.4

If $\phi (1-\frac{1}{\cdot })\in \mathrm {RV}_{-\alpha }$ for some $\alpha >1$ and $f_{n}$ is a positive deterministic function converging to 0 as $n\rightarrow \infty$, then

$$\begin{aligned} \log \phi (1-f_{n})\sim \alpha \log (f_{n}). \end{aligned}$$

Proof

By Proposition B.1.9(1) of de Haan and Ferreira (2007), $\phi \in \mathrm {RV}_{\alpha }(1)$ implies that

$$\begin{aligned} \log \phi (1-x)\sim \alpha \log (x) \end{aligned}$$

as $x\rightarrow 0$. $\square$

The following proof is motivated by the proof of Theorem 3 in Bassamboo et al. (2008).

Proof of Lemma 5.1

Let

$$\begin{aligned} \hat{L}=\prod _{j\le |\mathcal {W}|}\left( \frac{p_{j}}{p_{j}^{*}}\right) ^{n_{j}Y_{j}}\left( \frac{1-p_{j}}{1-p_{j}^{*}}\right) ^{n_{j}(1-Y_{j} )}. \end{aligned}$$

Note that if $\mathbb {E}\left[ L_{n}\left| V=\frac{v}{\phi (1-f_{n} )}\right. \right] <nb$, $p_{j}^{*}=p_{\theta ^{*}}(V\phi (1-f_{n}),j)$ where $\theta ^{*}$ is chosen by solving $\Lambda _{L_{n}|V}^{\prime } (\theta )=nb$; otherwise $p_{j}^{*}=p\left( V\phi (1-f_{n}),j\right)$ by setting $\theta ^{*}=0$. Besides, (5.8) shows $\hat{L}$ can be written as follows.

$$\begin{aligned} \hat{L}=\exp (-\theta ^{*}L_{n}|V+\Lambda _{L_{n}|V}(\theta ^{*})). \end{aligned}$$

Then it follows that, for any v,

$$\begin{aligned} 1_{\left\{ L_{n}>nb,V=\frac{v}{\phi (1-f_{n})}\right\} }\hat{L} \le 1_{\left\{ L_{n}>nb,V=\frac{v}{\phi (1-f_{n})}\right\} }\exp (-\theta ^{*}nb+\Lambda _{L_{n}|V}(\theta ^{*}))\qquad \text {a.s.} \end{aligned}$$

Since $\Lambda _{L_{n}|V}(\theta )$ is a strictly convex function, one can observe that $-\theta nb+\Lambda _{L_{n}|V}(\theta )$ is minimized at $\theta ^{*}$ and equals 0 at $\theta =0$. Hence, the following relation

$$\begin{aligned} 1_{\left\{ L_{n}>nb,V=\frac{v}{\phi (1-f_{n})}\right\} }\hat{L} \le 1_{\left\{ L_{n}>nb,V=\frac{v}{\phi (1-f_{n})}\right\} }\qquad \text {a.s.} \end{aligned}$$

(A.10)

holds for any v.

To prove the theorem, now we re-express

$$\begin{aligned} \mathbb {E}^{*}\left[ 1_{\{L_{n}>nb\}}L^{*^{2}}\right]&=\mathbb {E}^{*}\left[ 1_{\left\{ L_{n}>nb,V\le \frac{v_{\delta }^{*} }{\phi (1-f_{n})}\right\} }L^{*^{2}}\right] +\mathbb {E}^{*}\left[ 1_{\left\{ L_{n}>nb,V>\frac{v_{\delta }^{*}}{\phi (1-f_{n})}\right\} }L^{*^{2}}\right] \\&=K_{1}+K_{2}, \end{aligned}$$

where $v_{\delta }^{*}$ is the unique solution to the equation $r(v)=b-\delta$.

The remaining part of proof will be divided into three steps.

Step 1. In this step, we show

$$\begin{aligned} K_{1}=o(f_{n}). \end{aligned}$$

By Lemma A.3, for sufficiently large n, there exists a finite positive constant C such that

$$\begin{aligned} \frac{f_{V}(v)}{f_{V}^{*}(v)}\le C\left( -\log \phi (1-f_{n})\right) \end{aligned}$$

for all v. From (A.10), it then follows that

$$\begin{aligned} 1_{\left\{ L_{n}>nb,V\le \frac{v_{\delta }^{*}}{\phi (1-f_{n})}\right\} }L^{*^{2}}\le C\left( -\log \phi (1-f_{n})\right) \left( 1_{\left\{ L_{n}>nb,V\le \frac{v_{\delta }^{*}}{\phi (1-f_{n})}\right\} }L^{*}\right) \qquad \text {a.s.} \end{aligned}$$

Therefore, $K_{1}$ is upper bounded by

$$\begin{aligned} \mathbb {E}^{*}\left[ 1_{\left\{ L_{n}>nb,V\le \frac{v_{\delta }^{*} }{\phi (1-f_{n})}\right\} }L^{*^{2}}\right]&\le C\left( -\log \phi (1-f_{n})\right) \left( \mathbb {E}^{*}\left[ 1_{\left\{ L_{n}>nb,V\le \frac{v_{\delta }^{*}}{\phi (1-f_{n})}\right\} }L^{*}\right] \right) \\&=C\left( -\log \phi (1-f_{n})\right) \left( \mathbb {P}\left( L_{n}>nb,V\le \frac{v_{\delta }^{*}}{\phi (1-f_{n})}\right) \right) \\&\le C\left( -\log \phi (1-f_{n})\right) \exp (-\beta n). \end{aligned}$$

The last step is due to step 1 in the proof of Theorem 4.1. Moreover, by Lemma A.4, $-\log \phi (1-f_{n})\sim \alpha \log \left( \frac{1}{f_{n}}\right) =o\left( \frac{1}{f_{n}}\right)$. Note $f_{n}$ has a sub-exponential decay rate, it implies $\frac{1}{f_{n}}\exp (-\beta n/2)\rightarrow 0$. Therefore, $K_{1}$ is still $o(f_{n})$.

Step 2. We show that

$$\begin{aligned} \limsup _{n\rightarrow \infty }\frac{\log K_{2}}{\log f_{n}}\le 2. \end{aligned}$$

(A.11)

By Jensen’s inequality,

$$\begin{aligned} \mathbb {E}^{*}\left[ 1_{\left\{ L_{n}>nb,V>\frac{v_{\delta }^{*}}{\phi (1-f_{n})}\right\} }L^{*^{2}}\right]&\ge \left( \mathbb {E} ^{*}\left[ 1_{\left\{ L_{n}>nb,V>\frac{v_{\delta }^{*}}{\phi (1-f_{n} )}\right\} }L^{*}\right] \right) ^{2}\\&=\left( \mathbb {P}\left( L_{n}>nb,V>\frac{v_{\delta }^{*}}{\phi (1-f_{n})}\right) \right) ^{2}\\&\sim f_{n}^{2}\left( \frac{(v^{*})^{-1/\alpha }}{\Gamma (1-1/\alpha )}\right) ^{2}, \end{aligned}$$

where the last step is due to Theorem 4.1. Then (A.11) follows by applying the logarithm function on both sides and using the fact that $\log \left( f_{n}\right) <0$ for all sufficiently large n.

Step 3. We show that

$$\begin{aligned} \liminf _{n\rightarrow \infty }\frac{\log K_{2}}{\log f_{n}}\ge 2. \end{aligned}$$

(A.12)

First note that, on the set $\left\{ L_{n}>nb,V>\frac{v_{\delta }^{*}}{\phi (1-f_{n})}\right\}$, by (A.10) the likelihood ratio $L^{*}$ is upper bounded by $\frac{f_{V}(v)}{f_{V}^{*}(v)}$ and hence by (A.9), with sufficiently large n, it holds for all $v>\frac{v_{\delta }^{*}}{\phi (1-f_{n})}$ that

$$\begin{aligned} \frac{f_{V}(v)}{f_{V}^{*}(v)}&<\frac{C_{0}}{\overline{F}_{V}(x_{0} )}x_{0}^{1/\log \phi (1-f_{n})}\left( -\log \phi (1-f_{n})\right) v^{-1/\alpha -\frac{1}{\log \phi (1-f_{n})}+\varepsilon }\\&\le C\left( -\log \phi (1-f_{n})\right) \left( \frac{v_{\delta }^{*} }{\phi (1-f_{n})}\right) ^{-1/\alpha -\frac{1}{\log \phi (1-f_{n})}+\varepsilon }\\&<C\left( -\log \phi (1-f_{n})\right) \left( \phi (1-f_{n})\right) ^{1/\alpha +\frac{1}{\log \phi (1-f_{n})}-\varepsilon }. \end{aligned}$$

Multiplying it with the indicator and taking expectation under $\mathbb {E} ^{*}$, we obtain

$$\begin{aligned} \mathbb {E}^{*}\left[ 1_{\left\{ L_{n}>nb,V>\frac{v_{\delta }^{*}}{\phi (1-f_{n})}\right\} }L^{*^{2}}\right] \le C^{2}\left( -\log \phi (1-f_{n})\right) ^{2}\left( \phi (1-f_{n})\right) ^{2/\alpha +\frac{2}{\log \phi (1-f_{n})}-2\varepsilon }. \end{aligned}$$

Then, taking logarithms on both sides, dividing by $\log f_{n}$ and by Lemma A.4, we obtain

$$\begin{aligned} \liminf _{n\rightarrow \infty }\frac{\log \mathbb {E}^{*}\left[ 1_{\left\{ L_{n}>nb,V>\frac{v_{\delta }^{*}}{\phi (1-f_{n})}\right\} }L^{*^{2} }\right] }{\log f_{n}}\ge 2-2\alpha \varepsilon . \end{aligned}$$

Finally, (A.12) is yield by letting $\varepsilon \downarrow 0$.

Combining Step 1, Step 2 and Step 3, the desired result asserted in the theorem is obtained. $\square$

The following two proofs are motivated by Chan and Kroese (2010). Lemma A.5 below will be used in proving Lemma 6.1.

Lemma A.5

Let $R_{1},\ldots ,R_{n}$ be an i.i.d. sequence of standard exponential random variables. Suppose $R_{(k)}$ is the kth order statistic and $\lim _{n\rightarrow \infty }\frac{k}{n}=a<1$. Then, for every $\varepsilon >0$, there exists a constant $\beta >0$ such that the following inequality

$$\begin{aligned} \mathbb {P}\left( \left| R_{(k)}-\log \left( \frac{1}{1-a}\right) \right| \ge \varepsilon \right) \le \frac{\beta }{n}. \end{aligned}$$

holds for all sufficiently large n.

Proof

For i.i.d. standard exponential random variables $R_{i},i=1,\ldots ,n$, it follows from Rényi (1953) that

$$\begin{aligned} R_{(k)}\overset{d}{=}\sum _{j=1}^{k}\frac{R_{j}}{n-j+1}. \end{aligned}$$

Then,

$$\begin{aligned} \mathbb {E}[R_{(k)}]=\sum _{j=1}^{k}\frac{1}{n-j+1}=H_{n}-H_{n-k}\rightarrow \log \left( \frac{1}{1-a}\right) , \quad \text {as }n\rightarrow \infty , \end{aligned}$$

(A.13)

where $H_{n}$ denotes the nth harmonic number, i.e., $H_{n}=1+\frac{1}{2}+\cdots +\frac{1}{n}$ for $n\ge 1$. (A.13) is verified by noting the following asymptotic expansion; see, e.g., Berndt (1998),

$$\begin{aligned} H_{n}\sim \log (n)+\gamma +O\left( \frac{1}{n}\right) , \end{aligned}$$

and $\gamma$ is the Euler’s constant. Similarly,

$$\begin{aligned} \mathrm {Var}[R_{(k)}]=\sum _{j=1}^{k}\left( \frac{1}{n-j+1}\right) ^{2} =H_{n}^{(2)}-H_{n-k}^{(2)}\sim \frac{a}{1-a}\frac{1}{n}, \quad \text {as } n\rightarrow \infty , \end{aligned}$$

(A.14)

where $H_{n}^{(2)}$ is the nth harmonic number of order 2, i.e., $H_{n}^{(2)}=1+\frac{1}{2^{2}}+\cdots +\frac{1}{n^{2}}$ for $n\ge 1$. (A.14) is derived by applying the asymptotic expansion of $H_{n}^{(2)}$; see, e.g., Berndt (1998),

$$\begin{aligned} H_{n}^{(2)}\sim \frac{\pi ^{2}}{6}-\frac{1}{n}+O\left( \frac{1}{n^{2}}\right) . \end{aligned}$$

Then, by Chebyshev’s inequality, it follows that, for every $n>0$,

$$\begin{aligned} \mathbb {P}\left( \vert R_{(k)}-\mathbb {E}[R_{(k)}]\vert \ge \varepsilon \right) \le \frac{\mathrm {Var}[R_{(k)}]}{\varepsilon ^{2}}. \end{aligned}$$

Due to (A.13) and (A.14), there exists N, such that for all $n\ge N$,

$$\begin{aligned} \mathbb {P}\left( \left| R_{(k)}-\log \left( \frac{1}{1-a}\right) \right| \ge \varepsilon \right) \le \frac{\beta }{n}, \end{aligned}$$

where $\beta$ only depends on $\varepsilon$ and a. $\square$

Proof of Lemma 6.1

Recall that $O_{i}=\frac{R_{i}}{\phi (1-l_{i}f_{n})}$, for all $i=1,\ldots ,n$. Then the order statistic $O_{(k)}$ is almost surely lower bounded by

$$\begin{aligned} \frac{R_{(k)}}{\phi \left( 1-\max \limits _{j\le |\mathcal {W}|}l_{j} f_{n}\right) }. \end{aligned}$$

Since $k=\min \{l:\sum _{i=1}^{l}c_{(i)}>nb\}$, we have

$$\begin{aligned} \liminf _{n\rightarrow \infty }\frac{k}{n}\ge \frac{b}{\max \limits _{j\le |\mathcal {W}|}c_{j}}:=b^{\prime }. \end{aligned}$$

Fix $\varepsilon >0$. For all sufficiently large n, $\mathbb {E}\left[ S^{2}(\mathbf {R})\right]$ can be bounded as follows,

$$\begin{aligned} \mathbb {E}\left[ S^{2}(\mathbf {R})\right]&\le \mathbb {E}\left[ \mathbb {P}\left( V>\frac{R_{(\lfloor nb^{\prime }\rfloor )}}{\phi \left( 1-\max \limits _{j\le |\mathcal {W}|}l_{j}f_{n})\right) }\right) ^{2}\right] \\&\le \mathbb {E}\left[ \left( \mathbb {P}\left( V>\frac{R_{(\lfloor nb^{\prime }\rfloor )}}{\phi \left( 1-\max \limits _{j\le |\mathcal {W}|}l_{j} f_{n})\right) },R_{(\lfloor nb^{\prime }\rfloor )}\ge \log \left( \frac{1}{1-b^{\prime }}\right) -\varepsilon \right) \right. \right. \\&\quad +\, \left. \left. \mathbb {P}\left( V>\frac{R_{(\lfloor nb^{\prime }\rfloor )}}{\phi \left( 1-\max \limits _{j\le |\mathcal {W}|}l_{j}f_{n})\right) },R_{(\lfloor nb^{\prime }\rfloor )}<\log \left( \frac{1}{1-b^{\prime }}\right) -\varepsilon \right) \right) ^{2}\right] \\&\le \left( \mathbb {P}\left( V>\frac{\log \left( \frac{1}{1-b^{\prime } }\right) -\varepsilon }{\phi \left( 1-\max \limits _{j\le |\mathcal {W}|} l_{j}f_{n})\right) }\right) +\mathbb {P}\left( R_{(\lfloor nb^{\prime }\rfloor )}<\log \left( \frac{1}{1-b^{\prime }}\right) -\varepsilon \right) \right) ^{2}. \end{aligned}$$

Then,

$$\begin{aligned} \limsup _{n\rightarrow \infty }\frac{\mathbb {E}\left[ S^{2}(\mathbf {R})\right] }{f_{n}^{2}}&\le \left( \limsup _{n\rightarrow \infty }\frac{\mathbb {P} \left( V>\frac{\log \left( \frac{1}{1-b^{\prime }}\right) -\varepsilon }{\phi \left( 1-\max \limits _{j\le |\mathcal {W}|}l_{j}f_{n})\right) }\right) }{f_{n}}+\limsup _{n\rightarrow \infty }\frac{\mathbb {P}\left( R_{(\lfloor nb^{\prime }\rfloor )}<\log \left( \frac{1}{1-b^{\prime }}\right) -\varepsilon \right) }{f_{n}}\right) ^{2}\\&\le \left( \max \limits _{j\le |\mathcal {W}|}l_{j}\frac{\left( \log \left( \frac{1}{1-b^{\prime }}\right) -\varepsilon \right) ^{-1/\alpha }}{\Gamma (1-1/\alpha )}+M\right) ^{2}<\infty . \end{aligned}$$

The last step is due to the regular variation of V, Lemma A.5 and the condition that $\frac{1}{n}=O(f_{n})$. $\square$