2. Main results

We assume without loss of generality that 0 belongs to the interior of T and that $\sigma (0)=\sigma _{\ast }=\sup_{t\in T}\sigma (t)$ . As in [Reference Liu and Xu5, Reference Liu and Xu6], we introduce a discretization scheme on T. For any positive $S_{u}$ , let $G_{S_{u}}$ be a subset of $\mathbb{R}^{d}$ defined by

\begin{equation*}G_{S_{u}}=\left\{ \left( \frac{i_{1}}{S_{u}},\frac{i_{2}}{S_{u}},\ldots,\frac{i_{d}}{S_{u}}\right) \,:\,i_{1},\ldots,i_{d}\in \mathbb{Z}\right\} ,\end{equation*}

where $\mathbb{Z}$ is the set of integers, i.e. $G_{S_{u}}$ is a regular lattice on $\mathbb{R}^{d}$ . For each $t=(t_{1},\ldots,t_{d})\in G_{s_{u}}$ , define $T_{S_{u}}\left( t\right) = \{ (u_{1},\ldots,u_{d})\in T\,:\,u_{j}\in(t_{j}-1/(2S_{u});\ t_{j}+1/(2S_{u})]$ for $j=1,\ldots,d\}$ , i.e. the intersection of T and the $(1/S_{u})$ -cube centered at t. Furthermore, let $T_{S_{u}}=\left\{ t\in G_{S_{u}}\,:\, \text{mes}\left( T_{S_{u}}\left( t\right) \right)>0\right\}.$ Since T is compact, $T_{S_{u}}$ is a finite set. We enumerate the elements in $T_{S_{u}}=\left\{ t_{0},t_{1},\ldots,t_{M_{u}}\right\} $ , where $t_{0}=0$ and $M_{u}= \# T_{S_{u}}-1$ . Note that $M_{u}\sim \text{mes}(T)S_{u}^{d}$ as $u\rightarrow \infty $ .

The approximation of $\mathcal{I}\left( T\right) $ by a discrete sum is given by

\begin{equation*}\mathcal{I}_{M_{u}}\left( T\right) =\sum_{i=0}^{M_{u}}m_{i,u}\text{e}^{\mu(t_{i})+\sigma (t_{i})f(t_{i})} , \end{equation*}

where $m_{i,u} = \text{mes}\left( T_{S_{u}}\left( t_{i}\right) \right) $ and we let $w_{M_{u}}(u)=\mathbb{P}\left( \mathcal{I}_{M_{u}}\left( T\right) >u\right)$ .

The following proposition helps us to understand how to control the difference between $w_{M_{u}}(u)$ and w(u), and therefore how to control the bias for an estimator of w(u). It gathers two results from [Reference Liu and Xu5, Reference Liu and Xu6].

Proposition 1. Under Assumptions 1, for any $0<\varepsilon <1/2$ there exists a constant $\kappa _{0}$ such that, for any $\eta >0$ , $u>\text{e}$ , and lattice size $S_{u}=\kappa _{0}^{1/\beta } \left\vert \log \eta \right\vert ^{1/\beta }\eta ^{-1/\beta }\left( \log u\right)^{2(1+\varepsilon )/\beta }$ , we have

\begin{equation*}\frac{|w_{M_{u}}(u)-w(u)|}{w(u)}<\eta .\end{equation*}

Under Assumptions 2, for any $\varepsilon >0$ there exists a constant $\kappa _{0}$ such that, for any $\eta \in (0,1)$ , $u>2$ , and lattice size $S_{u}=\kappa _{0}\eta ^{-(1+\varepsilon )}\left( \log u\right)^{2+\varepsilon }$ , we have

\begin{equation*}\frac{|w_{M_{u}}(u)-w(u)|}{w(u)}<\eta .\end{equation*}

For the proof under Assumptions 1, see [Reference Liu and Xu5, Theorem 2.4]; under Assumptions 2, see [Reference Liu and Xu6, Theorem 10].

We now explain how we construct our conditional Monte Carlo type estimator of $w_{M_{u}}(u)$ . We mainly modify the Asmussen–Kroese estimator introduced in [Reference Kortschak and Hashorva3] in the following way. Let $\omega_{i,u}=m_{i,u}\text{e}^{\mu (t_{i})}$ and $X_{i,u}=\text{e}^{\sigma (t_{i})f(t_{i})}$ for $i=0,\ldots,M_{u}$ . We have $\mathcal{I}_{M_{u}}\left( T\right) =$ $\sum_{i=0}^{M_{u}}m_{i,u}\text{e}^{\mu(t_{i})}X_{i,u}=\sum_{i=0}^{M_{u}}\omega _{i,u}X_{i,u}$ . Let $\bar{\omega}_{u}=\sum_{i=0}^{M_{u}}\omega _{i,u}$ and define $v_{1,M_{u}}(u) = \mathbb{P}\big( \bar{\omega}_{u}\exp$ $\big(\sum_{i=0}^{M_{u}}\omega _{i,u}\log X_{i,u}/\bar{\omega}_{u}\big)>u\big)$ , $v_{2,M_{u}}(u) = \sum_{j=0}^{M_{u}}\Psi _{j}\left( u\right)$ , where

\begin{align*} \Psi_{j}\left( u\right) & = \mathbb{P}\Bigg( \sum_{i=0}^{M_{u}}\omega _{i,u}X_{i,u}>u,\ \bar{\omega}_{u}\exp \left( \sum_{i=0}^{M_{u}}\omega_{i,u}\log X_{i,u}/\bar{ \omega}_{u}\right) \leq u, \bigvee_{i=0}^{M_{u}}\omega_{i,u}X_{i,u}=\omega_{j,u}X_{j,u}\Bigg).\end{align*}

Since, by Jensen’s inequality, we have

\begin{equation*}\exp \left( \sum_{i=0}^{M_{u}}\omega _{i,u}\log X_{i,u}/\bar{\omega}_{u}\right) \leq \frac{1}{\bar{\omega}_{u}}\sum_{i=0}^{M_{u}}\omega_{i,u}X_{i,u},\end{equation*}

we deduce that

\begin{equation*}\mathbb{P}( \mathcal{I}_{M_{u}}( T) >u) =v_{1,M_{u}}(u)+\mathbb{P}\Bigg(\sum_{i=0}^{M_{u}}\omega _{i,u}X_{i,u}>u,\bar{\omega}_{u}\exp \left(\sum_{i=0}^{M_{u}}\omega _{i,u}\log X_{i,u}/\bar{\omega}_{u}\right) \leq u\Bigg) ,\end{equation*}

and we finally get the following decomposition of $w_{M_{u}}(u)$ :

\begin{equation*}w_{M_{u}}(u)=\mathbb{P}\left( \mathcal{I}_{M_{u}}\left( T\right) >u\right)=v_{1,M_{u}}(u)+v_{2,M_{u}}(u). \end{equation*}

Such a decomposition is interesting for several reasons. First, note that $v_{1,M_{u}}(u)$ may be numerically computed since the random variable $\sum_{i=0}^{M_{u}}\omega _{i,u}\log X_{i,u}$ has a Gaussian distribution. Second, we have noted in experiments that the variance of the estimator from [Reference Kortschak and Hashorva3] may be large when the Gaussian random variables are highly correlated and when $\bar{\omega}_{u}\exp \big( \sum_{i=0}^{M_{u}}\omega_{i,u}\log X_{i,u}/\bar{\omega}_{u}\big) >u$ . Consequently, it is more efficient to replace the probability of such an event by its value.

For $j=0,1,\ldots,M_{u}$ , let $N_{j,u}$ be a vector of $M_{u}+1$ independent standard Gaussian random variables, and $A_{j,u}$ be a lower non-singular triangular matrix of size $M_{u}+1$ such that

\begin{equation*}\Bigg(\,\begin{array}{c} f(t_{j}) \\[5pt]f(t_{(-j)}) \end{array}\Bigg) =A_{j,u}N_{j,u},\end{equation*}

with $f(t_{(-j)})=\left( f(t_{i})\right) _{i\neq j}$ . We introduce the following unbiased estimator of $\Psi _{j}\left( u\right) $ :

\begin{align*}Z_{j}\left( u\right) & = \mathbb{P}\Bigg( \sum_{i=0}^{M_{u}}\omega _{i,u}X_{i,u}>u,\ \bar{\omega}_{u}\exp \left( \sum_{i=0}^{M_{u}}\omega _{i,u}\log X_{i,u}/\bar{\omega}_{u}\right) \leq u, \\ & \qquad \quad \bigvee_{i=0}^{M_{u}}\omega _{i,u}X_{i,u}=\omega_{j,u}X_{j,u} \, \Big \vert \, N_{j,u}^{(-1)}\Bigg) ,\end{align*}

where $N_{j,u}^{(-1)}=\left( N_{j,u,2},\ldots,N_{j,u,M_{u}+1}\right) ^{\prime }$ . We have

\begin{align*}&\left\{ \sum_{i=0}^{M_{u}}\omega _{i,u}X_{i,u}>u,\bar{\omega}_{u}\exp\left( \sum_{i=0}^{M_{u}}\omega _{i,u}\log X_{i,u}/\bar{\omega}_{u}\right)\leq u,\bigvee_{i=0}^{M_{u}}\omega _{i,u}X_{i,u}=\omega_{j,u}X_{j,u}\right\} \\[5pt]&\quad =\left\{ \sum_{i=0}^{M_{u}}\omega _{i,u}X_{i,u}>u\right\} \bigcap \left\{\bar{\omega}_{u}\exp \left( \sum_{i=0}^{M_{u}}\omega _{i,u}\log X_{i,u}/\bar{\omega}_{u}\right) \leq u\right\} \\[5pt] & \qquad \bigcap_{i\neq j}\left\{ \omega_{i,u}X_{i,u}\leq \omega _{j,u}X_{j,u}\right\} .\end{align*}

It is noteworthy that all the events in the previous intersection may be written as an event that $N_{j,u,1}$ is smaller or greater than a threshold that only depends on $N_{j,u}^{(-1)}$ , $A_{j,u}$ , $\left( \omega_{i,u}\right) _{i}$ , and $\left( \sigma (t_{i})\right) _{i}$ . The threshold for the event $\big\{\sum_{i=0}^{M_{u}}\omega _{i,u}X_{i,u}>u\big\}$ has to be computed numerically, while the others have analytical expressions. The computational cost for $Z_{j}\left( u\right) $ is mainly due to the cost of computing the matrix $A_{j,u}$ through a Cholesky decomposition: the serial version of the Cholesky algorithm is of cubic complexity, i.e. the cost of decomposition of the $(M_{u}+1)\times (M_{u}+1)$ matrix is of order $O\left(M_{u}^{3}\right) $ .

Let $\mathcal{J}_{u}$ be a random variable independent of $\left(N_{j,u}\right) _{j=0,\ldots,M_{u}}$ , such that

(2) \begin{equation}\mathbb{P}\left( \mathcal{J}_{u}=j\right) =\frac{\mathbb{P}\left( X_{j,u}>u\right) }{\sum_{i=0}^{M_{u}}\mathbb{P}\left( X_{i,u}>u\right) }. \end{equation}

Our unbiased estimator of $w_{M_{u}}(u)$ is finally given by

\begin{equation*}Z_{M_{u}}\left( u\right) =v_{1,M_{u}}(u)+\left( \sum_{i=0}^{M_{u}}\mathbb{P}\left(X_{i,u}>u\right) \right) \sum_{j=0}^{M_{u}}\textbf{1}_{\{\mathcal{J}_{u}=j\}}\frac{Z_{j}\left( u\right) }{\mathbb{P}\left( X_{j,u}>u\right) }. \end{equation*}

Compared to [Reference Kortschak and Hashorva3], we split $w_{M_{u}}(u)$ into two parts, of which one part ( $v_{1,M_{u}}(u)$ ) can be computed without simulation, which will increase the accuracy of our estimator.

2.1. Assumptions 1

Given that the tail probability w(u) converges to zero, it is usually meaningful to consider the relative error of a Monte Carlo estimator Z(u) with respect to w(u). A well-accepted efficiency concept is so-called weak efficiency, also known as logarithmic efficiency [Reference Asmussen and Glynn2, Chapter VI].

An unbiased estimator Z(u) of w(u) is said to be logarithmically efficient if

\begin{equation*}\lim_{u\rightarrow \infty }\frac{\log \mathbb{E}\left\{(Z(u)-w(u))^{2}\right\} }{\log w^{2}(u)}=1.\end{equation*}

The following proposition establishes that the unbiased estimator $Z_{M_{u}}(u)$ of $w_{M_{u}}(u)$ is an asymptotically logarithmic efficient estimator, and that the relative bias to estimate w(u) is well controlled with an appropriate choice of $S_{u}$ .

Proposition 2. Assume that Assumptions 1 are satisfied. Let $\left( \eta _{u}\right) $ be a sequence of positive constants such that $\lim_{u\rightarrow \infty }\eta _{u}=0$ and $\lim_{u\rightarrow \infty }\log \eta _{u}/\log (w(u))=0$ . If $S_{u}$ is chosen such that $S_{u}=O\big( \left\vert \log \eta _{u}\right\vert ^{1/\beta }\eta_{u}^{-1/\beta }\left( \log u\right) ^{2(1+\varepsilon )/\beta }\big)$ for some $0<\varepsilon <1/2$ , then

(3) \begin{equation}\lim_{u\rightarrow \infty }\frac{\log \mathbb{E}\left\{(Z_{M_{u}}(u)-w(u))^{2}\right\} }{\log w^{2}(u)}=1.\end{equation}

It is shown in [Reference Liu and Xu5] that the importance sampling estimator introduced in this paper also satisfies $\left( 3\right) $ .

2.2. Assumptions 2

A slightly stronger efficiency concept is polynomial efficiency. An unbiased estimator Z(u) of w(u) is said to be polynomially efficient of order q if

\begin{equation*}\lim_{u\rightarrow \infty }\frac{\mathbb{E}\left\{ (Z(u)-w(u))^{2}\right\} }{|\log \left( w(u)\right) |^{q}w^{2}(u)}<\infty .\end{equation*}

When $q=0$ , Z(u) is also said to be strongly efficient.

Proposition 3. Assume that Assumptions 2 are satisfied. Let $\left( \eta _{u}\right) $ be a sequence of positive constants such that $\lim_{u\rightarrow \infty }\eta _{u}=0$ and $\lim \inf_{u\rightarrow \infty}\eta _{u}\left( \log u\right) ^{\nu }>0$ for some positive constant $\nu $ . If $S_{u}$ is chosen such that $S_{u}=O\big( \eta _{u}^{-(1+\varepsilon )}\left( \log u\right) ^{2+\varepsilon }\big)$ for some $\varepsilon >0$ , then

\begin{equation*}\lim_{u\rightarrow \infty }\frac{\mathbb{E}\left\{(Z_{M_{u}}(u)-w(u))^{2}\right\} }{|\log \left( w(u)\right)|^{[(2+\varepsilon )+(1+\varepsilon )\nu ]d}w^{2}(u)}<\infty .\end{equation*}

It is shown in [Reference Liu and Xu6] that the importance sampling estimator introduced in this paper is strongly efficient, which is slightly better than our estimator (from a theoretical point of view at least).

We should point out that, contrary to [Reference Kortschak and Hashorva3], our estimator has a number of terms ( $M_{u}$ ) in its construction that depend on the threshold u (due to the discrete approximation of the integral). Since $M_{u}$ tends to infinity as u tends to infinity, this adds several technical issues and leads to a slower rate of convergence than in [Reference Kortschak and Hashorva3].

3. Simulation

In this section we present numerical examples to study the performance of our estimator (denoted here by $Z_{\text{NR}}$ ) compared to the importance sampling estimator of [Reference Liu and Xu5] (denoted by $Z_{\text{LX}}$ ). We only consider the importance sampling estimator of [Reference Liu and Xu5] and not [Reference Liu and Xu6] because we do not necessarily want to assume that f is a differentiable Gaussian random field. To understand the advantage of splitting the probability $w_{M_{u}}(u)$ into two parts in the construction of our estimator, we also decided to include the estimator of [Reference Kortschak and Hashorva3], denoted by $Z_{\text{AK}}$ . All the results are based on $N=10^{4}$ independent simulations for $Z_{\text{NR}}$ and $Z_{\text{LX}}$ . The CPU time to generate $10^{4}$ samples is less than one second for the LX estimator, but more than ten seconds for both our estimator and the AK estimator.

As in [Reference Liu and Xu5], we assume in this section that f is a zero-mean homogeneous Gaussian random field with covariance function $C(s,t)=\text{e}^{-|t-s|^{\alpha }}$ . When $\alpha =2$ , f is infinitely differentiable, whereas when $\alpha =1$ , f is non-differentiable. We take $T=[-1/2,1/2]$ and discretize it with discretization size $S_{u}=100$ following the procedure given in the previous section. The detailed simulation is described in the following steps.

1. i. Generate a random variable $\mathcal{J}_{u}$ according to the distribution $\left(2\right)$ . For $j=\mathcal{J}_{u}$ ,

2. ii. Generate a vector $N_{j,u}$ of $M_{u}+1$ independent standard Gaussian random variables; compute the matrix $A_{j,u}$ and the vector $A_{j,u}N_{j,u}$ .

3. iii. Compute the probability

   \begin{align*} & \mathbb{P}\Bigg( \sum_{i=0}^{M_{u}}\omega _{i,u}X_{i,u}>u,\bar{\omega}_{u}\exp\left( \sum_{i=0}^{M_{u}}\omega _{i,u}\log X_{i,u}/\bar{\omega}_{u}\right)\leq u, \nonumber\\& \qquad\bigvee_{i=0}^{M_{u}}\omega _{i,u}X_{i,u}=w_{j,u}X_{j,u} \, \Big\vert \,N_{j,u}^{(-1)}\Bigg). \end{align*}

4. iv. Compute

   \begin{equation*}Z_{M_{u}}\left( u\right) =v_{1,M_{u}}(u)+\left( \sum_{i=0}^{M_{u}}P\left(X_{i,u}>u\right) \right) \frac{Z_{j}\left( u\right) }{\mathbb{P}\left(X_{j,u}>u\right) }.\end{equation*}

The computational complexity for generating an estimate of $w_{M_{u}}(u)$ with our approach is the number of simulations multiplied by the cost for generating one copy of $Z_{M_{u}}\left( u\right) $ , which is mainly the cost of computing the matrix $A_{j,u}$ , i.e. of order $O\left( S_{u}^{3d}\right) $ . The overall computational cost is also a polynomial in $\eta ^{-1}$ and $\log \left( u\right)$ .

First, we consider the case where $\mu (t)=0$ and $\sigma (t)=1$ as in [Reference Liu and Xu5]. To validate the simulation results, [Reference Liu and Xu5] computed crude Monte Carlo estimators:

The estimated tail probabilities of w(u) are shown in Table 1 ( $u=\text{e}^{b}$ ). To compare the performance of the estimators, we provide their estimated coefficient of variation (CV, the ratio of the standard deviation over the mean) as well as the ratio of their variance per unit computer time over the variance per unit computer time of $Z_{\text{LX}}$ (RVCT). The variance per unit computer time for a given estimator Z equals $\tau (Z)\text{Var}(Z)$ , where $\tau(Z)$ is the expected time to generate Z.

Table 1.

Estimates of w(u) on $T=[-1/2,1/2]$ , $\mu (t)=0$ , and $\sigma(t)=1 $ for $u=\text{e}^{b}$ . The results for $Z_{\text{LX}}$ are from [Reference Liu and Xu5].

Comparing the simulation results for $\alpha =1,2$ and $b=3,5,7$ , we can see that the conditional Monte Carlo estimators have better performance for both measures (CV and RVCT) than the estimator of [Reference Liu and Xu5]. Our estimator is also the better of the two conditional Monte Carlo estimators. which proves that the split of the probability into two parts is very useful.

Next, we consider the case with $\mu (t)=2|t|$ and $\sigma (t)=1-t^{2}$ . To validate the simulation results, [Reference Liu and Xu5] computed crude Monte Carlo estimators:

The estimated tail probabilities of w(u) with their estimated coefficient of variation (CV) and their ratio of the variances per unit computer times (RVCT) are shown in Table 2 ( $u=\text{e}^{b}$ ).

As in [Reference Liu and Xu5], the simulation results show that the coefficients of variation increase as the tail probabilities become smaller. The coefficients of variation and the ratios of the variance per unit computer times of the conditional Monte Carlo estimators are also much smaller in this case than the LX estimator. Moreover, the continuity of the process affects the empirical performance of the three estimators: they admit smaller coefficients of variation when the process is more continuous (corresponding to a larger value of $\alpha $ ).

Table 2.

Estimates of w(u) on $T=[-1/2,1/2]$ , $\mu (t)=2|t|$ , and $\sigma(t)=1-t^{2}$ for $u=\text{e}^{b}$ . The results for $Z_{\text{LX}}$ are from [Reference Liu and Xu5].

4. Proofs of the main results

4.1 Proofs under Assumptions 1

Lemma 1. As $u\rightarrow \infty $ , for some positive constants $\tilde{c}_{1}$ and $\tilde{c}_{2}$ we have

\begin{align*} w_{M_{u}}(u) & \geq \exp \left( -\frac{(\log u)^{2}+\tilde{c}_{1}\log u\log S_{u}}{2\sigma _{\ast }^{2}}\right) , \\ w_{M_{u}}(u) & \leq \exp \left( -\frac{(\log u)^{2}-\tilde{c}_{2}(\log u)}{ 2\sigma _{\ast }^{2}}\right). \end{align*}

Proof. For the lower bound, we have

\begin{align*}w_{M_{u}}(u) &\geq P\left( w_{0,u}X_{0,u}>u\right) \\&= P\left( S_{u}^{-d}\text{e}^{\mu (0)+\sigma _{\ast }f(0)}>u\right) \\&= P\left( \sigma _{\ast }f(0)>\log u+d\log S_{u}-\mu (0)\right) \\&\geq \exp \left( -\frac{(\log u)^{2}+\tilde{c}_{1}\log u\log S_{u}}{2\sigma _{\ast }^{2}}\right)\end{align*}

for some positive constant $\tilde{c}_{1}$ .

For the upper bound, we have

\begin{align*}w_{M_{u}}(u) &\leq P\left( \left( M_{u}+1\right)\bigvee_{i=0}^{M_{u}}\omega _{i,u}X_{i,u}>u\right) \\&\leq P\left( \max_{t\in T}\text{e}^{\mu (t)+\sigma (t)f(t)}>\frac{u}{\text{mes}\left(T\right) }\right) \\&\leq P\left( \max_{t\in T}\sigma (t)f(t)>\log u-\log \left( \text{mes}\left(T\right) \right) -\max_{t\in T}\mu (t)\right) .\end{align*}

By the Borel–TIS lemma and for sufficiently large u, we deduce that

\begin{align*}w_{M_{u}}(u) &\leq \exp \left( -\frac{\left( \log u-\log \left( \text{mes}\left(T\right) \right) -\max_{t\in T}\mu (t)\right) ^{2}}{2\sigma _{\ast }^{2}}\right) \\&\leq \exp \left( -\frac{(\log u)^{2}-\tilde{c}_{2}(\log u)}{2\sigma_{\ast }^{2}}\right)\end{align*}

for some positive constant $\tilde{c}_{2}$ .

Lemma 2. Let

\begin{equation*}\sigma _{A}^{2}=\frac{\int_{T^{2}}\text{e}^{\mu (t)+\mu (s)}\sigma (t)\sigma(s)C\left( t,s\right) \, \text{d} t \, \text{d} s}{\int_{T^{2}}\text{e}^{\mu (t)+\mu (s)} \, \text{d} t \, \text{d} s} ,\end{equation*}

and $\epsilon >0$ be such that $\sigma _{A}^{2}\left( 1+\epsilon \right)<\sigma _{\ast }^{2}$ . We have, as $u\rightarrow \infty $ ,

\begin{equation*}v_{1,M_{u}}(u)\leq \exp \left( -\frac{\left( \log u-\log \left(\int_{T}\text{e}^{\mu (t)} \, \text{d} t\right) \right) ^{2}}{2\sigma _{A}^{2}\left( 1+\epsilon\right) }\right)\end{equation*}

and, for some positive constant $\tilde{c}_{1}$ ,

\begin{equation*}v_{2,M_{u}}(u)\geq \frac{1}{2}\exp \left( -\frac{(\log u)^{2}+\tilde{c}_{1}\log u\log S_{u}}{2\sigma _{\ast }^{2}}\right) .\end{equation*}

Therefore, we have

\begin{equation*}\lim_{u\rightarrow \infty }\frac{v_{1,M_{u}}(u)}{v_{2,M_{u}}(u)}=0.\end{equation*}

Proof. We have

\begin{equation*}v_{1,M_{u}}(u)=P\left( \bar{\omega}_{u}\exp \left( \sum_{i=0}^{M_{u}}\omega_{i,u}\log X_{i,u}/\bar{\omega}_{u}\right) >u\right) .\end{equation*}

By Riemann integrability, we know that $\lim_{u\rightarrow \infty }\bar{\omega}_{u}=\int_{T}\text{e}^{\mu (t)} \, \text{d} t$ . Moreover, the random variable $\frac{1}{\bar{\omega}_{u}}\sum_{i=0}^{M_{u}}\omega _{i,u}\log X_{i,u}=\frac{1}{\bar{\omega}_{u}}\sum_{i=0}^{M_{u}}\omega _{i,u}\sigma (t_{i})f(t_{i})$ is a centered Gaussian random variable with variance

\begin{equation*}\frac{1}{\bar{\omega}_{u}^{2}}\sum_{i=0}^{M_{u}}\sum_{j=0}^{M_{u}}\text{mes}\left(T_{S_{u}}\left( t_{i}\right) \right) \text{e}^{\mu (t_{i})}\sigma (t_{i})\text{mes}\left(T_{S_{u}}\left( t_{j}\right) \right) \text{e}^{\mu (t_{j})}\sigma (t_{j})C\left(t_{i},t_{j}\right)\end{equation*}

converging to

\begin{equation*}\sigma _{A}^{2}=\frac{\int_{T^{2}}\text{e}^{\mu (t)+\mu (s)}\sigma (t)\sigma(s)C\left( t,s\right) \, \text{d} t \, \text{d} s}{\int_{T^{2}}\text{e}^{\mu (t)+\mu (s)} \, \text{d} t \, \text{d} s}<\sigma_{\ast }^{2}\text{.}\end{equation*}

Therefore, for large u,

\begin{equation*}v_{1,M_{u}}(u)\leq \exp \left( -\frac{\left( \log u-\log \left(\int_{T}\text{e}^{\mu (t)} \, \text{d} t\right) \right) ^{2}}{2\sigma _{A}^{2}\left( 1+\epsilon\right) }\right) .\end{equation*}

Now,

\begin{align*}v_{2,M_{u}}(u) &= P\left( \sum_{i=0}^{M_{u}}\omega _{i,u}X_{i,u}>u,\bar{\omega}_{u}\exp \left( \sum_{i=0}^{M_{u}}\omega _{i,u}\log X_{i,u}/\bar{\omega}_{u}\right) \leq u\right) \\&= P\left( \sum_{i=0}^{M_{u}}\omega _{i,u}X_{i,u}>u\right) -v_{1,M_{u}}(u) \\&\geq P\left( w_{0,u}X_{0,u}>u\right) -v_{1,M_{u}}(u).\end{align*}

Moreover, in the same way as in Lemma 1,

\begin{equation*}P\left( w_{0,u}X_{0,u}>u\right) \geq \exp \left( -\frac{(\log u)^{2}+\tilde{c}_{1}\log u\log S_{u}}{2\sigma _{\ast }^{2}}\right)\end{equation*}

for some positive constant $\tilde{c}_{1}$ . It follows that, for large u,

\begin{align*}P( w_{0,u} & X_{0,u}>u) -v_{1,M_{u}}(u) \\& \geq \exp \left( -\frac{(\log u)^{2}+\tilde{c}_{1}\log u\log S_{u}}{2\sigma _{\ast }^{2}}\right) -\exp \left( -\frac{\left( \log u-\log \left(\int_{T}\text{e}^{\mu (t)} \, \text{d} t\right) \right) ^{2}}{2\sigma _{A}^{2}\left(1+\epsilon \right) }\right) \\&\geq \frac{1}{2}\exp \left( -\frac{(\log u)^{2}+\tilde{c}_{1}\log u\log S_{u}}{2\sigma _{\ast }^{2}}\right)\end{align*}

and

\begin{equation*}\qquad\qquad\ \ \qquad\qquad\qquad\qquad\qquad\lim_{u\rightarrow \infty }\frac{v_{1,M_{u}}(u)}{v_{2,M_{u}}(u)}=0. \qquad\qquad\qquad\qquad\qquad\qquad\qquad\Box\end{equation*}

Lemma 3.

\begin{equation*}\mathbb{E}\left\{ \left( Z_{M_{u}}(u)-v_{1,M_{u}}(u)\right) ^{2}\right\}=\sum_{j=0}^{M_{u}}\frac{\mathbb{E}\left\{ Z_{j}\left( u\right) ^{2}\right\}}{P\left( \mathcal{J}_{u}=j\right) }.\end{equation*}

Proof. As in [Reference Liu and Xu5, Lemma A.3], the proof follows by straightforward calculations.

Proof of Proposition 2. It is easily seen that

\begin{equation*}\mathbb{E}\left\{ \left( Z_{M_{u}}(u)\right) ^{2}\right\} =\mathbb{E}\left\{\left( Z_{M_{u}}(u)-v_{1,M_{u}}(u)\right) ^{2}\right\} +v_{1,M_{u}}(u)\left(v_{1,M_{u}}(u)+2v_{2,M_{u}}(u)\right) .\end{equation*}

By Lemma 2, we have, for large u,

\begin{equation*}v_{1,M_{u}}(u)\leq \exp \left( -\frac{\left( \log u-\log \left(\int_{T}\text{e}^{\mu (t)} \, \text{d} t\right) \right) ^{2}}{2\sigma _{A}^{2}\left( 1+\epsilon\right) }\right) ,\end{equation*}

and, by Lemma 1,

\begin{equation*}v_{2,M_{u}}(u)\leq w_{M_{u}}\left( u\right) \leq \exp \left( -\frac{(\log u)^{2}-\tilde{c}_{2}(\log u))}{2\sigma _{\ast }^{2}}\right) .\end{equation*}

Moreover, we have

\begin{align*}\mathbb{E}\{ (Z_{M_{u}}(u)-w(u))^{2}\} &= \mathbb{E}\{(Z_{M_{u}}(u))^{2}\} -w_{M_{u}}^{2}(u)+(w_{M_{u}}(u)-w(u))^{2} \\&= \mathbb{E}\{ \left( Z_{M_{u}}(u)-v_{1,M_{u}}(u)\right) ^{2}\}+v_{1,M_{u}}(u)\left( v_{1,M_{u}}(u)+2v_{2,M_{u}}(u)\right) \\& \quad -w_{M_{u}}^{2}(u)+(w_{M_{u}}(u)-w(u))^{2}\end{align*}

and, for large u,

(4) \begin{equation}\mathbb{E}\left\{ (Z_{M_{u}}(u)-w(u))^{2}\right\} \leq \mathbb{E}\left\{\left( Z_{M_{u}}(u)-v_{1,M_{u}}(u)\right) ^{2}\right\}+3v_{1,M_{u}}(u)w(u)+\eta _{u}^{2}w(u)^{2} \end{equation}

by Proposition 1.

By Lemma 3, we deduce that

\begin{equation*}\mathbb{E}\left\{ \left( Z_{M_{u}}(u)-v_{1,M_{u}}(u)\right) ^{2}\right\}=\sum_{j=0}^{M_{u}}\frac{\mathbb{E}\left\{ Z_{j}\left( u\right) ^{2}\right\}}{P\left( \mathcal{J}_{u}=j\right) }=\sum_{j=0}^{M_{u}}P\left( \mathcal{J}_{u}=j\right) \frac{\mathbb{E}\left\{ Z_{j}\left( u\right) ^{2}\right\} }{P\left( \mathcal{J}_{u}=j\right) ^{2}}.\end{equation*}

It is important to note that $Z_{j}\left( u\right) $ is bounded in the following way:

\begin{equation*}Z_{j}\left( u\right) \leq P\left( X_{j,u}>\frac{u}{\text{e}^{\mu (t_{j})}\text{mes}\left(T\right) }\right) .\end{equation*}

Since

\begin{equation*}P\left( \mathcal{J}_{u}=j\right) =\frac{P\left( X_{j,u}>u\right) }{\sum_{i=0}^{M_{u}}P\left( X_{i,u}>u\right) },\end{equation*}

it follows that, for large u,

\begin{align*}\frac{\mathbb{E}\left\{ Z_{j}\left( u\right) ^{2}\right\} }{P\left( \mathcal{J}_{u}=j\right) ^{2}} &\leq \frac{P\left( X_{j,u}>\frac{u}{\text{e}^{\mu(t_{j})}\text{mes}\left( T\right) }\right) ^{2}}{P\left( X_{j,u}>u\right) ^{2}}\left( \sum_{i=0}^{M_{u}}P\left( X_{i,u}>u\right) \right) ^{2} \\&\leq \frac{(\log u)^{2}}{\sigma ^{2}(t_{j})}\exp \left( -\frac{(\log u-\mu(t_{j})-\log \left( \text{mes}\left( T\right) \right) )^{2}}{\sigma ^{2}(t_{j})}+\frac{(\log u)^{2}}{\sigma ^{2}(t_{j})}\right) \\ & \quad \times \left(\sum_{i=0}^{M_{u}}P\left( X_{i,u}>u\right) \right) ^{2} \\&\leq C(\log u)^{2}\exp \left( 2\frac{\mu (t_{j})+\log \left( \text{mes}\left(T\right) \right) }{\sigma ^{2}(t_{j})}\log u\right) \left(\sum_{i=0}^{M_{u}}P\left( X_{i,u}>u\right) \right) ^{2} \\& = C(\log u)^{2}u^{2(\mu (t_{j})+\log \left( \text{mes}\left( T\right) \right)/\sigma ^{2}(t_{j})}\left( \sum_{i=0}^{M_{u}}P\left( X_{i,u}>u\right)\right) ^{2}\end{align*}

for some positive constant C. Then we have

\begin{align*}\sum_{j=0}^{M_{u}}\frac{\mathbb{E}\left\{ Z_{j}\left( u\right) ^{2}\right\}}{P\left( \mathcal{J}_{u}=j\right) } & = \sum_{j=0}^{M_{u}}P\left( \mathcal{J}_{u}=j\right) \frac{\mathbb{E}\left\{ Z_{j}\left( u\right) ^{2}\right\} }{P\left( \mathcal{J}_{u}=j\right) ^{2}} \\&\leq C(\log u)^{2}\sum_{j=0}^{M_{u}}P\left( \mathcal{J}_{u}=j\right)u^{2\left( \max_{T}\mu (t)+\log \left( \text{mes}\left( T\right) \right) \right)/\min_{T}\sigma ^{2}(t)} \\ & \quad \times \left( \sum_{i=0}^{M_{u}}P\left( X_{i,u}>u\right)\right) ^{2} \\&\leq C(\log u)^{2}u^{2\left( \max_{T}\mu (t)+\log \left( \text{mes}\left(T\right) \right) \right) /\min_{T}\sigma ^{2}(t)}\left(\sum_{i=0}^{M_{u}}P\left( X_{i,u}>u\right) \right) ^{2}.\end{align*}

Moreover, we have

\begin{equation*}\sum_{i=0}^{M_{u}}P\left( X_{i,u}>u\right) \leq (1+M_{u})P\left(\bigvee_{i=0}^{M_{u}}X_{i,u}>u\right)\leq (1+M_{u})P\left( \text{e}^{\max_{t\in T}\sigma (t)f(t)}>u\right) ,\end{equation*}

where $P\left( \exp\left\{\max_{t\in T}\sigma (t)f(t)\right\}>u\right) =P\left( \max_{t\in T}\sigma (t)f(t)>\log u\right)$ . By the Borel–TIS lemma, we also have

\begin{equation*}P\left( \max_{t\in T}\sigma (t)f(t)>\log u\right) \leq \exp \left( -\frac{(\log u)^{2}}{2\sigma _{\ast }^{2}}\right) .\end{equation*}

It follows that

\begin{multline*}\mathbb{E}\left\{ \left( Z_{M_{u}}(u)-v_{1,M_{u}}(u)\right) ^{2}\right\}\leq \\C(\log u)^{2}u^{2\left( \max_{T}\mu (t)+\log \left( \text{mes}\left( T\right)\right) \right) /\min_{T}\sigma ^{2}(t)}(1+M_{u})^{2}\exp \left( -\frac{(\log u)^{2}}{\sigma _{\ast }^{2}}\right) .\end{multline*}

We know from [Reference Liu and Xu5, Lemma 4.3] that there exist constants $c_{0}$ , $c_{1}$ , $c_{2}$ such that

\begin{equation*}-\frac{\left( \log u\right) ^{2}+c_{1}\log u\log \left( \log u\right) +c_{0}}{2\sigma _{\ast }}\leq \log w(u)\leq -\frac{\left( \log u\right) ^{2}}{2\sigma _{\ast }}+c_{2}\log u .\end{equation*}

We therefore deduce from (4) that

\begin{equation*}\qquad\qquad\qquad\qquad\qquad\qquad\qquad\lim_{u\rightarrow \infty }\frac{\log \mathbb{E}\left\{Z_{M_{u}}(u)^{2}\right\} }{\log w^{2}(u)}=1. \qquad\qquad\qquad\qquad\qquad\qquad\Box\end{equation*}

4.2. Proofs under Assumptions 2

Lemma 4. Let

\begin{equation*}\sigma _{A}^{2}=\sigma ^{2}\frac{\int_{T^{2}}C\left( t-s\right) \, \text{d} t \, \text{d} s}{\text{mes}(T)^{2}}<\sigma ^{2} ,\end{equation*}

and $\epsilon >0$ be such that $\sigma _{A}^{2}\left( 1+\epsilon \right)<\sigma _{\ast }^{2}$ . We have, as $u\rightarrow \infty $ ,

\begin{equation*}v_{1,M_{u}}(u)\leq \exp \left( -\frac{\left( \log u-\log \left(\int_{T}\text{e}^{\mu (t)} \, \text{d} t\right) \right) ^{2}}{2\sigma _{A}^{2}\left( 1+\epsilon\right) }\right)\end{equation*}

and, for some positive constant $\tilde{c}_{1}$ ,

\begin{equation*}v_{2,M_{u}}(u)\geq \frac{1}{2}\exp \left( -\frac{(\log u)^{2}+\tilde{c}_{1}\log u\log S_{u}}{2\sigma _{\ast }^{2}}\right) .\end{equation*}

Therefore, we have

\begin{equation*}\lim_{u\rightarrow \infty }\frac{v_{1,M_{u}}(u)}{v_{2,M_{u}}(u)}=0.\end{equation*}

Proof. The proof proceeds in the same way as for Lemma 2.

Lemma 5. For some positive constant C, we have, for large u, $\mathbb{E}\{ Z_{0}\left( u\right) ^{2}\} \leq Cw^{2}\left(u\right)$ .

Proof. Let $c_{u}$ be defined as

\begin{equation*}c_{u}=\left( 1+\varepsilon \right) \sqrt{4\frac{\log \left( 1+M_{u}\right) }{\sigma ^{2}}}\end{equation*}

with $\varepsilon >0$ .

Note that there exist standard Gaussian random variables $\tilde{N}_{j}$ for $j\neq 0$ (independent of f(0), i.e. $N_{0,u,1}$ ) such that $f(t_{j})=\rho _{j}\,f(0)+\sqrt{1-\rho _{j}^{2}}\tilde{N}_{j}$ , $j\neq 0$ , where $\rho _{j}=C(t_{j})$ .

We have

\begin{equation*}\mathbb{E}\big\{ Z_{0}\left( u\right) ^{2}\big\} =\mathbb{E}\left\{Z_{0}\left( u\right) ^{2}\textbf{1}_{\left\{\bigvee_{i=1}^{M_{u}}\left\vert \tilde{N}_{i}\right\vert >c_{u}\sqrt{\log (u)}\right\} }\right\} +\mathbb{E}\left\{ Z_{0}\left( u\right) ^{2}\textbf{1}_{\left\{ \bigvee_{i=1}^{M_{u}}\left\vert \tilde{N}_{i}\right\vert \leq c_{u}\sqrt{\log (u)}\right\} }\right\} .\end{equation*}

Let us consider the first component of the previous sum. We have

\begin{align*}\mathbb{E}\left\{ Z_{0}\left( u\right) ^{2}\textbf{1}_{\left\{\bigvee_{i=1}^{M_{u}}\left\vert \tilde{N}_{i}\right\vert >c_{u}\sqrt{\log (u)}\right\} }\right\} &\leq \sum_{i=1}^{M_{u}}\mathbb{E}\left\{Z_{0}\left( u\right) ^{2}\textbf{1}_{\left\{ \left\vert \tilde{N}_{i}\right\vert >c_{u}\sqrt{\log (u)}\right\} }\right\} \\&\leq M_{u}P^{2}\bigg( \text{e}^{\sigma f(t_{0})}>\frac{u}{\text{mes}\left( T\right) }\bigg) P\big( |\tilde{N}_{i}|>c_{u}\sqrt{\log (u)}\big) .\end{align*}

Moreover, for large u, we have

\begin{equation*}M_{u}P\left( |\tilde{N}_{i}|>c_{u}\sqrt{\log (u)}\right) \sim \frac{\text{mes}(T)S_{u}^{d}}{c_{u}\sqrt{\log (u)}\,}\exp \left( -\frac{c_{u}^{2}}{2}\log (u)\right)\end{equation*}

and

\begin{align*}P^{2}\bigg( \text{e}^{\sigma f(0)}>\frac{u}{\text{mes}\left( T\right) }\bigg)&= P^{2}\left( f(0)>\log u/\sigma \right) \\&\sim \frac{\sigma ^{2}}{\left( \log u-\log \left( \text{mes}\left( T\right)\right) \right)^{2}}\exp \left( -\frac{\left( \log u-\log \left( \text{mes}\left(T\right) \right) \right) ^{2}}{\sigma ^{2}}\right) .\end{align*}

Therefore, using the definition of $c_{u}$ , there exists a positive constant $\alpha $ such that, for large u,

\begin{multline*}\mathbb{E}\left\{ Z_{0}\left( u\right) ^{2}\textbf{1}_{\left\{\bigvee_{i=1}^{M_{u}}\left\vert \tilde{N}_{i}\right\vert >c_{u}\sqrt{\log (u)}\right\} }\right\} \leq \\CS_{u}^{d}\frac{1}{c_{u}( \log (u))^{3/2}}\exp \bigg( {-}\frac{(\log u)^{2}+2\log ( 1+M_{u}) \log (u)}{\sigma ^{2}}-\alpha\log ( 1+M_{u}) \log (u)\bigg) .\end{multline*}

Note that, for some positive constant C,

\begin{align*}P\left( \omega _{0,u}X_{0,u}>u\right) ^{2} & \sim P\left( f(0)>\left( \log u+\log M_{u}+\log \text{mes}(T)\right) /\sigma \right) ^{2} \\&\sim C\frac{1}{(\log u)^{2}}\exp \left( -\frac{(\log u)^{2}+2\log \left(M_{u}\right) \log (u)}{\sigma ^{2}}\right) .\end{align*}

It follows that

\begin{equation*}\mathbb{E}\left\{ Z_{0}\left( u\right) ^{2}\textbf{1}_{\left\{\bigvee_{i=1}^{M_{u}}\left\vert \tilde{N}_{i}\right\vert >c_{u}\sqrt{\log (u)}\right\} }\right\} =o\left( P\left( w_{0,u}X_{0,u}>u\right) ^{2}\right) ,\end{equation*}

and also

\begin{equation*}\mathbb{E}\left\{ Z_{0}\left( u\right) ^{2}\textbf{1}_{\left\{\bigvee_{i=1}^{M_{u}}\left\vert \tilde{N}_{i}\right\vert >c_{u}\sqrt{\log (u)}\right\} }\right\} =o\left( w^{2}\left( u\right) \right) ,\end{equation*}

since $w_{0,u}X_{0,u}\leq \mathcal{I}\left( T\right) $ .

Let us now consider the second component. Let

\begin{equation*}\Omega _{u}=\Bigg\{\! \sum_{i=0}^{M_{u}}\omega _{i,u}X_{i,u}>u,\bar{\omega}_{u}\exp \! \Bigg(\! \sum_{i=0}^{M_{u}}\omega _{i,u}\log X_{i,u}/\bar{\omega}_{u}\Bigg)\! \leq u,\bigvee_{i=0}^{M_{u}}\omega _{i,u}X_{i,u}=\omega_{0,u}X_{0,u}\Bigg\} .\end{equation*}

We have

\begin{align*}& \mathbb{E}\left\{ Z_{0}\left( u\right) ^{2}\textbf{1}_{\left\{\bigvee_{i=1}^{M_{u}}\left\vert \tilde{N}_{i}\right\vert \leq c_{u}\sqrt{\log (u)}\right\} }\right\} \\[3pt]& = \mathbb{E}\left\{ \left( P\left( \left. \Omega _{u}\right\vert N_{0,u}^{(-1)}\right) \textbf{1}_{\left\{ \bigvee_{i=1}^{M_{u}}\left\vert \tilde{N}_{i}\right\vert \leq c_{u}\sqrt{\log (u)}\right\} }\right)^{2}\right\} \\[3pt]& = \mathbb{E}\left\{ P\left( \Omega_{u},\bigvee_{i=1}^{M_{u}}\left\vert \tilde{N}_{i}\right\vert \leq c_{u}\sqrt{\log (u)} \, \Big\vert \, N_{0,u}^{(-1)}\right) ^{2}\right\} \\[3pt]& \leq \mathbb{E}\Bigg\{ P\Bigg( \sum_{i=0}^{M_{u}}\omega_{i,u}X_{i,u}>u,\bigvee_{i=0}^{M_{u}}\omega _{i,u}X_{i,u}=\omega_{0,u}X_{0,u}, \bigvee_{i=1}^{M_{u}}\left\vert \tilde{N}_{i}\right\vert \leq c_{u}\sqrt{\log (u)} \, \Big\vert \, N_{0,u}^{(-1)}\Bigg) ^{2}\Bigg\} \\[3pt]& \leq \mathbb{E}\Bigg\{ P\Bigg(w_{0,u}X_{0,u}+\sum_{i=1}^{M_{u}}\omega _{i,u}X_{i,u}>u, \omega _{0,u}X_{0,u}>\frac{u}{1+M_{u}}, \bigvee_{i=1}^{M_{u}}\left\vert \tilde{N}_{i}\right\vert \leq c_{u}\sqrt{\log (u)} \, \Big\vert \, N_{0,u}^{(-1)}\Bigg) ^{2}\Bigg\} .\end{align*}

Moreover, we have $\sum_{i=1}^{M_{u}}\omega _{i,u}X_{i,u}=\sum_{i=1}^{M_{u}}\omega_{i,u}\exp\big\{\sigma (t_{i})\big( \rho _{i}f(0)+\sqrt{1-\rho _{i}^{2}}\tilde{N}_{i}\big) \big\}$ and, if $\bigvee_{i=1}^{M_{u}}\big\vert \tilde{N}_{i}\big\vert \leq c_{u}\sqrt{\log (u)}$ , we deduce that

\begin{align*}\sum_{i=1}^{M_{u}}\omega _{i,u}X_{i,u} &\leq \sum_{i=1}^{M_{u}}\omega_{i,u}\exp\left\{\sigma (t_{i})\left( \rho _{i}f(0)+\sqrt{1-\rho _{i}^{2}}c_{u}\sqrt{\log (u)}\right) \right\} \\&=\sum_{i=1}^{M_{u}}\omega _{i,u}\exp\left\{\sigma (t_{i})f(0)\left( \rho _{i}+\sqrt{1-\rho _{i}^{2}}c_{u}\sqrt{\log (u)/f(0)^{2}}\right) \right\}.\end{align*}

If we also assume that $w_{0,u}X_{0,u}>u/(1+M_{u})$ , or equivalently $f(0)>(\log u-\log \left( \text{mes}(T)\right) )/\sigma $ , we get, for large u,

\begin{align*}c_{u}\sqrt{\log (u)/f(0)^{2}} &= \left( 1+\varepsilon \right) \sqrt{4\frac{\log \left( 1+M_{u}\right) }{\sigma ^{2}}}\sqrt{\frac{\log (u)}{(\log u-\log \left( \text{mes}(T)\right) )^{2}/\sigma ^{2}}} \\&\sim \left( 1+\varepsilon \right) \sqrt{4\frac{\log \left( 1+M_{u}\right)}{\log (u)}}.\end{align*}

Then we have, for some positive constant c and for large u,

\begin{equation*}\sum_{i=1}^{M_{u}}\omega _{i,u}X_{i,u}\leq \sum_{i=1}^{M_{u}}\omega_{i,u}\exp\left\{\sigma (t_{i})f(0)\left( \rho _{i}+\sqrt{1-\rho _{i}^{2}}c\sqrt{\log \left( 1+M_{u}\right) /\log (u)}\right) \right\}.\end{equation*}

It follows that

\begin{multline*}\left\{ \omega _{0,u}X_{0,u}+\sum_{i=1}^{M_{u}}\omega_{i,u}X_{i,u}>u,w_{0,u}X_{0,u}>\frac{u}{1+M_{u}},\bigvee_{i=1}^{M_{u}}\left\vert \tilde{N}_{i}\right\vert \leq c_{u}\sqrt{\log (u)}\right\} \\\subset \left\{ \omega _{0,u}X_{0,u}+\sum_{i=1}^{M_{u}}\omega_{i,u}\exp\left\{\sigma f(0)\left( \rho _{i}+\sqrt{1-\rho _{i}^{2}}c\sqrt{\log \left( 1+M_{u}\right) /\log (u)}\right) \right\}\right\} .\end{multline*}

Since $M_{u}\sim \text{mes}(T)S_{u}^{d}$ , we have

\begin{equation*}\lim_{u\rightarrow \infty }\left( \sup_{i=1,\ldots,M_{u}}\sqrt{1-\rho _{i}^{2}}\right) \sqrt{\log \left( 1+M_{u}\right) /\log (u)}=0\end{equation*}

and

\begin{align*}\omega _{0,u}X_{0,u}+\sum_{i=1}^{M_{u}}\omega _{i,u}\exp\left\{\sigma f(0)\left(\rho _{i}+\sqrt{1-\rho _{i}^{2}}c\sqrt{\log \left( 1+M_{u}\right) /\log (u)}\right) \right\} \sim \int_{T}\text{e}^{\sigma f(0)C(t)} \, \text{d} t.\end{align*}

Now, by the method of steepest descent we know that, for large v,

\begin{equation*}\int_{T}\text{e}^{\sigma vC(t)} \, \text{d} t\sim \left( \frac{2\pi }{\sigma v}\right)^{d/2}\text{e}^{\sigma v},\end{equation*}

and then, for large u, $P\left( \int_{T}\text{e}^{\sigma f(0)C(t)} \, \text{d} t>u\right) \sim P\left( f(0)>v\right)$ , where v is the unique solution to $(2\pi /\sigma )^{d/2}v^{-d/2}\text{e}^{\sigma v}=u$ . Since

\begin{equation*}P\left( f(0)>v\right) \sim \frac{1}{v}\exp \left( -\frac{1}{2}v^{2}\right)\sim \frac{1}{H\text{mes}\left( T\right) v^{d}}w\left( u\right) ,\end{equation*}

we deduce that, for some positive constant C,

\begin{equation*}\mathbb{E}\left\{ Z_{0}\left( u\right) ^{2}\textbf{1}_{\left\{\bigvee_{i=1}^{M_{u}}\left\vert \tilde{N}_{i}\right\vert \leq c_{u}\sqrt{\log (u)}\right\} }\right\} \leq Cw^{2}\left( u\right) ,\end{equation*}

and the result follows.

Proof of Proposition 3. Recall that $S_{u}$ is chosen as $S_{u}=\kappa _{0}\eta _{u}^{-(1+\varepsilon )}\left( \log u\right)^{2+\varepsilon }$ for some $\varepsilon >0$ , where $\left( \eta _{u}\right) $ is a sequence such that lim $_{u\rightarrow \infty }\eta _{u}=0$ , and that we have

\begin{equation*}\lim_{u\rightarrow \infty }\frac{w_{M_{u}}(u)}{w(u)}=1.\end{equation*}

Due to homogeneity, we have, for large u, $P\left( \mathcal{J}_{u}=j\right) \sim 1/M_{u}$ and, by Lemma 5, $\mathbb{E}\{Z_{j}\left( u\right) ^{2}\}\leq Cw(u)^{2}$ .

We have

\begin{equation*}\mathbb{E}\left\{ \left( Z_{M_{u}}(u)\right) ^{2}\right\} =\left[ \mathbb{E}\left\{ \left( Z_{M_{u}}(u)-v_{1,M_{u}}(u)\right) ^{2}\right\}+v_{1,M_{u}}(u)\left( v_{1,M_{u}}(u)+2v_{2,M_{u}}(u)\right) \right] .\end{equation*}

By Lemma 3,

\begin{equation*}\mathbb{E}\left\{ \left( Z_{M_{u}}(u)-v_{1,M_{u}}(u)\right) ^{2}\right\}=\sum_{j=0}^{M_{u}}\frac{\mathbb{E}\left\{ Z_{j}\left( u\right) ^{2}\right\}}{P\left( \mathcal{J}_{u}=j\right) }=\sum_{j=0}^{M_{u}}P\left( \mathcal{J}_{u}=j\right) \frac{\mathbb{E}\left\{ Z_{j}\left( u\right) ^{2}\right\} }{P\left( \mathcal{J}_{u}=j\right) ^{2}}.\end{equation*}

By Lemma 4,

\begin{equation*}v_{1,M_{u}}(u)\leq C\exp \left( -\frac{\left( \log u-\log \left(\int_{T}\text{e}^{\mu (t)} \, \text{d} t\right) \right) ^{2}}{2\sigma _{A}^{2}\left( 1+\epsilon\right) }\right) ,\end{equation*}

and

\begin{equation*}v_{2,M_{u}}(u)\leq w_{M_{u}}\left( u\right) \leq C\exp \left( -\frac{(\log u)^{2}-\tilde{c}_{2}(\log u))}{2\sigma _{\ast }^{2}}\right)\end{equation*}

as in Lemma 1. Moreover, we have

\begin{align*}\mathbb{E}\left\{ (Z_{M_{u}}(u)-w(u))^{2}\right\} &= \mathbb{E}\left\{(Z_{M_{u}}(u))^{2}\right\} -w_{M_{u}}^{2}(u)+(w_{M_{u}}(u)-w(u))^{2} \\&= \mathbb{E}\big\{ ( Z_{M_{u}}(u)-v_{1,M_{u}}(u)) ^{2}\big\}+v_{1,M_{u}}(u)( v_{1,M_{u}}(u)+2v_{2,M_{u}}(u)) \\& \quad -w_{M_{u}}^{2}(u)+(w_{M_{u}}(u)-w(u))^{2}\end{align*}

and, for large u, $\mathbb{E}\left\{ (Z_{M_{u}}(u)-w(u))^{2}\right\} \leq \mathbb{E}\big\{\left( Z_{M_{u}}(u)-v_{1,M_{u}}(u)\right) ^{2}\big\}+3v_{1,M_{u}}(u)w(u)+\eta _{u}^{2}w(u)^{2}$ by Proposition 1.

Now,

\begin{equation*}\frac{\mathbb{E}\left\{ \left( Z_{M_{u}}(u)-v_{1,M_{u}}(u)\right)^{2}\right\} }{w_{M_{u}}(u)^{2}} \leq \left( \frac{w(u)^{2}}{w_{M_{u}}(u)^{2}}\right) \sum_{j=0}^{M_{u}}\frac{1}{P\left( \mathcal{J}_{u}=j\right) }\frac{\mathbb{E}\left\{ Z_{j}\left( u\right) ^{2}\right\} }{w(u)^{2}}\leq C\left( M_{u}\right) ^{2}\end{equation*}

as $u\rightarrow \infty $ . We deduce that, for large u,

\begin{equation*}\lim_{u\rightarrow \infty }\frac{\mathbb{E}\left\{(Z_{M_{u}}(u)-w(u))^{2}\right\} }{w_{M_{u}}^{2}(u)}\leq CM_{u}^{2}.\end{equation*}

Finally, $M_{u}\sim \text{mes}(T)S_{u}^{d}\sim \text{mes}(T)\eta _{u}^{-(1+\varepsilon )d}\left(\log u\right) ^{(2+\varepsilon )d}$ , and, for large u and some positive constants $c_{1}^{\prime }$ and $c_{2}^{\prime }$ , we have $c_{1}^{\prime }|\log w(u)|\leq \left( \log u\right) ^{2}\leq c_{2}^{\prime}|\log w(u)|$ . It follows that

\begin{equation*}\qquad\quad\qquad\qquad\qquad\qquad\lim_{u\rightarrow \infty }\frac{\mathbb{E}\left\{(Z_{M_{u}}(u)-w(u))^{2}\right\} }{|\log ( w(u))|^{[(2+\varepsilon )+(1+\varepsilon )\nu ]d}w^{2}(u)}<\infty. \qquad\qquad\qquad\qquad\Box\end{equation*}