Asymptotic Behavior of Tail Density for Sum of Correlated Lognormal Variables

Abstract

We consider the asymptotic behavior of a probability density function for the sum of any two lognormally distributed random variables that are nontrivially correlated. We show that both the left and right tails can be approximated by some simple functions. Furthermore, the same techniques are applied to determine the tail probability density function for a ratio statistic, and for a sum with more than two lognormally distributed random variables under some stricter conditions. The results yield new insights into the problem of characterization for a sum of lognormally distributed random variables and demonstrate that there is a need to revisit many existing approximation methods.

1. Introduction

The lognormal distribution has applications in many fields such as survival analysis [1], genetic studies [2, 3], financial modelling [4, 5], telecommunication studies [6, 7] amongst others. It has been found that many types of data can be modeled by lognormal distributions, which include human blood pressure, microarray data, stock options, survival rate for different groups of human beings, and the received power's long-term fluctuation. In these occasions, we wish to make some inferences based on the collected data involving the addition of a few lognormally distributed random variables (RVs). Deriving the statistical properties of a sum of lognormally distributed RVs is therefore desirable [6, 8]. Also note that the number of summands is so small in practice that the central limit theorem is not applicable.

Many research works assume that all the summands are independent, either justified by practical considerations or for the sake of simplicity. However, there are some applications (e.g., the Asian option pricing model [4]) in which correlations among the summands are inevitable. Our study will address the correlation problem.

In some cellular mobile systems (see [9]), the signal quality is largely dictated by signal to interference ratio (SIR). On a large scale, both useful signals and interfering signals experience lognormal shadow fadings. That is to say, SIR can be modeled by (𝑋₁+⋯+𝑋_𝑛)/(𝑌₁+⋯+𝑌_𝑚) where all the RVs 𝑋₁,…,𝑋_𝑛,𝑌₁,…,𝑌_𝑚 are lognormally distributed. SIR only characterizes the instantaneous quality. For ordinary users and network operators, one important factor to consider is the outage probability target to a certain SIR threshold. For example, for the users of a data transfer service, it is required that the outage probability 𝑃₀=𝑃𝑟(SIR<𝛾₀ such that BER(𝛾₀)=10⁻⁶) needs to be less than 0.01. Here BER stands for bit error rate that usually depends on SIR and other factors. In this paper, Theorem 2.8 provides an approximation to 𝑃𝑟(SIR<𝛾₀) when 𝑛=2, 𝑚=2.

In addition, Theorems 2.6 and 2.7 try to characterize the left and right tails of a sum of lognormally distributed RVs. The theorems are useful to construct a Padé approximation to the probability density function (PDF) of a sum of lognormally distributed RVs. For example, if that approximation is available, and if useful signals and interfering signals are independent, the outage probability can be numerically estimated as follows: Let 𝑍₁=log(∑^𝑛_𝑖=1𝑋_𝑖) and 𝑍₂=log(∑^𝑚_𝑗=1𝑌_𝑗), then 𝑃𝑟(𝑆𝐼𝑅<𝛾₀)=𝑃𝑟(𝑍₁−𝑍₂<log(𝛾₀))≃^log(𝛾0)_−∞ℝ𝑓^approx_𝑍1(𝑥−𝑦)𝑓^approx_𝑍2(−𝑦)𝑑𝑦𝑑𝑥. (1.1)

Since Fenton [10] addressed the problem, many methods have been developed, but none of them have been successful in finding a closed form representation for the PDF of a sum of multiple lognormally distributed RVs. These methods can be divided into three categories.

(i)The first type of methods attempt to characterize the PDF by calculating the moment generating function [11, 12] or the characteristic function [13, 14]. The results obtained can be used in the numerical computations of a PDF or a cumulative density function (CDF). To our knowledge, no work has succeeded in using the results of this category to describe the shape of a PDF or CDF. (ii)The second type of methods [15–18] use the bound technique for the CDF of an underlying statistic.(iii) The third type of methods focuses on finding a good approximation to either the PDF or CDF of the underlying statistic. Most published works belong to this category. The way to find the approximation can often be described as follows: first, assume a specific distribution that the sum (or the ratio of sum) of the lognormally distributed RV follows; then use a variety of methods to identify the parameters for that specific distribution. The specific distributions in the literature include lognormal [15, 19], reciprocal Gamma [4], log shifted Gamma [20], and user-defined PDF [21, 22]. In some works [23], only the CDF approximation is defined. Moment matching [10, 24, 25], moment generating function matching [19], and the least squares fitting [26, 27] are a few popular methods used to determine the parameters associated with the distribution.

In this paper, we will rigorously characterize the right and left tails behavior of a PDF for a random variable 𝑍=log(𝑒^𝑋1+𝑒^𝑋2), where (𝑋₁,𝑋₂) are jointly distributed with 𝑁(𝜇,Σ) as distribution. This is our first step towards understanding the more general problem: the characterization of the PDF of 𝑍=log(∑^𝑁_𝑘=1𝑒^𝑋𝑘) where (𝑋₁,…,𝑋_𝑁) are jointly distributed as 𝑁(𝜇,Σ). Note here we do not assume that the 𝑋_𝑘 are independent (except for Theorem 2.7), nor do we assume that 𝑋₁,…,𝑋_𝑁 have the same marginal distribution. We hope that our study can lead to a better solution to the works presented in [28] or [29].

Janos [15] is the first one to study the right tail probability of a sum of lognormals. More advanced and more general studies can be found in [30]. We have not found any theoretical results regarding the left tails. In addition, the right tails results we show cannot be deduced from the results in [5, 30–32].

Our results show that it is possible to find some elementary functions 𝑔_𝐿,𝑔_𝑅 such that lim_{𝑧→−∞}𝑓_𝑍(𝑧)𝑔_𝐿(𝑧)=1,lim_𝑧→+∞𝑓_𝑍(𝑧)𝑔_𝑅(𝑧)=1. (1.2) The explicit forms of 𝑔_𝐿 and 𝑔_𝑅 enable us to assess the performance of the existing approximation methods and to determine how to improve these methods. By Theorem 2.3 (see also the subsequent remark and Corollary 2.5), we can determine that at the left tail region, even under the independence assumption, given any function 𝑔_∗ within the families of PDFs such as lognormal, reciprocal Gamma or log shifted Gamma, either lim_{𝑧→−∞}𝑔_∗(𝑧)/𝑓_𝑍(𝑧) does not exist or lim_{𝑧→−∞}𝑔_∗(𝑧)/𝑓_𝑍(𝑧) can be only zero or ∞. No previous works have led to this discovery. Szyszkowicz [18] has pointed out that some precedent models are wrong in the tail region, but this work was based on a hypothesis that was only justified by the numerical results, and it still focused on finding the best lognormal type approximation. In view of our results, such efforts are unlikely to succeed.

Our characterization of the behavior of the tail of the PDF of a sum of two lognormals is complete in the sense that our results cover all nondegenerate covariance matrices. Our work regarding the ratio RV is obtained under more stringent conditions. This new result shows that the ratio RV is neither lognormal nor log Gamma. This indicates that others should be cautious with the method in [9] despite the successful examples demonstrated therein.

When the number of summands exceeds two, the situation for a PDF approximation becomes much more complicated. We are able to show some left tail and right tail results by imposing some conditions on the covariance matrix that covers the independent case. The result of Theorem 2.7 could be well-known to experts working with functions from the subexponential class. Unfortunately, we did not find any references that explicitly state the result, so we provide a short proof in Appendix F. Further in this line, [5] has presented the complete CDF approximation for the right tail with an arbitrary covariance matrix. However our results cannot be deduced trivially from the CDF behavior and are interesting in themselves. For example, for any polynomial growth continuous function ℎ, we can say that lim_𝑧→+∞𝐸[ℎ(𝑒^𝑍)1_𝑍>𝑧]∫^∞_𝑧ℎ(𝑒^𝑥)𝑔_𝑅(𝑥)𝑑𝑥=1, (1.3) whereas such an approximation cannot be a direct consequence from the result in [5].

In the following sections, we will first present our results followed by a numerical validation. We will then discuss some future studies that this paper does not cover. Also we present the proofs in the appendix.

2. Main Results

Let (𝑋₁,𝑋₂) be a jointly normally distributed random vector. Let 𝜌 be the correlation coefficient, and 𝜇_𝑖=𝐸(𝑋_𝑖), 𝜎²_𝑖=Var(𝑋_𝑖), for 𝑖=1,2. Then the joint PDF of (𝑒^𝑋1,𝑒^𝑋2) is given by 𝑓(𝑢,𝑣)≡ℎ(𝑢,𝑣)2𝜋𝜎₁𝜎₂1−𝜌²𝑢𝑣, (2.1) where 1ℎ(𝑢,𝑣)=exp{−2(1−𝜌²)[(log(𝑢)−𝜇₁𝜎₁)²−2𝜌(log(𝑢)−𝜇₁)(log(𝑣)−𝜇₂)𝜎₁𝜎₂+(log(𝑣)−𝜇₂𝜎₂)²]}. (2.2) We wish to study the left and right tail probabilities of 𝑍=log(𝑒^𝑋1+𝑒^𝑋2), which has PDF as 𝑓_𝑍(𝑧)≡𝑒^𝑧𝑔(𝑒^𝑧) with 𝑔(𝑧)≡^𝑧₀𝑓(𝑦,𝑧−𝑦)𝑑𝑦. (2.3) We hope to understand the asymptotic behavior of 𝑔(𝑒^𝑧) when 𝑧→±∞. Direct calculus yields 𝑔(𝑒^𝑧𝑒)=^−𝑧2𝜋𝜎₁𝜎₂1−𝜌²¹⁰𝑒^{−𝐴(𝑧,𝑡)}(1−𝑡)𝑡𝑑𝑡, (2.4) where the exponent 𝐴(𝑧,𝑡)=12(1−𝜌²)[(log(1−𝑡)+𝑧−𝜇₁𝜎₁)²+(log𝑡+𝑧−𝜇₂𝜎₂)²−2𝜌log(1−𝑡)+𝑧−𝜇₁𝜎₁log𝑡+𝑧−𝜇₂𝜎₂]. (2.5) Rewrite 𝐴 in three terms 𝐴(𝑡)=𝐴₁(𝑧)+2𝑧𝐴₂(𝑡)+A₃(𝑡), where the 𝐴_𝑖 are defined as 𝐴₁1(𝑧)=2(1−𝜌²){𝑧²(1𝜎²₁−2𝜌𝜎₁𝜎₂+1𝜎²₂𝜌)+2𝑧[(𝜎₁−1𝜎₂)𝜇₂𝜎₂𝜌+(𝜎₂−1𝜎₁)𝜇₁𝜎₁𝐴]},₂1(𝑡)=2(1−𝜌²)1[(𝜎₁−𝜌𝜎₂)log(1−𝑡)𝜎₁1+(𝜎₂−𝜌𝜎₁)log𝑡𝜎₂𝐴],₃1(𝑡)=2(1−𝜌²)[(log(1−𝑡)−𝜇₁)²𝜎²₁−2𝜌(log(1−𝑡)−𝜇₁)(log𝑡−𝜇₂)𝜎₁𝜎₂+(log𝑡−𝜇₂)²𝜎²₂]. (2.6)

Hence, by changing the variable 𝑡 to 𝑒^−𝑢, 𝑓_𝑍𝑒(𝑧)=^{−𝐴1(𝑧)}2𝜋𝜎₁𝜎₂1−𝜌^2∞0𝑒^{−2𝑧𝐴2(𝑒−𝑢)−𝐴3(𝑒−𝑢)}1−𝑒^−𝑢𝑑𝑢. (2.7)

We regroup the integrand in (2.7) in the form ^∞₀𝑒^{−2𝑧𝐴2(𝑒−𝑢)−𝐴3(𝑒−𝑢)}1−𝑒^−𝑢𝑑𝑢=^∞₀𝐻(𝑢)𝑒^{−𝑧𝐺(𝑢)}𝑑𝑢, (2.8)

with 1𝐺(𝑢)=1−𝜌²[(𝜎₂−𝜎₁𝜌)log(1−𝑒^−𝑢)𝜎²₁𝜎₂−(𝜎₁−𝜎₂𝜌)𝑢𝜎²₂𝜎₁𝑒],𝐻(𝑢)=^{−𝐴3(𝑒−𝑢)}1−𝑒^−𝑢. (2.9)Remark 2.1. Without loss of generality, in this paper, we always assume 0<𝜎₁≤𝜎₂,|𝜌|<1. (2.10) We also use the following notation.

Definition 2.2. We say that two functions 𝑓 and ℎ are equivalent near some point𝑎∈ℝ, denoted by 𝑓(𝑧)∼_𝑎ℎ(𝑧), if we havelim_𝑧→𝑎𝑓(𝑧)/ℎ(𝑧)=1. For the left tail, we have the following result.

Theorem 2.3. Let 𝑋_𝑖∼𝑁(𝜇_𝑖,𝜎²_𝑖) be defined as above for 𝑖=1,2 and 𝜌∈(−1,1) be the correlation coefficient. Let 𝑓_𝑍 be the PDF of 𝑍=log(𝑒^𝑋1+𝑒^𝑋2), then 𝑓_𝑍(𝑧)∼_−∞ℎ_𝐿(𝑧) as follows.(i)If 𝜌<𝜎₁/𝜎₂, one has ℎ_𝐿(𝑧)=𝐻(𝑢₀)𝑒^{−𝑧𝐺(𝑢0)−𝐴1(𝑧)}2𝜋𝑧(1−𝜌²)𝐺^′′(𝑢₀)𝜎₁𝜎₂with𝑢₀𝜎=log²¹+𝜎²₂−2𝜌𝜎₁𝜎₂𝜎²₁−𝜌𝜎₁𝜎₂. (2.11) Here the functions 𝐺, 𝐻, 𝐴₁ are defined by (2.9) and (2.6).(ii)If 𝜌=𝜎₁/𝜎₂, one has ℎ_𝐿(𝑧)=𝑁(𝜇₁,𝜎²₁)𝑒log(−𝑧)^{1/(2(𝜎22−𝜎21)){−(log𝑧𝜌−log(log𝑧𝜌)+𝜇2−𝜇1)2−2log𝑧𝜌}}with𝑧_𝜌=−𝑧(𝜌⁻²−1). (2.12)(iii)If 𝜌>𝜎₁/𝜎₂, one has ℎ_𝐿1(𝑧)=2𝜋𝜎²₁𝑒^{−(𝑧−𝜇1)2/(2𝜎21)}. (2.13)

In particular, when 𝜌=0, we find that 𝑓_𝑍∼_−∞𝐶₁𝑒^{−𝐶2𝑧−𝐶3𝑧2}/√𝑧, which means that any lognormal, reciprocal Gamma or log shifted Gamma cannot be used to fit the left tail, under the independence hypothesis. The situation for the right tail of 𝑓_𝑍 is simpler. It is interesting to remark that the result does not depend on the correlation coefficient 𝜌 (see also [5]). Here and later on, we employ the lexicographical order to the couple (𝜎,𝜇).

Theorem 2.4. Let 𝑍 and 𝑓 be defined as above, then 𝑓_𝑍(𝑧)∼_∞ℎ_𝑅(𝑧), where ℎ_𝑅(𝑧) is defined as follows.(i)If (𝜎₁,𝜇₁)≠(𝜎₂,𝜇₂), one has ℎ_𝑅(𝑧)=𝑁(𝜇,𝜎²)with(𝜎,𝜇)=max{(𝜎₁,𝜇₁),(𝜎₂,𝜇₂)}. (2.14)(ii) If (𝜎₁,𝜇₁)=(𝜎₂,𝜇₂)=(𝜎,𝜇), one has ℎ_𝑅(𝑧)=2/(𝜎√2𝜋)𝑒^{−(𝑧−𝜇)2/(2𝜎2)}.

The following corollary is an immediate consequence of Theorems 2.3 and 2.4.

Corollary 2.5. Let 𝑓_𝑉 be the PDF of 𝑉=𝑒^𝑋1+𝑒^𝑋2, where 𝑋₁,𝑋₂ are i.i.d. RVs following 𝑁(0,𝜎²) distributions. Then, one has lim_𝑧→0𝑓_𝑉(𝑧)𝑓_𝑉𝐿(𝑧)=1,where𝑓_𝑉𝐿(𝑧)=exp[−(log𝑧−log2)²/𝜎²]𝑧𝜎;−𝜋log𝑧lim_𝑧→+∞𝑓_𝑉(𝑧)𝑓_𝑉𝑅(𝑧)=1,where𝑓_𝑉𝑅√(𝑧)=2exp[−log²𝑧/(2𝜎²)]√.𝜋𝜎𝑧 (2.15) The results in Corollary 2.5 confirm those results reported in [18]. Furthermore, we can easily show that the models in [4, 21] will also fail in the tail regions.

Next we show that our left tail and right tail study can be extended for some special cases in higher dimension by using the Laplace methods.Theorem 2.6. Let (𝑋_𝑖)_{1≤𝑖≤𝑛} (𝑛≥3) be a joint normally distributed random variable with distribution 𝑁(𝜇,Σ). Let 𝐌=Σ⁻¹. Let 𝑓_𝑍 be the PDF of random variable 𝑍=log(∑_{1≤𝑖≤𝑛}𝑒^𝑋𝑖). If 𝐌=(𝑚_𝑖𝑗) satisfies 𝑚_𝑘=∑_{1≤𝑗≤𝑛}𝑚_𝑘𝑗>0 for all 𝑘=1,…,𝑛, then the left tail of 𝑓_𝑍 satisfies 𝑓_𝑍(𝑧)∼_−∞𝐻₁(𝑈₀)2𝜋|𝑧|^𝑛−1|detΣ||det𝐇_𝐹(𝑈₀)|exp[−𝑧^𝑛_𝑘=1𝑚_𝑘𝑚log^𝑘𝑚−12^𝑛_𝑖,𝑗=1(𝑧−𝜇_𝑖)𝑚_𝑖𝑗(𝑧−𝜇_𝑗)] (2.16) where 𝑚=∑_{1≤𝑗≤𝑛}𝑚_𝑗 , 𝐹(𝑢₂,…,𝑢_𝑛)=∑_{1≤𝑖≤𝑛}m_𝑖𝐶_𝑖, 𝐇_𝐹=(𝜕²_𝑖𝑗𝐹) denotes the Hessian matrix of 𝐹. Here, 𝐻₁(𝑢₂,…,𝑢_𝑛𝑒)≡^{−⟨𝐶,𝐌(𝐶−2𝜇)⟩/2}1−∑_{2≤𝑖≤𝑛}𝑒^𝑢𝑖,𝑈₀𝑚=(log^𝑘𝑚)_{2≤𝑘≤𝑛}∈ℝ^𝑛−1, (2.17) and 𝐶=(𝐶_𝑖)∈ℝ^𝑛 is given by 𝐶=(log(1−_{2≤𝑘≤𝑛}𝑒^𝑢𝑘),𝑢₂,𝑢₃,…,𝑢_𝑛). (2.18)Theorem 2.7. Let (𝑋₁,…,𝑋_𝑛) be independent normally distributed RVs, that is, 𝑋_𝑖∼𝑁(𝜇_𝑖,𝜎²_𝑖). Let 𝑍=log(∑^𝑛_𝑖=1𝑒^𝑋𝑖). Define (𝜎,𝜇)=max{(𝜎_𝑖,𝜇_𝑖),1≤𝑖≤𝑛} for the lexicographical order and 𝑚_𝑛 the number of maximum points, that is 𝑚_𝑛=#{1≤𝑖≤𝑛,𝑠.𝑡.(𝜎_𝑖,𝜇_𝑖)=(𝜎,𝜇)}. Then the PDF of 𝑍, 𝑓_𝑍(𝑥) satisfies lim_𝑥→∞𝑓_𝑍(𝑥)𝑁(𝜇,𝜎²)(𝑥)=𝑚_𝑛where𝑁(𝜇,𝜎²1)(𝑥)=√𝑒2𝜋𝜎^{−(𝑥−𝜇)2/(2𝜎2)}. (2.19) Finally, we show a result for the quotient of sums of i.i.d lognormal variables.Theorem 2.8. Let 𝑋₁,𝑋₂,𝑌₁,𝑌₂ be i.i.d random variables. Each of them follows 𝑁(0,𝜎²) distribution. Let 𝑊=log((𝑒^𝑋1+𝑒^𝑋2)/(𝑒^𝑌1+𝑒^𝑌2)) and 𝑓_𝑊(𝑤) be the PDF of 𝑊, then 𝑓_𝑊(𝑤)∼_±∞2𝑒^{−(|𝑤|+log2)2/(3𝜎2)}𝜎𝜋|𝑤|. (2.20) This result can be generalized to the case where 𝑋₁,𝑋₂ follow 𝑁(0,𝜎²_𝑥),𝑌₁,𝑌₂ follow 𝑁(0,𝜎²_𝑦), with any positive constants 𝜎_𝑥 and 𝜎_𝑦. Indeed, we can prove that 𝑓_𝑊(𝑤)∼_−∞2𝜎_𝑥√𝑒−𝜋𝑤^{−(𝑤−log2)2/(𝜎2𝑥+2𝜎2𝑦)},𝑓_𝑊(𝑤)∼_+∞2𝜎_𝑦√𝑒𝜋𝑤^{−(𝑤+log2)2/(𝜎2𝑦+2𝜎2𝑥)}. (2.21) For the sake of brevity, we only present the proof for the special case where both variances are equal.

3. Numerical Validation

We have validated the two-dimensional theoretical results by performing Monte-Carlo simulations. The curve generated by Monte Carlo method is obtained through bin-based density estimation. In all of the presented cases, we can see that our approximations match the numerical results closely.

For the simulation parameters of the 𝑍 statistic, in order to test our results in the extreme cases, we have chosen 𝜌=0.7,0.8,0.9. The mean values were arbitrarily set. The values of 𝜎 were chosen to be 9.6 and 12 so that their ratio 𝜎₁/𝜎₂ is 0.8.

For the parameters of the ratio statistic (denoted by 𝑊), we used (𝜎_𝑥,𝜎_𝑦)=(12,9.6) for two groups of normally distributed RVs. The mean for these RVs were set to 0. Due to the symmetry properties that 𝑊 has, it is sufficient to show the verification results for the left tail of 𝑓_𝑊.

4. Further Remarks

We have seen that our tail density approximations (for a sum of lognormal RVs) do not deal with an arbitrary covariance matrix. In the 2D proofs, we used the classical approximation technique for integrals, called the Laplace method (see Lemmas A.1, E.2). Then we divided the study into a few subcases and then proceeded in different ways. Comparing Theorems 2.3, 2.6 to Theorems 2.4, 2.7, it seems that in general the left tail behavior is more involved than the right tail case. We hope to adapt our approach to the higher-dimensional space, especially for the right tail behavior, which will lead to the result in [5]. It is also useful to perform a higher order approximation for both tails so that an efficient Padé approximation can be developed accordingly.

In view of the work in [32], it should be worthwhile to extend our lognormal work (at least for the right tail) to a more general family such as the subexponential class. The importance of this distribution family can be found in [5, 32, 33]. Perhaps future work for the sum of lognormals mentioned above may shed some insight on the subexponentionl class problem.

Figure 1 

Left tail 𝑓𝑍, 𝜌<𝜎1/𝜎2. g_1 is the approximation.

Figure 2 

Left tail 𝑓𝑍,𝜌=𝜎1/𝜎2. g_2 is the approximation.

Figure 3 

Left tail 𝑓𝑍,𝜌>𝜎1/𝜎2. g_3 is the approximation.

Figure 4 

Right tail 𝑓𝑍,𝜌<𝜎1/𝜎2. d_1 is the approximation.

Figure 5 

Right tail 𝑓𝑍,𝜌>𝜎1/𝜎2. d_2 is the approximation.

Figure 6 

Left tail 𝑓𝑊,(𝜎𝑥,𝜎𝑦)=(12,9.6). r_L is the approximation.

Appendices

A. Preliminaries

Here we list some basic lemmas useful in the subsequent discussion. Their proofs use standard techniques and hence are omitted.

Lemma A.1. Let 𝐻 be a positive, integrable function on an interval (𝑎,𝑏)⊂ℝ. Let 𝐺∈𝐶²(𝑎,𝑏) be concave such that 𝑥₀∈(𝑎,𝑏) verifies 𝐺^′(𝑥₀)=0, 𝐺^′′(𝑥₀)<0, then lim_𝑧→∞−𝑧𝐺^′′(𝑥₀)𝑒^{−𝑧𝐺(𝑥0)}√2𝜋𝐻(𝑥₀)×^𝑏_𝑎𝐻(𝑥)𝑒^{𝑧𝐺(𝑥)}𝑑𝑥=1. (A.1)

This result is the so-called Laplace method in modern analysis (see [34]), which is more often cited as saddle point approximation in other fields such as statistics or physics. Later on, we also give a higher dimensional version (see Lemma E.2).

Lemma A.2. Let 𝑥₀(𝑧) be a nonnegative function defined on ℝ, and lim_𝑧→𝑧0𝑥₀(𝑧)=∞. Assume that 𝑓(𝑧,𝑡) is a nontrivial, nonnegative function such that for any fixed 𝑧 near 𝑧₀∈ℝ, one has 𝑓(𝑧,𝑡)∈𝐿¹(ℝ) and for any 𝜖>0, ^𝑥0_𝑥^(𝑧)+𝜖_{0(𝑧)−𝜖}𝑓(𝑧,𝑡)𝑑𝑡∼_𝑧0‖𝑓(𝑧,⋅)‖_𝐿1. (A.2) If moreover 𝐺 is a bounded uniformly continuous function over ℝ, such that liminf_𝑥→∞𝐺(𝑥)>0, then _ℝ𝑓(𝑧,𝑡)𝐺(𝑡)𝑑𝑡∼_𝑧0𝐺(𝑥₀(𝑧))‖𝑓(𝑧,⋅)‖_𝐿1. (A.3)

In fact, we need a special case of lemma 16. When lim_𝑧→∞𝐺(𝑥)=𝐺₀≠0 exists, we can simply require that 𝐺 is a continuous function and replace the term 𝐺(𝑥₀(𝑧)) in (A.3) by 𝐺₀.

Lemma A.3. Let 𝐺 be a bounded measurable function defined on (𝑎,∞) with a∈ℝ∪{−∞}, such that lim_𝜉→∞𝐺(𝜉)=𝐺₀ exists. Let 𝜆 be a positive constant. Then lim_𝑧→∞𝑒^{−𝑧2/(4𝜆)∞}_𝑎𝐺(𝜉)𝑒^{𝑧𝜉−𝜆𝜉2}𝑑𝜉=𝐺₀𝜋𝜆. (A.4)

Using usual developments, we also have the following asymptotic expansion.

Lemma A.4. Fix 𝜆∈ℝ, if 𝑢(𝑧) satisfies 𝑧=(𝑒^𝑢(𝑧)−1)(𝑢(𝑧)+𝜆), then as 𝑧→∞, 𝑢(𝑧) is uniquely determined and 𝑢(𝑧)=log𝑧−loglog𝑧+loglog𝑧−𝜆log𝑧+𝑜(loglog𝑧log²𝑧). (A.5)

B. The Left Tail Behavior

In this section, we will prove Theorem 2.3. We discuss the cases 𝜌<𝜎₁/𝜎₂, 𝜌=𝜎₁/𝜎₂ and 𝜌>𝜎₁/𝜎₂, respectively. Recall that 0<𝜎₁≤𝜎₂ and |𝜌|<1.

B.1.

Case 1. 𝜌<𝜎₁/𝜎₂. Using the formulas (2.7) and (2.8), we need only to understand the behavior of 𝑇(𝑧)=^∞₀𝑒^{−2𝑧𝐴2(𝑒−𝑢)−𝐴3(𝑒−𝑢)}1−𝑒^−𝑢𝑑𝑢=^∞₀𝐻(𝑢)𝑒^{−𝑧𝐺(𝑢)}𝑑𝑢 (B.1) as 𝑧→−∞. Here 𝐺 and 𝐻 are defined by (2.9). Since 𝐺^′1(𝑢)=1−𝜌²(𝜎₂𝜌−𝜎₁𝜎₁𝜎²₂+𝜎₁𝜌−𝜎₂𝜎₂𝜎²₁𝑒^−𝑢𝑒^−𝑢−1),𝐺^′′(𝑢)=(𝜎₁𝜌−𝜎₂)𝑒^−𝑢𝜎₂𝜎²₁(1−𝜌²)(𝑒^−𝑢−1)². (B.2) Thus, 𝐺^′(𝑢)=0 has a unique solution 𝑢₀𝜎=log²¹+𝜎²₂−2𝜌𝜎₁𝜎₂𝜎²₁−𝜌𝜎₁𝜎₂. (B.3) We also have 𝐺^′′(𝑢)<0 in ℝ and 𝐻>0, integrable over ℝ₊. Hence, Lemma A.1 allows us to conclude 𝑓_𝑍(𝑧)∼_−∞𝐻(𝑢₀)𝑒^{−𝑧𝐺(𝑢0)−𝐴1(𝑧)}2𝜋𝑧(1−𝜌²)𝐺^′′(𝑢₀)𝜎₁𝜎₂. (B.4)

B.2.

Case 2. 𝜌=𝜎₁/𝜎₂. Here we can simplify 𝐴₁ and 𝐴₂ as 𝐴₁𝑧(𝑧)=²−2𝜇₁𝑧2𝜎²₁,𝐴₂(𝑡)=log(1−𝑡)2𝜎²₁. (B.5) Rewrite 𝑇(𝑧)=^∞₀𝑒^{𝐿(𝑧,𝑢)}𝐺₂(𝑢)𝑑𝑢 (B.6) with 𝐿(𝑧,𝑢)=−𝑧log(1−𝑒^−𝑢)𝜎²₁−(𝑢+𝜇₂)²−2𝜇₁𝑢2(1−𝜌²)𝜎²₂+𝜇₁𝜇₂(1−𝜌²)𝜎²₂, (B.7)𝐺₂1(𝑢)=1−𝑒^−𝑢1exp[−2(1−𝜌²)([log(1−𝑒^−𝑢)−𝜇₁]²𝜎²₁+2𝜌(𝑢+𝜇₂)log(1−𝑒^−𝑢)𝜎₁𝜎₂)]. (B.8) Clearly lim_𝑢→0+𝐺₂(𝑢)=0,lim_𝑢→∞G₂(𝑢)=𝑒^{−𝜇21/(2(1−𝜌2)𝜎21)}=𝐺₀>0. (B.9) Thus, 𝐺₂ is uniformly bounded and uniformly continuous over ℝ₊. Furthermore, we have 𝜕_𝑢𝑧𝐿(𝑧,𝑢)=−𝜎²₁1𝑒^𝑢−−1𝑢+𝜇₂−𝜇₁(1−𝜌²)𝜎²₂,𝜕²_𝑢𝑧𝐿(𝑧,𝑢)=𝜎²₁𝑒^𝑢(𝑒^𝑢−1)²−1𝜎²₂(1−𝜌²),𝜕³_𝑢𝑧𝐿(𝑧,𝑢)=−𝜎²₁𝑒^𝑢(𝑒^𝑢+1)(𝑒^𝑢−1)³. (B.10) If 𝑧<0, 𝜕²_𝑢𝐿(𝑧,𝑢)<0, 𝜕³_𝑢𝐿(𝑧,𝑢)>0 for any 𝑢∈ℝ₊. Let 𝑢_𝑧 be the unique solution of 𝜕_𝑢𝐿(𝑧,𝑢)=0. Obviously, 𝑢_𝑧 satisfies the equation (𝑒^𝑢𝑧−1)(𝑢_𝑧+𝜇₂−𝜇₁)=−(𝜌⁻²−1)𝑧. According to Lemma A.4, we obtain lim_{𝑧→−∞}𝑢_𝑧=lim_{𝑧→−∞}−𝑧𝑒^−𝑢𝑧=∞ and 𝜕²_𝑢𝐿(𝑧,𝑢_𝑧)∼_−∞−log(−𝑧)𝜎²₂−𝜎²₁,𝑒^{𝐿(𝑧,𝑢𝑧)∼−∞𝑒2221𝜌𝜌21]2𝜌21}1/(2(𝜎−𝜎)){−[log𝑧−log(log𝑧)+𝜇−𝜇−2log𝑧+𝜇} (B.11) where 𝑧_𝜌=−(𝜌⁻²−1)𝑧.
The situation here is more delicate than in Lemma A.1, but we can follow the same idea. For any 𝜖∈(0,1) fixed, there exists 𝑅>0 such that |𝐺₂(𝑥)/𝐺₀−1|≤𝜖, 𝜕²_𝑢𝐿(𝑧,𝑥)<0 for any 𝑥>𝑅. Choosing now 𝑧 near −∞ such that 𝑢_𝑧>𝑅+1 and
𝜕1≤^2𝑢𝐿(𝑧,𝑢_𝑧−𝜖)𝜕²_𝑢𝐿(𝑧,𝑢_𝑧)≤1+𝐶𝜖. (B.12)
We will decompose the integral into three parts: First we consider the integral of 𝑒^{𝐿(𝑧,𝑢)}𝐺₂(𝑢) over [𝑢_𝑧−𝜖,𝑢_𝑧+𝜖]. Using Taylor expansion and the monotonicity of 𝜕²_𝑢𝐿, we get
^𝑢𝑧_𝑢^+𝜖_𝑧−𝜖𝑒^{𝐿(𝑧,𝑢)}𝐺₂(𝑢)𝑑𝑢≥(1−𝜖)𝐺₀^𝑢𝑧_𝑢^+𝜖_𝑧−𝜖𝑒^{𝐿(𝑧,𝑢𝑧)+𝜕2𝑢𝐿(𝑧,𝑢𝑧−𝜖)(𝑢−𝑢z)2/2}𝑑𝑢. (B.13) By (B.12), for 𝜖>0 small enough, liminf_{𝑧→−∞}𝑒^{−𝐿(𝑧,𝑢𝑧)}−𝜕²_𝑢𝐿(𝑧,𝑢_𝑧)𝐺₀√2𝜋^{𝑢𝑧𝑢+𝜖}_𝑧^−𝜖𝑒^{𝐿(𝑧,𝑢)}𝐺₂(𝑢)𝑑𝑢≥1−𝜖√1+𝐶𝜖. (B.14) Using 𝐿(𝑧,𝑢_𝑧+𝜖), we also have (for small 𝜖>0) limsup_{𝑧→−∞}𝑒^{−𝐿(𝑧,𝑢𝑧)}−𝜕²_𝑢𝐿(𝑧,𝑢_𝑧)𝐺₀√2𝜋^{𝑢𝑧𝑢+𝜖}_𝑧^−𝜖𝑒^{𝐿(𝑧,𝑢)}𝐺₂(𝑢)𝑑𝑢≤1+𝜖√1−𝐶𝜖. (B.15)
Consider now the integral of 𝑒^{𝐿(𝑧,𝑢)}𝐺₂(𝑢) on [𝑢_𝑧+𝜖,∞). Since 𝐿(𝑧,𝑢) is strictly concave in 𝑢, 1‖𝐺₂‖_∞^∞_{𝑢𝑧+𝜖}𝑒^{𝐿(𝑧,𝑢)}𝐺₂(𝑢)𝑑𝑢≤^∞_{𝑢𝑧+𝜖}𝑒^{𝐿(𝑧,𝑢𝑧+𝜖)+𝜕𝑢𝐿(𝑧,𝑢𝑧+𝜖)(𝑢−𝑢𝑧−𝜖)}𝑒𝑑𝑢=^{𝐿(𝑧,𝑢𝑧+𝜖)}𝜕_𝑢𝐿(𝑧,𝑢_𝑧+𝜖). (B.16)
Moreover, 𝜕_𝑢𝐿(𝑧,𝑢_𝑧+𝜖)≥𝜖𝜕²_𝑢𝐿(𝑧,𝑢_𝑧), 𝐿(𝑧,𝑢_𝑧+𝜖)<𝐿(𝑧,𝑢_𝑧), so we get lim_{𝑧→−∞}𝑒^{−𝐿(𝑧,𝑢𝑧)}−𝜕²_𝑢𝐿(𝑧,𝑢_𝑧)^∞_{𝑢𝑧+𝜖}𝑒^{𝐿(𝑧,𝑢)}𝐺₂(𝑢)𝑑𝑢=0. (B.17) Similarly, lim_z→−∞𝑒^{−𝐿(𝑧,𝑢𝑧)}−𝜕²_𝑢𝐿(𝑧,𝑢_𝑧)^𝑢𝑧₀^−𝜖𝑒^{𝐿(𝑧,𝑢)}𝐺₂(𝑢)𝑑𝑢=0. (B.18) Combining all these estimates, we deduce 1+𝜖√1−𝐶𝜖≥limsup_{𝑧→−∞}𝑒^{−𝐿(𝑧,𝑢𝑧)}−𝜕²_𝑢𝐿(𝑧,𝑢_𝑧)𝐺₀√2𝜋^∞0𝑒^{𝐿(𝑧,𝑢)}𝐺₂(𝑢)𝑑𝑢≥liminf_{𝑧→−∞}𝑒^{−𝐿(𝑧,𝑢𝑧)}−𝜕²_u𝐿(𝑧,𝑢_𝑧)𝐺₀√2𝜋^∞0𝑒^{𝐿(𝑧,𝑢)}𝐺₂(𝑢)𝑑𝑢≥1−𝜖√.1+𝐶𝜖 (B.19) As 𝜖>0 can be arbitrarily small, by (2.7), 𝑓_𝑍(𝑧)∼_−∞𝐺₀𝑒^{𝐿(𝑧,𝑢𝑧)−𝐴1(𝑧)}−2𝜋(1−𝜌²)𝜕²_𝑢𝐿(𝑧,𝑢_𝑧)𝜎₁𝜎₂. (B.20) Applying (B.5), (B.9) and (B.11), we complete the proof.

B.3.

Case 3. 𝜌>𝜎₁/𝜎₂. Rewrite 𝐴₂(𝑒^−𝑢)=𝜉(𝑢)=𝐵₁log(1−𝑒^−𝑢)+𝐵₂𝑢 (B.21) with two positive constants 𝐵₁=𝜎₂−𝜎₁𝜌2(1−𝜌²)𝜎²₁𝜎₂,𝐵₂=𝜎₂𝜌−𝜎₁2(1−𝜌²)𝜎²₂𝜎₁. (B.22) Thus, ^∞₀𝑒^{−2𝑧𝐴2(𝑒−𝑢)−𝐴3(𝑒−𝑢)}1−𝑒^−𝑢𝑑𝑢=^∞₀𝑒^{𝐿2(𝑧,𝑢)}𝐺₂(𝑢)𝑑𝑢 (B.23) with 𝐺₂ given by (B.8) and 𝐿₂(𝑧,𝑢)=−2𝑧[𝐵₁log(1−𝑒^−𝑢)+𝐵₂𝑢]−(𝑢+𝜇₂)²2(1−𝜌²)𝜎²₂+𝜌𝜇₁(𝑢+𝜇₂)(1−𝜌²)𝜎₁𝜎₂=−2𝑧[𝐵₁log(1−𝑒^−𝑢)+𝐵₂1𝑢]−2(1−𝜌²)[(𝑢+𝜇₂𝜎₂−𝜌𝜇₁𝜎₁)²−𝜇²₁𝜌²𝜎²₁]. (B.24) Notice that 𝜉 is 𝐶^∞ diffeomorphism from (0,∞) into ℝ, with 𝜉^′(𝑢)=𝐵₁(𝑒^𝑢−1)⁻¹+𝐵₂>0 in (0,∞). Let 𝜂 be the inverse function of 𝜉, namely 𝑢=𝜂(𝜉). We have the following properties: lim_𝜉→+∞𝜂(𝜉)=∞,lim_𝜉→+∞𝜂^′1(𝜉)=𝐵₂,lim_𝜉→+∞𝜉−𝐵₂𝜂(𝜉)−𝐵₁𝑒^{−𝜂(𝜉)}=1. (B.25) The change of variable 𝑢=𝜂(𝜉) yields ^∞₀𝑒^{−2𝑧𝐴2(𝑒−𝑢)−𝐴3(𝑒−𝑢)}1−𝑒^−𝑢𝑑𝑢=_ℝ𝐺₂(𝜂(𝜉))𝜂^′(𝜉)𝑒^{𝐿2(𝑧,𝜂(𝜉))}𝑑𝜉. (B.26) For 𝑧<0, as 𝜂^′(𝜉)‖𝐺₂‖_∞∈𝐿¹(ℝ₋), 𝐿₂(𝑧,𝜂(𝜉))≤𝐶 in ℝ₋ and lim_{𝑧→−∞}𝐿₂(𝑧,𝜂(𝜉))=−∞ for 𝜉<0, we obtain lim_{𝑧→−∞ℝ−}𝐺₂(𝜂(𝜉))𝜂^′(𝜉)𝑒^{𝐿2(𝑧,𝜂(𝜉))}𝑑𝜉=0. (B.27) Furthermore, _ℝ+𝐺₂(𝜂(𝜉))𝜂^′(𝜉)𝑒^{𝐿2(𝑧,𝜂(𝜉))}𝑑𝜉=_ℝ+𝐻₂(𝜉,𝜂(𝜉))𝑒^{−2𝑧𝜉−1/(2(1−𝜌2))(𝜉/(𝐵2𝜎2)+𝜇2/𝜎2−𝜌𝜇1/𝜎1)2}𝑑𝜉, (B.28) where 𝐻₂(𝜉,𝜂(𝜉))=𝐺₂(𝜂(𝜉))𝜂^′(𝜉)𝑒^{𝜇21𝜌2/(2(1−𝜌2)𝜎21)}×𝑒^{[(𝜉/(𝐵2𝜎2)+𝜇2/𝜎2−𝜌𝜇1/𝜎1)2−(𝜂(𝜉)/𝜎2+𝜇2/𝜎2−𝜌𝜇1/𝜎1)2]/(2(1−𝜌2))} (B.29) is a bounded function in ℝ₊ by properties of 𝜂 and 𝐺₂. Otherwise, using (B.9) and (B.25) lim_𝜉→∞𝐻₂𝐺(𝜉,𝜂(𝜉))=⁰𝐵₂𝑒^{𝜇21𝜌2/(2(1−𝜌2)𝜎21)}=𝑒^{−𝜇21/(2𝜎21)}𝐵₂>0. (B.30) Applying Lemma A.3, we get _ℝ+𝐺₂(𝜂(𝜉))𝜂^′(𝜉)𝑒^{𝐿2(𝑧,𝜂(𝜉))}𝑑𝜉∼_−∞𝜎₂2𝜋(1−𝜌²)𝑒^{−𝜇21/(2𝜎21)}𝑒^{2𝐵22𝜎22(1−𝜌2)𝑧2+2𝐵2(𝜇2−𝜌𝜇1𝜎2/𝜎1)𝑧}. (B.31) Finally, combining (B.27), (B.31), (B.22) and (2.7) 𝑓_𝑍(𝑧)∼_−∞1𝜎₁√𝑒2𝜋^{−𝜇21/(2𝜎21)}𝑒^{−𝐴1(𝑧)+2𝐵22𝜎22(1−𝜌2)𝑧2+2𝐵2(𝜇2−𝜌𝜇1𝜎2/𝜎1)𝑧}=1𝜎₁√𝑒2𝜋^{−(𝑧−𝜇1)2/(2𝜎21)}, (B.32) which is just the claimed result.

C. The Right Tail Behavior

Here we prove Theorem 2.4. We begin with the formulae (2.7), (2.6) and divide the study into two cases: 𝜌<𝜎₁/𝜎₂ and 𝜌≥𝜎₁/𝜎₂. Since the arguments are often similar to the previous consideration and the situation is simpler, we will proceed with less details.

C.1.

Case 1. 𝜌<𝜎₁/𝜎₂. We have 𝐴₂(𝑒^−𝑢)=𝜉(𝑢)=𝐵₁log(1−𝑒^−𝑢)+𝐵₂𝑢 with 𝐵_𝑖 given by (B.22). Since 𝐵₁>0 and 𝐵₂<0, it's clear that −𝜉^′′(𝑢)=𝐵₁𝑒^𝑢(𝑒^𝑢−1)⁻²>0 in (0,∞) and 𝑢₀=log(1−𝐵₁/𝐵₂) is the unique solution of 𝜉^′(𝑢)=0. Hence 𝜉 is decreasing on (0,𝑢₀), increasing on (𝑢₀,∞) and lim_𝑢→0⁺𝜉(𝑢)=lim_𝑢→∞𝜉(𝑢)=−∞. Thus for any 𝑡∈(−∞,𝜉(𝑢₀)), there exist exactly two solutions 𝜂₁(𝜉)<𝜂₂(𝜉) for 𝜉(𝑢)=𝑡. Before proceeding, we list some properties of 𝜂_𝑖lim_{𝜉→−∞}𝜂₁(𝜉)=lim_{𝜉→−∞}𝜉𝜂₁(𝜉)=0,lim_{𝜉→−∞}𝜂^′₁(𝜉)1−𝑒^{−𝜂1(𝜉)}=1𝐵₁,lim_{𝜉→−∞}𝜂₂(𝜉)=∞,lim_{𝜉→−∞}𝜂^′₂1(𝜉)=𝐵₂,lim_{𝜉→−∞}𝜉−𝐵₂𝜂₂(𝜉)−𝐵₁𝑒^{−𝜂2(𝜉)}=1. (C.1)(C.2)

Note that 𝐼(𝑧)=^∞_𝑢0𝑒^{−2𝑧𝐴2(𝑒−𝑢)−𝐴3(𝑒−𝑢)}1−𝑒^−𝑢𝑑𝑢=−^𝜉(𝑢0)_−∞𝐺₂(𝜂₂(𝜉))𝜂^′₂(𝜉)𝑒^{𝐿2(𝑧,𝜂2(𝜉))}𝑑𝜉=−^𝜉(𝑢0)_−∞𝐻₂(𝜉,𝜂₂(𝜉))𝑒^{−2𝑧𝜉−1/(2(1−𝜌2))(𝜉/(𝐵2𝜎2)+𝜇2/𝜎2−𝜌𝜇1/𝜎1)2}𝑑𝜉, (C.3) where 𝐺₂, 𝐿₂ and 𝐻₂ are defined by (B.8), (B.24), (B.29) respectively. Then we can repeat the above proof for the left tail (the third case), substituting the function 𝜂 by 𝜂₂, using the properties in (C.2), we conclude that 𝐼(𝑧)∼_∞𝜎₂2𝜋(1−𝜌²)𝑒^{𝐴1(𝑧)−(𝑧−𝜇1)2/(2𝜎21)}. (C.4) For the integral over (0,𝑢₀), we have 2(1−𝜌²)𝐴₃(𝑒^{−𝜂1(𝜉)})=[log(1−𝑒^{−𝜂1(𝜉)})−𝜇₁]²𝜎²₁−2𝜌[log(1−𝑒^{−𝜂1(𝜉)})−𝜇₁](−𝜂₁(𝜉)−𝜇₂)𝜎₁𝜎₂+(−𝜂₁(𝜉)−𝜇₂)²𝜎²₂=[𝜉−𝐵₂𝜂₁(𝜉)−𝐵₁𝜇₁𝐵₁𝜎₁]²+2𝜌(𝜂₁(𝜉)+𝜇₂)𝐵₁𝜎₁𝜎₂[𝜉−𝐵₂𝜂₁(𝜉)−𝐵₁𝜇₁]+(𝜂₁(𝜉)+𝜇₂)²𝜎²₂𝜉=(𝐵₁𝜎₁−𝜇₁𝜎₁+𝜌𝜇₂𝜎₂)²+𝐺₁(𝜉,𝜂₁𝜇(𝜉))+²²(1−𝜌²)𝜎²₂, (C.5) with 𝐺₁(𝜉,𝜂₁(𝜉))=−2(𝜉−𝐵₁𝜇₁)𝐵₂𝜂₁(𝜉)𝐵²₁𝜎²₁+𝐵²₂𝜂²₁(𝜉)𝐵²₁𝜎²₁+2𝜌𝜂₁(𝜉)𝐵₁𝜎₁𝜎₂[𝜉−𝐵₂𝜂₁(𝜉)−𝐵₁𝜇₁−𝐵₂𝜇₂𝜂]+²¹(𝜉)+2𝜂₁(𝜉)𝜇₂𝜎²₂. (C.6)

On the other hand, 𝐽(𝑧)=^𝑢0₀𝑒^{−2𝑧𝐴2(𝑒−𝑢)−𝐴3(𝑒−𝑢)}1−𝑒^−𝑢𝑑𝑢=^𝜉(𝑢0)_−∞𝐻₁(𝜉,𝜂₁(𝜉))𝑒^{−2𝑧𝜉−1/(2(1−𝜌2))(𝜉/(𝐵1𝜎1)−𝜇1/𝜎1+𝜌𝜇2/𝜎2)2}𝑑𝜉, (C.7) where 𝐻₁(𝜉,𝜂₁𝜂(𝜉))=^′1(𝜉)1−𝑒^{−𝜂1(𝜉)}exp[−𝐺₁(𝜉,𝜂₁(𝜉))2(1−𝜌²)−𝜇²₂2𝜎²₂]. (C.8)

Using the properties (C.1), lim_{𝜉→−∞}𝐻₁(𝜉,𝜂₁𝑒(𝜉))=^{−𝜇22/(2𝜎22)}𝐵₁. (C.9) Using Lemma A.3, we get the behavior of 𝐽 as 𝑧→∞, and a simplification leads to 𝐽(𝑧)∼_∞𝜎₁2𝜋(1−𝜌²)𝑒^{𝐴1(𝑧)−(𝑧−𝜇2)2/(2𝜎22)}. (C.10)

Since 2𝜋𝜎₁𝜎₂1−𝜌²𝑓_𝑍(𝑧)=𝑒^{−𝐴1(𝑧)}[𝐼(𝑧)+𝐽(𝑧)], the dominant term of 𝑓_𝑍(𝑧) is clearly given by 𝐽(𝑧) if 𝜎₁<𝜎₂, However when 𝜎₁=𝜎₂, we need to compare 𝜇₁ and 𝜇₂. Finally, 𝑓_𝑍(𝑧)∼_∞𝑁(𝜇,𝜎²)with(𝜎,𝜇)=max{(𝜎_𝑖,𝜇_𝑖)},if(𝜎₁,𝜇₁)≠(𝜎₂,𝜇₂);2𝑁(𝜇,𝜎²),if(𝜎₁,𝜇₁)=(𝜎₂,𝜇₂)=(𝜎,𝜇). (C.11)

C.2.

Case 2. 𝜌≥𝜎₁/𝜎₂. We always have 𝐵₁>0, but now 𝐵₂≥0. Thus, 𝜉^′(𝑢)>0 and 𝜉^′′(𝑢)<0 for 𝑢>0, so 𝜉 is a diffeomorphism from (0,∞) to ℝ. Denote 𝜂(𝜉) as the inverse function of 𝜉. 𝜂(𝜉) satisfies (C.1) and lim_𝜉→∞𝜂(𝜉)=∞,lim_𝜉→∞𝜂^′1(𝜉)=𝐵₂if𝐵₂>0,lim_𝜉→∞𝜂^′(𝜉)𝑒^𝜂(𝜉)=1−1𝐵₁if𝐵₂=0. (C.12)

Let 𝐻₁ and 𝐺₁ be defined in (C.8), (C.6) respectively. Replace 𝜂₁ by 𝜂. It is not difficult to prove that we always have lim_𝜉→∞𝐺₁(𝜉,𝜂(𝜉))=∞ and ^∞₀𝑒^{−2𝑧𝐴2(𝑒−𝑢)−𝐴3(𝑒−𝑢)}1−𝑒^−𝑢𝑑𝑢=_ℝ𝐻₁(𝜉,𝜂(𝜉))𝑒^{−2𝑧𝜉−1/(2(1−𝜌2))(𝜉/(𝐵1𝜎1)−𝜇1/𝜎1+𝜌𝜇2/𝜎2)2}𝑑𝜉. (C.13) Moreover, when 𝑧 tends to ∞, the behavior of 𝑓_𝑍(𝑧) is dominated by the integral over ℝ₋. By (C.1) and (C.12), 𝐻₁(𝜉,𝜂(𝜉)) is uniformly bounded in ℝ. Using Lemma A.3 again, as for 𝐽(𝑧) in Case 1, we come to the conclusion 𝑓_𝑍(𝑧)∼_∞𝑁(𝜇₂,𝜎²₂).

D. Quotient of Sum of Lognormal

Here is the proof for Theorem 2.8. Let 𝑍₁=log(𝑒^𝑋1+𝑒^𝑋2),𝑍₂=log(𝑒^𝑌1+𝑒^𝑌2), as before, 𝑓_𝑍1(𝑡)=𝑓_𝑍21(𝑡)=2𝜋𝜎²¹₀1(1−𝑢)𝑢exp[−(𝑡+log𝑢)²+(𝑡+log(1−𝑢))²2𝜎²]𝑑𝑢. (D.1) Denoting Ω₁=[0,1]×[0,1]×ℝ, we get 𝑓_𝑍(𝑧)=_ℝ𝑓_𝑍1(𝑡)𝑓_𝑍2=1(𝑡−𝑧)𝑑𝑡.4𝜋²𝜎⁴_Ω1𝑒^{−1/(2𝜎2)[(log𝑢+𝑡)2+(log(1−𝑢)+𝑡)2+(log𝑣+𝑡−𝑧)2+(log(1−𝑣)+𝑡−𝑧)2]}𝑢(1−𝑢)𝑣(1−𝑣)𝑑𝑢𝑑𝑣𝑑𝑡. (D.2) Direct calculus yields (log𝑢+𝑡)²+(log(1−𝑢)+𝑡)²+(log𝑣+𝑡−𝑧)²+(log(1−𝑣)+𝑡−𝑧)²=(2𝑡−𝑧+log[𝑢𝑣(1−𝑢)(1−𝑣)]2)²+𝑧²+𝑧log[𝑢(1−𝑢)𝑣(1−𝑣)]+𝑅(𝑢,𝑣), (D.3)with 𝑅(𝑢,𝑣)=log²𝑢+log²(1−𝑢)+log²𝑣+log²(1−𝑣)−log²[𝑢𝑣(1−𝑢)(1−𝑣)]4. (D.4) Thus, 4𝜋²𝜎⁴𝑓_𝑍(𝑧)=¹₀¹_0ℝ𝑒^{−1/(2𝜎2){(2𝑡−𝑧+log[𝑢𝑣(1−𝑢)(1−𝑣)]/2)2+𝑧2+𝑧log[𝑢(1−𝑢)/(𝑣(1−𝑣))]+𝑅(𝑢,𝑣)}}𝑢(1−𝑢)𝑣(1−𝑣)𝑑𝑢𝑑𝑣𝑑𝑡=¹₀¹_0ℝ𝑒^{−1/(2𝜎2){4𝑡2+𝑧2+𝑧log[𝑢(1−𝑢)/(𝑣(1−𝑣))]+𝑅(𝑢,𝑣)}}=𝜎√𝑢(1−𝑢)𝑣(1−𝑣)𝑑𝑢𝑑𝑣𝑑𝑡𝜋𝑒^{−𝑧2/(2𝜎2)}√2¹⁰¹⁰𝑒^{−1/(2𝜎2){𝑧log[𝑢(1−𝑢)/(𝑣(1−𝑣))]+𝑅(𝑢,𝑣)}}√𝑢(1−𝑢)𝑣(1−𝑣)𝑑𝑢𝑑𝑣=22𝜋𝜎𝑒^{−𝑧2/(2𝜎2)}₀^1/2₀^1/2𝑒^{−1/(2𝜎2){𝑧log[𝑢(1−𝑢)/(𝑣(1−𝑣))]+𝑅(𝑢,𝑣)}}𝑢(1−𝑢)𝑣(1−𝑣)𝑑𝑢𝑑𝑣. (D.5) The last equality is obtained by the symmetry of our integrand between 𝑢, 𝑣 and 1−𝑢, 1−𝑣. Since 𝜁(𝑢)=log[𝑢(1−𝑢)] is increasing in (0,1/2], clearly for any 𝑧<0, 𝐼₁(𝑧)=₀^1/2_𝑢^1/2𝑒^{−1/(2𝜎2){𝑧log[𝑢(1−𝑢)/(𝑣(1−𝑣))]+𝑅(𝑢,𝑣)}}𝑢(1−𝑢)(𝑣(1−𝑣))𝑑𝑢𝑑𝑣≤₀^1/2_𝑢^1/2𝑒^{−𝑅(𝑢,𝑣)/(2𝜎2)}𝑢(1−𝑢)(𝑣(1−𝑣))𝑑𝑢𝑑𝑣=𝑎₀<∞ (D.6) Denote by 𝛾∶(−∞,−log4]→(0,1/2] the inverse function of 𝜁, then √𝛾(𝑠)=1−1−4𝑒^𝑠2,𝛾^′𝑒(𝑠)=^𝑠√1−4𝑒^𝑠. (D.7) Moreover, by change of variables 𝑢=𝛾(𝑠), 𝑣=𝛾(𝑡), 𝐽₁(𝑧)=₀^1/2𝑢₀𝑒^{−1/(2𝜎2){𝑧log[𝑢(1−𝑢)/(𝑣(1−𝑣))]+𝑅(𝑢,𝑣)}}𝑢(1−𝑢)𝑣(1−𝑣)𝑑𝑢𝑑𝑣=^𝑠0_−∞^𝑠_−∞𝑒^{−1/(2𝜎2)[𝑧(𝑠−𝑡)+(3𝑠2+3𝑡2−2𝑠𝑡)/4−2log(𝛾(𝑠))log(1−𝛾(𝑠))−2log(𝛾(𝑡))log(1−𝛾(𝑡))]}√1−4𝑒^𝑠√1−4𝑒^𝑡𝑑𝑠𝑑𝑡, (D.8) where 𝑠₀=−log4. Making a new change of variables 𝑥=−𝑠+𝑠₀, 𝑦=𝑠−𝑡, we then get 𝐽₁(𝑧)=^∞₀𝑒^{−1/(2𝜎2)(𝑧𝑦+𝑦2/2)∞}₀𝑒^{−(𝑥−𝑠0+𝑦/2)2/(2𝜎2)}𝐻(𝑥)𝐻(𝑥+𝑦)𝑑𝑥𝑑𝑦, (D.9) with 1𝐻(𝑥)=√1−𝑒^−𝑥1exp[𝜎²√log(1−1−𝑒^−𝑥2√)log(1+1−𝑒^−𝑥2)]. (D.10) Obviously, lim_𝑥→0+𝐻(𝑥)√𝑥𝑒^{−(log2)2/𝜎2}=lim_𝑥→∞𝐻(𝑥)=1. (D.11)

Consider 𝑃(𝑦)=^∞₀𝑒^{−(𝑥+𝑦/2−𝑠0)2/(2𝜎2)}𝐻(𝑥)𝐻(𝑥+𝑦)𝑑𝑥, (D.12) we claim that lim_𝑦→∞√𝑦√𝑒2𝜋𝜎^{((𝑦−2𝑠0)2−8(log2)2)/(8𝜎2)}𝑃(𝑦)=1. (D.13)

Indeed, for any 𝑦>0√𝑦𝑒^{((𝑦−2𝑠0)2−8(log2)2)/(8𝜎2)}=√𝑃(𝑦)𝑦𝑒^{−(log2)2/𝜎2∞}₀𝑒^{−(𝑥2+(𝑦−2𝑠0)𝑥)/(2𝜎2)}=√𝐻(𝑥)𝐻(𝑥+𝑦)𝑑𝑥𝑦𝑒^{−(log2)2/𝜎2}𝑦−2𝑠₀^∞₀𝑒^{−1/(2𝜎2)[𝑢+𝑢2/(𝑦−2𝑠0)2]}√𝑢𝑢𝐻(𝑦−2𝑠₀+𝑦)×𝑢𝑦−2𝑠₀𝑢𝐻(𝑦−2𝑠₀)𝑑𝑢. (D.14) We use 𝑥=𝑢/(𝑦−2𝑠₀) for the last equality. Using dominated convergence and (D.11), the claim (D.13) is obtained immediately. Defining 𝑄(𝑦)=√𝑦𝑒^{(𝑦−2𝑠0)2/(8𝜎2)}𝑃(𝑦), we finally get 𝐽₁(𝑧)=^∞₀𝑒^{−1/(2𝜎2)[𝑧𝑦+𝑦2/2+(𝑦/2−𝑠0)2]}√𝑦𝑄(𝑦)𝑑𝑦=𝑒^{−𝑠20/(2𝜎2)∞}₀𝑒^{−1/(2𝜎2)(𝑧𝑦−𝑠0𝑦+3𝑦2/4)}√𝑦𝑄(𝑦)𝑑𝑦. (D.15) As 𝑧→−∞, we need to understand the behavior of 𝑊(𝑏) when 𝑏 tends to ∞. More precisely, for 𝜆>0, 𝑊(𝑏)=^∞₀𝑒^{𝑏𝑦−𝜆𝑦2}√𝑦𝑄(𝑦)𝑑𝑦=2√𝑏^∞₀𝑒^{𝑏2(𝑥2−𝜆𝑥4)}𝑄(𝑏𝑥²)𝑑𝑥. (D.16) Here, 𝐺(𝑥)=𝑥²−𝜆𝑥⁴ is not concave in [0,∞). But notice that 𝐺 has a unique global maximum point at 𝑥₀=1/√2𝜆 in [0,∞) and 𝐺^′′(𝑥₀)=−4. So the spirit of the Laplace method works. This implies 𝑊(𝑏)∼_∞2𝜋𝑏𝑒^{𝑏2/(4𝜆)}×lim_𝑦→∞𝑄(𝑦). (D.17) Hence by (D.13), 𝐽₁(𝑧)∼_−∞2√2𝜋𝜎²√𝑒−𝑧^{(𝑧−𝑠0)2/(6𝜎2)−(log2)2/𝜎2}. (D.18) Finally, using 𝑓_𝑍𝑒(𝑧)=^{−𝑧2/(2𝜎2)}√2𝜋³𝜎³[𝐼₁(𝑧)+𝐽₁(𝑧)], (D.19) we conclude that 𝑓_𝑍(𝑧)∼_−∞𝑟(𝑧) where 𝑟(𝑧)=2𝑒^{−𝑧2/(2𝜎2)+(|𝑧|+𝑠0)2/(6𝜎2)−(log2)2/𝜎2}𝜎=𝜋|𝑧|2𝑒^{−(|𝑧|+log2)2/(3𝜎2)}𝜎𝜋|𝑧|. (D.20) The proof is complete, since 𝑓_𝑍 is an even function.

E. A Higher Dimension Left Tail Result

We will prove Theorem 2.6 in this section. Let (𝑋₁,𝑋₂,…,𝑋_𝑛)_𝑛≥3 be a joint normal variable with distribution 𝑁(𝜇,Σ), where 𝜇 is the mean vector (𝜇₁,…,𝜇_𝑛), and Σ is a non-singular covariance matrix. Let 𝑍=log(∑_{1≤𝑖≤𝑛}𝑒^𝑋𝑖). We wish to study the left tail of 𝑓_𝑍(𝑧), and the PDF of 𝑧. Let 𝑓 be the PDF of (𝑋₁,𝑋₂,…,𝑋_𝑛), that is, 𝑓(𝑥₁,…,𝑥_𝑛1)=(2𝜋)^𝑛|detΣ|exp[−(𝑥−𝜇)Σ⁻¹(𝑥−𝜇)^𝑡2]. (E.1) Let 𝐌=(𝑚_𝑖𝑗)=Σ⁻¹. By change of variables, it is clear that 𝑓_𝑍1(𝑧)=(2𝜋)^𝑛|detΣ|^Ω₀11−∑_{2≤𝑖≤𝑛}𝑒^𝑢𝑖∑exp(−^{1≤𝑖,𝑗≤𝑛}𝑚_𝑖𝑗𝐴_𝑖𝐴_𝑗2)𝑑𝑢₂…𝑑𝑢_𝑛, (E.2) where Ω₀={(𝑢_𝑖)_{2≤𝑖≤𝑛}∈ℝ^𝑛−1,∑_{2≤𝑖≤𝑛}𝑒^𝑢𝑖<1}, and 𝐴∈ℝ^𝑛 is a vector defined by 𝐴_𝑖=𝑧+𝐶_𝑖−𝜇_𝑖 with 𝐶 the vector given by (2.18). Consequently, _{1≤𝑖,𝑗≤𝑛}𝑚_𝑖𝑗𝐴_𝑖𝐴_𝑗=_{1≤𝑖,𝑗≤𝑛}𝑚_𝑖𝑗𝐶_𝑖(𝐶_𝑗−2𝜇_𝑗)+2𝑧𝐹(𝑢)+_{1≤𝑖,𝑗≤𝑛}𝑚_𝑖𝑗(𝑧−𝜇_𝑖)(𝑧−𝜇_𝑗), (E.3) with 𝐹 in Theorem 2.6. Therefore, if 𝐹(𝑢) has only one saddle point, we can expect to apply the Laplace method to the higher dimensional case. One necessary and sufficient condition to ensure that is summarized in the following lemma.

Lemma E.1. The function 𝐹(𝑢₂,…,𝑢_𝑛)=∑_{1≤𝑖≤𝑛}𝑚_i𝐶_𝑖 has a unique critical point within the set Ω₀, if and only if for any1≤𝑘≤𝑛, 𝑚_𝑘>0. Moreover, the Hessian of 𝐹, 𝐇_𝐹=(𝜕²_𝑖𝑗𝐹)_{2≤𝑖,𝑗≤𝑛} is negative definite over Ω₀ when 𝑚₁>0.

Indeed, it is easy to see that 𝜕𝐹𝜕𝑢_𝑘=𝑚_𝑘−𝑚₁𝑒^𝑢𝑘1−𝑆₁,∀2≤𝑘≤𝑛, (E.4) where 𝑆₁=∑_{2≤𝑖≤𝑛}𝑒^𝑢𝑖. Since 𝑚_𝑖≢0, the solution to the system ∇𝐹=0 would satisfy 𝑒^𝑢𝑘=𝑚_𝑘𝑚⁻¹ for all 2≤𝑘≤𝑛. Thus a solution exists in Ω₀ if and only if all 𝑚_𝑖’s are positive. As 𝜕²𝐹𝜕𝑢_𝑘𝜕𝑢_𝑗𝑚=−¹1−𝑆₁[𝑒^𝑢𝑘𝛿_𝑘𝑗+𝑒^𝑢𝑘𝑒^𝑢𝑗1−𝑆₁], (E.5) with 𝛿_𝑘𝑗 the Kronecker symbol, for any 𝐯=(𝑣_𝑖)∈ℝ^𝑛−1, ⟨𝐯,𝐇_𝐹𝑚𝐯⟩=−¹1−𝑆₁(_{2≤𝑘≤𝑛}𝑒^𝑢𝑘𝑣²_𝑘+_{2≤𝑘,𝑗≤𝑛}𝑒^𝑢𝑘𝑒^𝑢𝑗𝑣_𝑘𝑣_𝑗1−𝑆₁𝑚)=−¹1−𝑆₁[_{2≤𝑘≤𝑛}𝑒^𝑢𝑘𝑣²_𝑘+(∑_{2≤𝑘≤𝑛}𝑒^𝑢𝑘𝑣_𝑘)²1−𝑆₁]. (E.6) It is easy to conclude as long as 𝑚₁>0 and 𝑆₁<1. The following lemma is our key argument.

Lemma E.2. Let Ω⊂ℝ^𝑛 be a convex domain. Let 𝐻,𝐺 be two continuous functions defined on Ω. We further assume 𝐻is a positive integrable function over Ω; 𝐺 is 𝐶², strictly concave and 𝐺 has a unique critical point 𝑥₀ in Ω. Then lim_𝑧→∞Ω𝐻(𝑥)𝑒^{𝑧𝐺(𝑥)}𝑑𝑥×𝑧^𝑛|det𝐇_𝐺(𝑥₀)|𝐻(𝑥₀)𝑒^{𝑧𝐺(𝑥0)}(2𝜋)^𝑛/2=1 (E.7) where 𝐇_𝐺 denotes the Hessian matrix of 𝐺.

This lemma is an extension to higher dimension of Lemma A.1 and the proof is very similar, so we leave the details to interested readers. Returning to our proof, since all 𝑚_𝑖’s are positive, we can verify that

(i)Ω₀ is convex, since ∑_{2≤𝑖≤𝑛}𝑒^𝑢𝑖 is a convex function.(ii)The function 𝐹 has only one critical point 𝑈₀=log(𝑚_𝑘/𝑚)∈Ω₀ and 𝐹(𝑈₀)=∑_{1≤𝑘≤𝑛}𝑚_𝑘log(𝑚_𝑘/𝑚).(iii)𝐇_𝐹(𝑢₂,…,𝑢_𝑛) is negative definite for any point (𝑢₂,…,𝑢_𝑛)∈Ω₀.(iv) The function 𝐻 is clearly positive in Ω₀. The integrability of 𝐻 over Ω₀ is ensured by the fact that the matrix 𝐌 is positive definite. Because |𝐶|²=log²(1−𝑆₁)+∑_{2≤𝑖≤𝑛}𝑢²_𝑖, 𝐻 is bounded over Ω₀. Finally, as 𝑓_𝑍1(𝑧)=(2𝜋)^𝑛𝑒|detΣ|^{−(∑1≤𝑖,𝑗≤𝑛(𝑧−𝜇𝑖)𝑚𝑖𝑗(𝑧−𝜇𝑗))/2}×_Ω0𝐻(𝑢)𝑒^{−𝑧𝐹(𝑢)}𝑑𝑢₂…𝑑𝑢_𝑛, (E.8) a straightforward application of Lemma E.2 results in Theorem 2.6.

F. A Higher Dimension Right Tail Result

We will prove Theorem 2.7 by induction on the number 𝑛. It is trivially true for 𝑛=1, since 𝑍=𝑋₁. Suppose that the result holds for 𝑍=log(∑^𝑛_𝑖=1𝑒^𝑋𝑖). Consider now 𝑍₁=log(^𝑛+1_𝑖=1𝑒^𝑋𝑖)=log(𝑒^𝑍+𝑒^𝑋𝑛+1), (F.1) where (𝑋₁,…,𝑋_𝑛+1) is a normally distributed random vector whose covariance matrix is diagonal. Without loss of generality, we can assume that the sequence (𝜎_𝑖,𝜇_𝑖)_{1≤𝑖≤𝑛+1} is nonincreasing. There exist three possible cases:

(i)𝜎_𝑛+1<𝜎=𝜎₁;(ii)𝜎_𝑛+1=𝜎but𝜇_𝑛+1<𝜇=𝜇₁;(iii)(𝜎_𝑛+1,𝜇_𝑛+1)=(𝜎,𝜇).

By translation, we can assume that 𝜇=0.

Let us denote the PDF of (𝑒^𝑍+𝑒^𝑋𝑛+1) by 𝑔, then 𝑓_𝑍1(𝑧)=𝑒^𝑧𝑔(𝑒^𝑧) is the PDF of 𝑍₁. Consider the asymptotic behavior of 𝑔(𝑧) as 𝑧→∞. First, 𝑔(𝑧)=^𝑧₀𝑓_𝑍(log𝑢)𝑢×1√2𝜋𝜎_𝑛+1𝑒(𝑧−𝑢)^{−(log(𝑧−𝑢)−𝜇𝑛+1)2/(2𝜎2𝑛+1)}𝑑𝑢. (F.2) For any 𝐴>0 fixed (to be chosen later), as the function 1ℎ(𝑡)=√2𝜋𝜎_𝑛+1𝑡𝑒^{−(log𝑡−𝜇𝑛+1)2/(2𝜎2𝑛+1)} (F.3) is decreasing near +∞, there exists 𝑧₁>𝐴 such that ℎ(𝑧−𝑢)≤ℎ(𝑧−𝐴) on [0,𝐴] for all 𝑧≥𝑧₁. Hence for such 𝑧, 𝐼_𝐴(𝑧)=^𝐴₀𝑓_𝑍(log𝑢)𝑢ℎ(𝑧−𝑢)𝑑𝑢≤ℎ(𝑧−𝐴)×^𝐴₀𝑓_𝑍(log𝑢)𝑢𝑑𝑢=ℎ(𝑧−𝐴)×^log𝐴_−∞𝑓_𝑍(𝑣)𝑑𝑣≤ℎ(𝑧−𝐴). (F.4) Noting that [log(𝑧−𝐴)−𝜇_𝑛+1]²2𝜎²_𝑛+1=[log𝑧+log(1−𝐴/𝑧)−𝜇_𝑛+1]²2𝜎²_𝑛+1∼_∞(log𝑧)²2𝜎²_𝑛+1, (F.5) for the case (i), we immediately see 𝐼_𝐴1(𝑧)≤ℎ(𝑧−𝐴)=𝑜(𝑧𝑒^{−(log𝑧)2/(2𝜎2)})𝑎𝑠𝑧→∞. (F.6) For the case (ii), since 𝜇_𝑛+1<𝜇=0, there exists 𝑧₂>𝑧₁ such that for all 𝑧≥𝑧₂, 𝐴log𝑧+log(1−𝑧)−𝜇_𝑛+1𝜇≥log𝑧−^𝑛+12. (F.7) With such 𝑧, we have ℎ(𝑧−𝐴)×𝑧𝑒^{(log𝑧)2/(2𝜎2)}≤1√2𝜋𝜎_𝑛+1×𝑧𝑒𝑧−𝐴^{(4𝜇𝑛+1log𝑧−𝜇2𝑛+1)/(8𝜎2)}. (F.8) Recall that 𝜇_𝑛+1<0. So the estimate (F.6) still holds. Consider now 𝐽_𝐴=𝑔(𝑧)−𝐼_𝐴. For any 𝜖>0, by hypothesis of induction, we can fix 𝐴>0 large enough such that |𝑓_𝑍(𝑧)𝑚_𝑛𝑁(0,𝜎²)(𝑧)−1|≤𝜖in[log𝐴,∞). (F.9) Consequently |𝐽_𝐴/𝐾_𝐴−1|≤𝜖 for 𝑧≥𝐴 where 𝐾_𝐴(𝑧)=^𝑧_𝐴𝑚_𝑛𝑁(0,𝜎²)(log𝑢)𝑢×ℎ(𝑧−𝑢)𝑑𝑢. (F.10) Using exactly the same proof as for Theorem 2.4 (the case 𝜌<𝜎₁/𝜎₂), we get that 𝐾_𝐴(𝑧)∼_∞𝑚_𝑛𝑁(0,𝜎²)(log𝑧)𝑧. (F.11) Finally, 𝑚_𝑛(1+𝜖)≥limsup_𝑧→∞𝐽_𝐴(𝑧)×√2𝜋𝜎𝑧𝑒^{(log𝑧)2/(2𝜎2)}≥liminf_𝑧→∞𝐽_𝐴(𝑧)×√2𝜋𝜎𝑧𝑒^{(log𝑧)2/(2𝜎2)}≥𝑚_𝑛(1−𝜖) (F.12)

(F.6) implies that 𝐼_𝐴=𝑜(𝐽_𝐴) when 𝑧 tends to ∞. As 𝑔(𝑧)=𝐼_𝐴+𝐽_𝐴, we conclude 𝑚_𝑛(1+𝜖)≥limsup_𝑧→∞𝑔(𝑧)×√2𝜋𝜎𝑧𝑒^{(log𝑧)2/(2𝜎2)}≥liminf_𝑧→∞𝑔(𝑧)×√2𝜋𝜎𝑧𝑒^{(log𝑧)2/(2𝜎2)}≥𝑚_𝑛(1−𝜖) (F.13) As 𝜖 is arbitrary, the proof is completed for the cases (i) and (ii) because 𝑚_𝑛+1=𝑚_𝑛.

We now consider the case (iii). First we observe that (iii) simply means that (𝑋_𝑖)_{1≤𝑖≤𝑛+1} are i.i.d. following 𝑁(0,𝜎²) so that 𝑚_𝑛=𝑛 and 𝑚_𝑛+1=𝑛+1. The estimate (F.12) is always true for any 𝐴>0 satisfying (F.9). However, we need to estimate 𝐼_𝐴 differently. For any 𝜖>0, we fix 𝐴 such that (F.9) holds and ^𝐴₀𝑓_𝑍(log𝑢)𝑢=^log𝐴_−∞𝑓_𝑍(𝑣)𝑑𝑣∈(1−𝜖,1). (F.14) Because in case (iii), 𝐼_𝐴×√2𝜋𝜎𝑧𝑒^{(log𝑧)2/(2𝜎2)}−^𝐴₀𝑓_𝑍(log𝑢)𝑢𝑑𝑢=^𝐴₀𝑓_𝑍(log𝑢)𝑢𝑅(𝑢)𝑑𝑢, (F.15) with 𝑧𝑅(𝑢)=𝑒𝑧−𝑢^{−1/(2𝜎2)[2log𝑧log(1−𝑢/𝑧)+log2(1−𝑢/𝑧)]}−1, (F.16) which converges uniformly to 0 in [0,𝐴], we obtain 1≥limsup_𝑧→∞𝐼_𝐴(𝑧)×√2𝜋𝜎𝑧𝑒^{(log𝑧)2/(2𝜎2)}≥liminf_𝑧→∞𝐼_𝐴(𝑧)×√2𝜋𝜎𝑧𝑒^{(log𝑧)2/(2𝜎2)}≥1−𝜖. (F.17)

Combining with (F.12), it is easy to conclude.

Acknowlegment

The authors would like to thank the Editor, the anonymous referee for their valuable comments and suggestions.