# Asymptotic Behavior of Tail Density for Sum of Correlated Lognormal Variables

Authors: Xin Gao; Hong Xu; Dong Ye
Journal: International Journal of Mathematics and Mathematical Sciences
 (2009)

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## 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.


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## Body

## 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+⋯+Xn)/(Y1+⋯+Ym) where all the RVs X1,…,Xn,Y1,…,Ym 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, Theorems2.6 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(∑i=1nXi) and Z2=log(∑j=1mYj), then
(1.1)Pr(SIR<γ0)=Pr(Z1-Z2<log(γ0))≃∫-∞log(γ0)∫ℝfZ1approx(x-y)fZ2approx(-y)dydx.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 variableZ=log(eX1+eX2), 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(∑k=1NeXk) where (X1,…,XN) are jointly distributed as N(μ,Σ). Note here we do not assume that the Xk are independent (except for Theorem 2.7), nor do we assume that X1,…,XN 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 functionsgL,gR such that
(1.2)limz→-∞fZ(z)gL(z)=1,limz→+∞fZ(z)gR(z)=1.
The explicit forms of gL and gR 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 limz→-∞g*(z)/fZ(z) does not exist or limz→-∞g*(z)/fZ(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 Theorem2.7 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
(1.3)limz→+∞E[h(eZ)1Z>z]∫z∞h(ex)gR(x)dx=1,
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 μi=E(Xi), σi2=Var(Xi), for i=1,2. Then the joint PDF  of (eX1,eX2) is given by
(2.1)f(u,v)≡h(u,v)2πσ1σ21-ρ2uv,
where
(2.2)h(u,v)=exp{-12(1-ρ2)[(log(u)-μ1σ1)2-2ρ(log(u)-μ1)(log(v)-μ2)σ1σ2+(log(v)-μ2σ2)2]}.
We wish to study the left and right tail probabilities of Z=log(eX1+eX2), which has PDF as fZ(z)≡ezg(ez) with
(2.3)g(z)≡∫0zf(y,z-y)dy.
We hope to understand the asymptotic behavior of g(ez) when z→±∞. Direct calculus yields
(2.4)g(ez)=e-z2πσ1σ21-ρ2∫01e-A(z,t)(1-t)tdt,
where the exponent A(z,t)=(2.5)12(1-ρ2)[(log(1-t)+z-μ1σ1)2+(logt+z-μ2σ2)2-2ρlog(1-t)+z-μ1σ1logt+z-μ2σ2].
Rewrite A in three terms A(t)=A1(z)+2zA2(t)+A3(t), where the Ai are defined as
(2.6)A1(z)=12(1-ρ2){z2(1σ12-2ρσ1σ2+1σ22)+2z[(ρσ1-1σ2)μ2σ2+(ρσ2-1σ1)μ1σ1]},A2(t)=12(1-ρ2)[(1σ1-ρσ2)log(1-t)σ1+(1σ2-ρσ1)logtσ2],A3(t)=12(1-ρ2)[(log(1-t)-μ1)2σ12-2ρ(log(1-t)-μ1)(logt-μ2)σ1σ2+(logt-μ2)2σ22].Hence, by changing the variablet to e-u,
(2.7)fZ(z)=e-A1(z)2πσ1σ21-ρ2∫0∞e-2zA2(e-u)-A3(e-u)1-e-udu.We regroup the integrand in (2.7) in the form
(2.8)∫0∞e-2zA2(e-u)-A3(e-u)1-e-udu=∫0∞H(u)e-zG(u)du,with(2.9)G(u)=11-ρ2[(σ2-σ1ρ)log(1-e-u)σ12σ2-(σ1-σ2ρ)uσ22σ1],H(u)=e-A3(e-u)1-e-u.Remark 2.1.
Without loss of generality, in this paper, we always assume(2.10)0<σ1≤σ2,|ρ|<1.
We also use the following notation.Definition 2.2.
We say that two functionsf and h are equivalent near some pointa∈ℝ¯,  denoted by f(z)~ah(z), if we havelimz→af(z)/h(z)=1.           For the left tail, we have the following result.Theorem 2.3.
LetXi~N(μi,σi2) be defined as above for i=1,2 and ρ∈(-1,1) be the correlation coefficient. Let fZ be the PDF of Z=log(eX1+eX2), then fZ(z)~-∞hL(z) as follows.(i)
Ifρ<σ1/σ2, one has
(2.11)hL(z)=H(u0)e-zG(u0)-A1(z)2πz(1-ρ2)G′′(u0)σ1σ2withu0=logσ12+σ22-2ρσ1σ2σ12-ρσ1σ2.
Here the functions G, H, A1 are defined by (2.9) and (2.6).(ii)
Ifρ=σ1/σ2, one has
(2.12)hL(z)=N(μ1,σ12)log(-z)e1/(2(σ22-σ12)){-(logzρ-log(logzρ)+μ2-μ1)2-2logzρ}withzρ=-z(ρ-2-1).(iii)
Ifρ>σ1/σ2, one has
(2.13)hL(z)=12πσ12e-(z-μ1)2/(2σ12).In particular, whenρ=0, we find that fZ~-∞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 fZ 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.
LetZ and f be defined as above, then fZ(z)~∞hR(z), where hR(z) is defined  as follows.(i)
If(σ1,μ1)≠(σ2,μ2), one has
(2.14)hR(z)=N(μ,σ2)with(σ,μ)=max{(σ1,μ1),(σ2,μ2)}.(ii)
If(σ1,μ1)=(σ2,μ2)=(σ,μ), one has  hR(z)=2/(σ2π)e-(z-μ)2/(2σ2).The following corollary is an immediate consequence of Theorems2.3 and 2.4.Corollary 2.5.
LetfV be the PDF of V=eX1+eX2, where X1,X2 are i.i.d.  RVs following N(0,σ2) distributions. Then, one has
(2.15)limz→0fV(z)fVL(z)=1,wherefVL(z)=exp[-(logz-log2)2/σ2]zσ-πlogz;limz→+∞fV(z)fVR(z)=1,wherefVR(z)=2exp[-log2z/(2σ2)]πσz.       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(Xi)1≤i≤n (n≥3) be a joint normally distributed random variable with distribution N(μ,Σ). Let M=Σ-1. Let fZ be the  PDF  of random variable Z=log(∑1≤i≤neXi). If M=(mij) satisfies mk=∑1≤j≤nmkj>0 for all k=1,…,n, then the left tail of fZ satisfies
(2.16)fZ(z)~-∞H1(U0)2π|z|n-1|detΣ||detHF(U0)|exp[-z∑k=1nmklogmkm-12∑i,j=1n(z-μi)mij(z-μj)]
where m=∑1≤j≤nmj , F(u2,…,un)=∑1≤i≤nmiCi, HF=(∂ij2F) denotes the Hessian matrix of F. Here,
(2.17)H1(u2,…,un)≡e-〈C,M(C-2μ)〉/21-∑2≤i≤neui,U0=(logmkm)2≤k≤n∈ℝn-1,
and C=(Ci)∈ℝn   is given by
(2.18)C=(log(1-∑2≤k≤neuk),u2,u3,…,un).Theorem 2.7.
Let(X1,…,Xn) be independent normally distributed RVs, that is, Xi~N(μi,σi2). Let Z=log(∑i=1neXi). Define (σ,μ)=max{(σi,μi),1≤i≤n} for the lexicographical order and mn the number of maximum points, that is mn=#{1≤i≤n,s.t.(σi,μi)=(σ,μ)}. Then the PDF of Z, fZ(x) satisfies
(2.19)limx→∞fZ(x)N(μ,σ2)(x)=mnwhereN(μ,σ2)(x)=12πσe-(x-μ)2/(2σ2).      Finally, we show a result for the quotient of sums of i.i.d lognormal variables.Theorem 2.8.
LetX1,X2,Y1,Y2 be i.i.d random variables. Each of them follows N(0,σ2) distribution. Let W=log((eX1+eX2)/(eY1+eY2)) and fW(w) be the   PDF  of W, then
(2.20)fW(w)~±∞2e-(|w|+log2)2/(3σ2)σπ|w|.     This result can be generalized to the case where X1,X2 follow N(0,σx2),Y1,Y2 follow N(0,σy2), with any positive constants σx and σy. Indeed, we can prove that
(2.21)fW(w)~-∞2σx-πwe-(w-log2)2/(σx2+2σy2),fW(w)~+∞2σyπwe-(w+log2)2/(σy2+2σx2).
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 theZ 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 byW), we used (σx,σy)=(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 fW.


## 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 LemmasA.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 tailfZ, ρ<σ1/σ2. g_1 is the approximation.Figure 2
Left tailfZ,ρ=σ1/σ2. g_2 is the approximation.Figure 3
Left tailfZ,ρ>σ1/σ2. g_3 is the approximation.Figure 4
Right tailfZ,ρ<σ1/σ2. d_1 is the approximation.Figure 5
Right tailfZ,ρ>σ1/σ2. d_2 is the approximation.Figure 6
Left tailfW,(σx,σy)=(12,9.6). r_L is the approximation.



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