Asymptotic Behavior of Tail Density for Sum of Correlated Lognormal Variables

Recommended by Michael Evans 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.


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 2 International Journal of Mathematics and Mathematical Sciences 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 X 1 · · · X n / Y 1 · · · Y m where all the RVs X 1 , . . . , X n , Y 1 , . . . , 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 P 0 Pr SIR < γ 0 such that BER γ 0 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 < γ 0 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 Z 1 log n i 1 X i and Z 2 log m j 1 Y j , then 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. In this paper, we will rigorously characterize the right and left tails behavior of a PDF for a random variable Z log e X 1 e X 2 , where X 1 , X 2 are jointly distributed with N μ, Σ International Journal of Mathematics and Mathematical Sciences 3 as distribution. This is our first step towards understanding the more general problem: the characterization of the PDF of Z log N k 1 e X k where X 1 , . . . , 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 X 1 , . . . , 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 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 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.
where the exponent A z, t where the A i are defined as

2.6
International Journal of Mathematics and Mathematical Sciences 5 Hence, by changing the variable t to e −u , We regroup the integrand in 2.7 in the form

2.9
Remark 2.1. Without loss of generality, in this paper, we always assume We also use the following notation.
Definition 2.2. We say that two functions f and h are equivalent near some point a ∈ R, denoted by f z ∼ a h z , if we have lim z → a f z /h z 1.
For the left tail, we have the following result.

2.11
Here the functions G, H, A 1 are defined by 2.9 and 2.6 .

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iii In particular, when ρ 0, we find that f Z ∼ −∞ C 1 e −C 2 z−C 3 z 2 / √ 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 .
The following corollary is an immediate consequence of Theorems 2.3 and 2.4.

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.
and C C i ∈ R n is given by Theorem 2.7. Let X 1 , . . . , X n be independent normally distributed RVs, that is, Finally, we show a result for the quotient of sums of i.i.d lognormal variables.
This result can be generalized to the case where X 1 , X 2 follow N 0, σ 2 x , Y 1 , Y 2 follow N 0, σ 2 y , with any positive constants σ x and σ y . Indeed, we can prove that For the sake of brevity, we only present the proof for the special case where both variances are equal.

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 σ 1 /σ 2 is 0.8.  Figure 1: Left tail f Z , ρ < σ 1 /σ 2 . g 1 is the approximation. 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 .

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.

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 ⊂ 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 x 0 z be a nonnegative function defined on R, and lim z → z 0 x 0 z ∞. Assume that f z, t is a nontrivial, nonnegative function such that for any fixed z near z 0 ∈ R, one has f z, t ∈ L 1 R and for any > 0, If moreover G is a bounded uniformly continuous function over R, such that lim inf x → ∞ G x > 0, then In fact, we need a special case of lemma 16. When lim z → ∞ G x G 0 / 0 exists, we can simply require that G is a continuous function and replace the term G x 0 z in A.3 by G 0 .

Lemma A.3. Let G be a bounded measurable function defined on
A.4 Using usual developments, we also have the following asymptotic expansion.

B.1.
Case 1. ρ < σ 1 /σ 2 . Using the formulas 2.7 and 2.8 , we need only to understand the behavior of Here G and H are defined by 2.9 . Since Thus, G u 0 has a unique solution We also have G u < 0 in R and H > 0, integrable over R . Hence, Lemma A.1 allows us to conclude

B.2.
Case 2. ρ σ 1 /σ 2 . Here we can simplify A 1 and A 2 as International Journal of Mathematics and Mathematical Sciences Thus, G 2 is uniformly bounded and uniformly continuous over R . Furthermore, we have e u e u 1 e u − 1 3 .

B.10
If z < 0, ∂ 2 u L z, u < 0, ∂ 3 u L z, u > 0 for any u ∈ R . Let u z be the unique solution of ∂ u L z, u 0. Obviously, u z satisfies the equation e u z − 1 u z μ 2 − μ 1 − ρ −2 − 1 z. According to Lemma A.4, we obtain lim z → −∞ u z lim z → −∞ − ze −u z ∞ and The situation here is more delicate than in Lemma A.1, but we can follow the same idea. For any ∈ 0, 1 fixed, there exists R > 0 such that |G 2 x /G 0 − 1| ≤ , ∂ 2 u L z, x < 0 for any x > R. Choosing now z near −∞ such that u z > R 1 and

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We will decompose the integral into three parts: First we consider the integral of e L z,u G 2 u over u z − , u z . Using Taylor expansion and the monotonicity of ∂ 2 u L, we get By B.12 , for > 0 small enough, Using L z, u z , we also have for small > 0 lim sup Consider now the integral of e L z,u G 2 u on u z , ∞ . Since L z, u is strictly concave in u, Moreover, ∂ u L z, u z ≥ ∂ 2 u L z, u z , L z, u z < L z, u z , so we get Similarly, Combining all these estimates, we deduce International Journal of Mathematics and Mathematical Sciences

B.3.
with two positive constants Thus, with G 2 given by B.8 and

B.24
Notice that ξ is C ∞ diffeomorphism from 0, ∞ into R, with ξ u B 1 e u − 1 −1 B 2 > 0 in 0, ∞ . Let η be the inverse function of ξ, namely u η ξ . We have the following properties: The change of variable u η ξ yields For Furthermore, where is a bounded function in R by properties of η and G 2 . Otherwise, using B.9 and B.25 Applying Lemma A.3, we get 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 < σ 1 /σ 2 and ρ ≥ σ 1 /σ 2 . 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 < σ 1 /σ 2 . We have A 2 e −u ξ u B 1 log 1−e −u B 2 u with B i given by B.22 . Since B 1 > 0 and B 2 < 0, it's clear that −ξ u B 1 e u e u − 1 −2 > 0 in 0, ∞ and u 0 log 1 − B 1 /B 2 is the unique solution of ξ u 0. Hence ξ is decreasing on 0, u 0 , increasing on u 0 , ∞ and lim u → 0 ξ u lim u → ∞ ξ u −∞. Thus for any t ∈ −∞, ξ u 0 , there exist exactly two solutions η 1 ξ < η 2 ξ for ξ u t. Before proceeding, we list some properties of η i where G 2 , L 2 and H 2 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 η 2 , using the properties in C.2 , we conclude that For the integral over 0, u 0 , we have

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On the other hand, C.8 Using the properties C.1 , Using Lemma A.3, we get the behavior of J as z → ∞, and a simplification leads to Since 2πσ 1 σ 2 1 − ρ 2 f Z z e −A 1 z I z J z , the dominant term of f Z z is clearly given by J z if σ 1 < σ 2 , However when σ 1 σ 2 , we need to compare μ 1 and μ 2 . Finally, C.11

C.2.
Case 2. ρ ≥ σ 1 /σ 2 . We always have B 1 > 0, but now B 2 ≥ 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 Let H 1 and G 1 be defined in C.8 , C.6 respectively. Replace η 1 by η. It is not difficult to prove that we always have lim ξ → ∞ G 1 ξ, η ξ ∞ and Moreover, when z tends to ∞, the behavior of f Z z is dominated by the integral over R − . By C.1 and C.12 , H 1 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 μ 2 , σ 2 2 .

D. Quotient of Sum of Lognormal
Here is the proof for Theorem 2.8. Let Z 1 log e X 1 e X 2 , Z 2 log e Y 1 e Y 2 , as before, Denoting

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The last equality is obtained by the symmetry of our integrand between u, v and 1 − u, 1 − v.

D.14
We use x u/ y − 2s 0 for the last equality. Using dominated convergence and D.11 , the claim D.13 is obtained immediately. Defining Q y √ ye y−2s 0 2 / 8σ 2 P y , we finally get

D.15
As z → −∞, we need to understand the behavior of W b when b tends to ∞. More precisely, for λ > 0, Here, G x x 2 −λx 4 is not concave in 0, ∞ . But notice that G has a unique global maximum point at x 0 1/ √ 2λ in 0, ∞ and G x 0 −4. So the spirit of the Laplace method works. This implies 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 X 1 , X 2 , . . . , X n n≥3 be a joint normal variable with distribution N μ, Σ , where μ is the mean vector μ 1 , . . . , μ n , and Σ is a non-singular covariance matrix. Let Z log 1≤i≤n e X i . We wish to study the left tail of f Z z , and the PDF of z. Let f be the PDF of X 1 , X 2 , . . . , X n , that is, 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.