Sum Of Lognormal Random Variables Research Articles

Approximating the linear combination of log-normal RVs via pearson type IV distribution for UWB performance analysis

Approximating the sum of log-normal random variables (RVs) is a long-standing open issue and many studies in the old and recent literature resort to assume that the "power sum" can be well modeled via another log-normal RV. Such a log-normal approximation can be effectively used for accurate outage probability (Pout) analysis and modeling of signal-to-(noise+interference) ratio (SNIR) behaviors in the context of radio resource management in wireless systems; nevertheless, it is recognized that i) an adequate accuracy can only be obtained for a sub-range of values taken by the distribution; ii) the achievable accuracy depends on the parameters of the set of RVs; iii) log-normal approximation may not provide a satisfactory accuracy for estimating bit error probabilities. Moreover, all contributions in the open technical literature are referred to model the "power sum" distribution, i.e., the weighted linear combination of log-normal RVs with weights that may take only positive values. While this setup is consistent with typical scenarios where summands add incoherently, it has recently been shown that in some high frequency-selective multipath channels with log-normal distributed shadow- and fast-fading, e.g., in impulse radio ultra wide band (IR-UWB) systems, the signal- to-noise ratio (SNR) is more accurately modeled by the weighted linear combination of log-normal RVs with weights that may assume both positive and negative values. Motivated by the above considerations, and moving from recent results, which show that approximating the "power sum" of log-normal RVs can boil down to a model selection problem within the pearson system of distributions, the specific contribution of this paper can be summarized as follows, i) We show that the pearson type IV distribution can not only be used to accurately approximate the probability density function (PDF) of the "power sum" of log-normal RVs with positive weights, but it can also accurately model the scenario with both positive and negative weights, as well as the scenario with generically correlated RVs. ii) For the scenario with independent RVs, we propose a numerical technique, which is based on the moment generating function (MGF) matching principle and uses a least- squares criterion, to accurately estimate the four parameters of the pearson type IV distribution, iii) A byproduct of this analysis.

Bounds and approximations for sums of dependent log-elliptical random variables

Dhaene, Denuit, Goovaerts, Kaas and Vyncke [Dhaene, J., Denuit, M., Goovaerts, M.J., Kaas, R., Vyncke, D., 2002a. The concept of comonotonicity in actuarial science and finance: theory. Insurance Math. Econom. 31 (1), 3–33; Dhaene, J., Denuit, M., Goovaerts, M.J., Kaas, R., Vyncke, D., 2002b. The concept of comonotonicity in actuarial science and finance: Applications. Insurance Math. Econom. 31 (2), 133–161] have studied convex bounds for a sum of dependent random variables and applied these to sums of log-normal random variables. In particular, they have shown how these convex bounds can be used to derive closed-form approximations for several of the risk measures of such a sum. In this paper we investigate to which extent their general results on convex bounds can also be applied to sums of log-elliptical random variables which incorporate sums of log-normals as a special case. Firstly, we show that unlike the log-normal case, for general sums of log-ellipticals the convex lower bound does no longer result in closed-form approximations for the different risk measures. Secondly, we demonstrate how instead the weaker stop-loss order can be used to derive such closed-form approximations. We also present numerical examples to show the accuracy of the proposed approximations.

Lognormal Sum Approximation with a Variant of Type IV Pearson Distribution

In this paper, a variant of the Type IV Pearson distribution is proposed to approximate the distribution of the sum of lognormal random variables. Numerical and computer simulations show that independent of the statistical characteristics of the lognormal sum distribution, the Type IV Pearson variant outperforms the standard Type IV Pearson distribution and the normal variant distribution in accurately approximating the lognormal sum distribution for a whole probability range.

Approximating Lognormal Sum Distributions With Power Lognormal Distributions

In wireless communications, cochannel interference is usually characterized by a sum of lognormal random variables. Since the characteristic function of a lognormal distribution lacks explicit expression, and numerical calculation of a lognormal sum distribution is very challenging, lognormal distributions are often used to approximate lognormal sum distributions. However, it has been shown that a lognormal distribution can only capture a certain part of the body of a lognormal sum distribution. To improve the accuracy of approximation of lognormal sum distributions, one must resort to non-lognormal approximations. In this paper, we propose the use of power lognormal distributions to approximate lognormal sum distributions. To illustrate the superiority of the proposed model, some numerical experimental results are provided.

Asymptotics of sums of lognormal random variables with Gaussian copula

Let ( Y 1 , … , Y n ) have a joint n -dimensional Gaussian distribution with a general mean vector and a general covariance matrix, and let X i = e Y i , S n = X 1 + ⋯ + X n . The asymptotics of P ( S n > x ) as n → ∞ are shown to be the same as for the independent case with the same lognormal marginals. In particular, for identical marginals it holds that P ( S n > x ) ∼ n P ( X 1 > x ) no matter what the correlation structure is.

Lognormal Sum Approximation with Type IV Pearson Distribution

In this paper, the Type IV Pearson distribution is proposed to approximate the distribution of the sum of lognormal random variables, and the parameters of the Type IV Pearson distribution are derived through matching the mean, variance, skewness and kurtosis of the two distributions. Numerical simulations show that the Type IV Pearson distribution can accurately approximate distribution of the sum of lognormal random variables in a wide probability range.

Mixture Lognormal Approximations to Lognormal Sum Distributions

In wireless communication, co-channel interference is usually characterized by a sum of lognormal random variables. Since calculating the exact distribution of a lognormal sum has a lot of challenges, lognormal distributions are often used to approximate lognormal sum distributions. However, it has been shown that lognormal approximations can only capture a certain part of the body of a lognormal sum distribution, which implies that to accurately approximate a lognormal sum distribution, one has to resort to non-lognormal approximations. In this paper we propose to use a two-component mixture lognormal model to approximate lognormal sum distributions. Numerical examples are provided to compare the proposed mixture lognormal approximation with the existing ones.

Estimating the Ergodic Capacity of Log-Normal Channels

In this paper, we first provide a very accurate estimation of the capacity of a single-input single-output system operating in a log-normal environment. Then, hinging on the fact that the sum of log-normal Random Variables (RV) is well approximated by another log-normal RV, we apply the obtained results to find the capacity of Maximum Ratio Combining and Equal Gain Combining in a log-normal environment. The capacity in an interference-limited environment is also investigated in this paper. The analytical expressions obtained match perfectly the capacity given by simulations.

Distributed Sensor Networks Deployment Using Fuzzy Logic Systems

The effectiveness of distributed wireless sensor networks highly depends on the sensor deployment scheme. Given a finite number of sensors, optimizing the sensor deployment will provide sufficient sensor coverage and ameliorate the quality of communications. In this paper, we apply fuzzy logic systems to optimize the sensor placement after an initial random deployment. We use the outage probability due to co-channel interference to evaluate the communication quality. Fenton–Wilkinson method is applied to approximate the sum of log-normal random variables. Our algorithm is compared against the existing distributed self-spreading algorithm. Simulation results show that our approach achieves faster and stabler deployment and maximizes the sensor coverage with less energy consumption. Outage probability, as a measure of communication quality gets effectively decreased in our algorithm but it was not taken into consideration in the distributed self-spreading algorithm.

Least Squares Approximations to Lognormal Sum Distributions

In this paper, the least squares (LS) approximation approach is applied to solve the approximation problem of a sum of lognormal random variables (RVs). The LS linear approximation is based on the widely accepted assumption that the sum of lognormal RVs can be approximated by a lognormal RV. We further derive the solution for the LS quadratic (LSQ) approximation, and our results show that the LSQ approximation exhibits an excellent match with the simulation results in a wide range of the distributions of the summands. Using the coefficients obtained from the LSQ method, we present the explicit closed-form expressions of the coefficients as a function of the decibel spread and the number of the summands by applying an LS curve fitting technique. Closed-form expressions for the cumulative distribution function and the probability density function for the sum RV, in both the linear and logarithm domains, are presented

Estimation of typical sum of lognormal random variables using log shifted gamma approximation

The sum of lognormal distributions is a well-known problem that no closed-form expression exists and it is difficult to evaluate numerically. In this paper, log shifted gamma (LSG) approximation method is proposed to represent the sum of lognormal distributions and to derive a closed-form expression of the typical value of the sum. Illustrative results show that the LSG model provides much more accurate approximation than other previous methods for a wide range of lognormal variances.

An error bound for moment matching methods of lognormal sum distributions

To evaluate the distribution function of a sum of lognormal random variables. it is common to use approximation methods based on moment matching. These include the classical and simple Fenton-Wilk ...

Highly Accurate Simple Closed-Form Approximations to Lognormal Sum Distributions and Densities

Sums of lognormal random variables occur extensively in wireless communications, in part, because a shadowing environment is well modeled by a lognormal distribution. A closed-form expression does not exist for the sum distribution and, furthermore, it is difficult to numerically calculate the distribution. Numerous approximations exist that are based on approximating a sum of lognormal random variables as another lognormal random variable. A new paradigm to calculate an approximation to the lognormal sum distribution, based on curve fitting on lognormal probability paper, is introduced in this letter. Highly accurate, simple closed-form approximations to lognormal sum distributions are presented.

The log-normal approximation in financial and other computations

Sums of log-normals frequently appear in a variety of situations, including engineering and financial mathematics. In particular, the pricing of Asian or basket options is directly related to finding the distributions of such sums. There is no general explicit formula for the distribution of sums of log-normal random variables. This paper looks at the limit distributions of sums of log-normal variables when the second parameter of the log-normals tends to zero or to infinity; in financial terms, this is equivalent to letting the volatility, or maturity, tend either to zero or to infinity. The limits obtained are either normal or log-normal, depending on the normalization chosen; the same applies to the reciprocal of the sums of log-normals. This justifies the log-normal approximation, much used in practice, and also gives an asymptotically exact distribution for averages of log-normals with a relatively small volatility; it has been noted that all the analytical pricing formulae for Asian options perform poorly for small volatilities. Asymptotic formulae are also found for the moments of the sums of log-normals. Results are given for both discrete and continuous averages. More explicit results are obtained in the case of the integral of geometric Brownian motion.

The log-normal approximation in financial and other computations

Sums of log-normals frequently appear in a variety of situations, including engineering and financial mathematics. In particular, the pricing of Asian or basket options is directly related to finding the distributions of such sums. There is no general explicit formula for the distribution of sums of log-normal random variables. This paper looks at the limit distributions of sums of log-normal variables when the second parameter of the log-normals tends to zero or to infinity; in financial terms, this is equivalent to letting the volatility, or maturity, tend either to zero or to infinity. The limits obtained are either normal or log-normal, depending on the normalization chosen; the same applies to the reciprocal of the sums of log-normals. This justifies the log-normal approximation, much used in practice, and also gives an asymptotically exact distribution for averages of log-normals with a relatively small volatility; it has been noted that all the analytical pricing formulae for Asian options perform poorly for small volatilities. Asymptotic formulae are also found for the moments of the sums of log-normals. Results are given for both discrete and continuous averages. More explicit results are obtained in the case of the integral of geometric Brownian motion.

A Simple Bound on the Outage Probability With Lognormally Distributed Interferers

We consider the problem of determining the outage probability in a cellular radio system, where the interference is modeled by a sum of lognormal random variables. As no closed form expression is known, bounds can serve as a quick way of evaluating performance. By applying the arithmetic-geometric mean inequality, we give a lower bound, computationally simple as it is expressed by a single Q-function. The bound includes the case of correlated interferers and background noise. Numerical evaluation for a range of parameters valid for practical applications, shows good accuracy for moderate variances and improvement over a previously suggested bound. This is especially emphasized when the number of interferers is large.

An Optimal Lognormal Approximation to Lognormal Sum Distributions

Sums of lognormal random variables occur in many problems in wireless communications because signal shadowing is well modeled by the lognormal distribution. The lognormal sum distribution is not known in the closed form and is difficult to compute numerically. Several approximations to the distribution have been proposed and employed in applications. Some widely used approximations are based on the assumption that a lognormal sum is well approximated by a lognormal random variable. Here, a new paradigm for approximating lognormal sum distributions is presented. A linearizing transform is used with a linear minimax approximation to determine an optimal lognormal approximation to a lognormal sum distribution. The accuracies of the new method are quantitatively compared to the accuracies of some well-known approximations. In some practical cases, the optimal lognormal approximation is several orders of magnitude more accurate than previous approximations. Efficient numerical computation of the lognormal characteristic function is also considered.

Capacity of wireless optical communications

We consider the ergodic capacity and capacity-versus-outage probability of direct-detection optical communication through a turbulent atmosphere using multiple transmit and receive apertures. We assume shot-noise-limited operation in which detector outputs are doubly stochastic Poisson processes whose rates are proportional to the sum of the transmitted powers, scaled by lognormal random fades, plus a background noise. In the high and low signal-to-background ratio regimes, we show that the ergodic capacity of this fading channel equals or exceeds that for a channel with deterministic path gains. Furthermore, knowledge of these path gains is not necessary to achieve capacity when the signal-to-background ratio is high. In the low signal-to-background ratio regime, path-gain knowledge provides minimal capacity improvement when using a moderate number of transmit apertures. We also develop expressions for the capacity-versus-outage probability in the high and low signal-to-background ratio regimes, by means of a moment-matching approximation to the distribution for the sum of lognormal random variables. Monte Carlo simulations show that these capacity-versus-outage approximations are quite accurate for moderate numbers of apertures.

Broad distribution effects in sums of lognormal random variables

The lognormal distribution describing, e.g., exponentials of Gaussian random variables is one of the most common statistical distributions in physics. It can exhibit features of broad distributions that imply qualitative departure from the usual statistical scaling associated to narrow distributions. Approximate formulae are derived for the typical sums of lognormal random variables. The validity of these formulae is numerically checked and the physical consequences, e.g., for the current flowing through small tunnel junctions, are pointed out.

Bounds on the distribution of a sum of independent lognormal random variables

The distribution function of a sum of lognormal random variables (RVs) appears in several communication problems. Approximations are usually used for such distribution as no closed form nor bounds exist. Bounds can be very useful in assessing the performance of any given system. In this letter, we derive upper and lower bounds on the distribution function of a sum of independent lognormal RVs. These bounds are given in a closed form and can be used in studying the performance of cellular radio and broadcasting systems.