Sum Of Lognormal Random Variables Research Articles

Theoretical approach for uncertainty quantification in probabilistic safety assessment using sum of lognormal random variables

Probabilistic safety assessment is widely used to quantify the risks of nuclear power plants and their uncertainties. When the lognormal distribution describes the uncertainties of basic events, the uncertainty of the top event in a fault tree is approximated with the sum of lognormal random variables after minimal cutsets are obtained, and rare-event approximation is applied. As handling complicated analytic expressions for the sum of lognormal random variables is challenging, several approximation methods, especially Monte Carlo simulation, are widely used in practice for uncertainty analysis. In this study, a theoretical approach for analyzing the sum of lognormal random variables using an efficient numerical integration method is proposed for uncertainty analysis in probability safety assessments. The change of variables from correlated random variables with a complicated region of integration to independent random variables with a unit hypercube region of integration is applied to obtain an efficient numerical integration. The theoretical advantages of the proposed method over other approximation methods are shown through a benchmark problem. The proposed method provides an accurate and efficient approach to calculate the uncertainty of the top event in probabilistic safety assessment when the uncertainties of basic events are described with lognormal random variables.

A fast method to seek the mean and variance of the summation of lognormal variables

The sum of lognormal variables has been a topic of interest in several fields of research such as engineering, biology and finance, among others. For example, in the field of telecommunications, the aggregate interference of radio frequency signals is modeled as a sum of lognormal variables. To date, there is no closed expression for the probability distribution function (PDF) of this sum. Several authors have proposed approximations for this PDF, with which they calculate the mean and variance. However, each method has limitations in its range of parameters for mean, variance and number of random variables to be added. In other cases, long approximations as power series are used, which makes the analytical treatment impractical and reduces the computational performance of numerical operations. This paper shows an alternative method for calculating the mean and variance of the sum of lognormal random variables from a computational performance approach. Our method has been evaluated extensively by Monte Carlo simulations. As a result, this method is computationally efficient and yields a low approximation error computation for a wide range of mean values, variances and number of random variables.

Pricing of Basket Options by Conditioning and Moment Matching

The price of basket options can be represented as an exact analytical part and an approximate part by using a conditional variable. The first part is calculated by conditioning on the price process of the underlying asset and the second part is calculated by moment matching approach. In order to calculate the second part, we find a new single random variable, which has an analytically known distribution, to approximate the sum of log-normal random variables and to obtain a closed form pricing formula. Our method can be viewed as a combination of conditioning and moment matching methods. Numerical studies demonstrated that our formula is more accurate in handling both homogeneous and heterogeneous lognormal random variable cases. TOPICS:Derivatives, options Key Findings • The price of basket options can be represented as an exact analytical part and an approximate part by using a conditional variable. In order to calculate the approximate part, we find a new single random variable, which has an analytically known distribution, to approximate the sum of log-normal random variables and to obtain a closed form pricing formula. • Numerical studies demonstrate that our formula is more accurate in handling both homogeneous and heterogeneous lognormal random variable cases.

Performance of Spatial Diversity DCO-OFDM in a Weak Turbulence Underwater Visible Light Communication Channel

The performance of underwater visible light communication (UVLC) system is severely affected by absorption, scattering and turbulence. In this article, we study the performance of spectral efficient DC-biased optical orthogonal frequency division multiplexing (DCO-OFDM) in combination with the transceiver spatial diversity in turbulence channel. Based on the approximation of the weighted sum of lognormal random variables (RVs), we derived a theoretical exact bit error rate (BER) for DCO-OFDM systems with spatial diversity. The simulation results are compared with the analytical prediction, confirming the validity of the analysis. It is shown that spatial diversity can effectively reduce the turbulence-induced channel fading. The obtained results can be useful for designing, predicting, and evaluating the DCO-OFDM UVLC system in a weak oceanic turbulence condition.

Asymptotic Outage Analysis on Dual-Branch Diversity Receptions Over Non-Identically Distributed Correlated Lognormal Channels

It is challenging to quantify the left tail behavior of the cumulative distribution function of a sum of lognormal random variables, especially when they are correlated and non-identically distributed. The mathematical intractability hampers the analysis on diversity receptions over lognormal fading channels. In this paper, a novel asymptotic technique is exploited to study non-identically distributed dual-branch lognormal fading channels with correlation. Closed-form asymptotic outage probability expressions are derived for maximum-ratio combining (MRC), equal-gain combining (EGC), and selection combining (SC). Three counterintuitive facts are revealed: 1) one of the two channels contributes no asymptotic performance gain to the diversity reception systems under certain conditions, implying that a link can be discarded without causing asymptotic performance loss; 2) SC outperforms EGC under certain conditions; and 3) the outage probability of SC is arbitrarily close to that of MRC under certain conditions. The new result reveals insights into the long-standing problem of asymptotic analysis for correlated lognormal fading channels, paving the way for analysis on more general channel models. This paper also provides an efficient approach to evaluate the non-identically distributed lognormal fading channels at a high signal-to-noise ratio. Practically, the analytical results prevent some redundant units that contribute no performance gain.

Outage Probability Bounds of EGC Over Dual-Branch Non-Identically Distributed Independent Lognormal Fading Channels With Optimized Parameters

Outage probability analysis of diversity reception over lognormal fading channels is challenging because the cumulative distribution function (CDF) of a sum of lognormal random variables (RVs) does not yield a closed-form expression. Furthermore, traditional asymptotic analysis techniques fail for lognormal fading channels. The well-applied approximation techniques, such as Fenton, Wilkinson, and Schwartz and Yeh approximations, become unreliable when the signal-to-noise ratio (SNR) is high. In this paper, we derive closed-form bounds of outage probability for dual-branch equal-gain combining system over non-identically distributed lognormal fading channels. The bounds are shown tight from low-to-high SNR regimes, which provides an efficient way to evaluate the EGC system without resorting to expensive Monte Carlo simulation or numerical integration. Furthermore, the new analytical expressions can be used to test at what SNR the existing approximation techniques become inaccurate, and can also be used as criteria for system design.

Capacity of Optical Wireless System over Log-Normal Channels with Spatial Diversity in Presence of Atmospheric Losses

Abstract In this paper, average channel capacity of optical wireless communication system is evaluated under the combined effect of geometrical loss, attenuation due to weather conditions and weak atmospheric turbulence using a simple closed form expression. Fading induced due to atmospheric turbulence is modeled by log-normal distribution. Considering the fact that the sum of log-normal random variables can be well approximated by another log-normal random variable, the proposed expression has been utilized to compute the channel capacity for spatial diversity reception employing maximum ratio combining and equal gain combining over uncorrelated turbulence-induced fading conditions. It is shown that spatial diversity is an effective technique to mitigate the impairments caused by various atmospheric conditions such as haze, rain and fog. The quantitative improvement in channel capacity achieved by using diversity techniques is investigated and compared. Accuracy of the results is validated with exact results computed using Monte Carlo simulation.

On the efficient simulation of the left-tail of the sum of correlated log-normal variates

Abstract The sum of log-normal variates is encountered in many challenging applications such as performance analysis of wireless communication systems and financial engineering. Several approximation methods have been reported in the literature. However, these methods are not accurate in the tail regions. These regions are of primordial interest as small probability values have to be evaluated with high precision. Variance reduction techniques are known to yield accurate, yet efficient, estimates of small probability values. Most of the existing approaches have focused on estimating the right-tail of the sum of log-normal random variables (RVs). Here, we instead consider the left-tail of the sum of correlated log-normal variates with Gaussian copula, under a mild assumption on the covariance matrix. We propose an estimator combining an existing mean-shifting importance sampling approach with a control variate technique. This estimator has an asymptotically vanishing relative error, which represents a major finding in the context of the left-tail simulation of the sum of log-normal RVs. Finally, we perform simulations to evaluate the performances of the proposed estimator in comparison with existing ones.

Performance Studies of Underwater Wireless Optical Communication Systems With Spatial Diversity: MIMO Scheme

In this paper, we analytically study the performance of multiple-input multiple-output underwater wireless optical communication (UWOC) systems with ON–OFF keying modulation. To mitigate turbulence-induced fading, which is amongst the major degrading effects of underwater channels on the propagating optical signal, we use spatial diversity over UWOC links. Furthermore, the effects of absorption and scattering are considered in our analysis. We analytically obtain the exact and an upper bound bit error rate (BER) expressions for both optimal and equal gain combining. In order to more effectively calculate the system BER, we apply Gauss-Hermite quadrature formula as well as approximation to the sum of lognormal random variables. We also apply the photon-counting method to evaluate the system BER in the presence of shot noise. Our numerical results indicate an excellent match between the exact and upper bound BER curves. Also, a good match between the analytical results and numerical simulations confirms the accuracy of our derived expressions. Moreover, our results show that spatial diversity can considerably improve the system performance, especially for channels with higher turbulence, e.g., a $3\times 1$ multiple-input single-output transmission in a 25 m coastal water link with a log-amplitude variance of 0.16 can introduce 8 dB performance improvement at the BER of 10−9.

Computing optimal rebalance frequency for log-optimal portfolios in linear time

The pure form of log-optimal investment strategies are often considered to be impractical due to the inherent need for continuous rebalancing. It is however possible to improve investor log utility by adopting a discrete-time periodic rebalancing strategy. Under the assumptions of geometric Brownian motion for assets and approximate log-normality for a sum of log-normal random variables, we find that the optimum rebalance frequency is a piecewise continuous function of investment horizon. One can construct this rebalance strategy function, called the optimal rebalance frequency function, up to a specified investment horizon given a limited trajectory of the expected log of portfolio growth when the initial portfolio is never rebalanced. We develop the analytical framework to compute the optimal rebalance strategy in linear time, a significant improvement from the previously proposed search-based quadratic time algorithm.

The Lognormal Distribution and Quantum Monte Carlo Data

AbstractQuantum Monte Carlo data are often afflicted with distributions that resemble lognormal probability distributions and consequently their statistical analysis cannot be based on simple Gaussian assumptions. To this extent a method is introduced to estimate these distributions and thus give better estimates to errors associated with them. This method entails reconstructing the probability distribution of a set of data, with given mean and variance, that has been assumed to be lognormal prior to undergoing a blocking or renormalization transformation. In doing so, we perform a numerical evaluation of the renormalized sum of lognormal random variables. This technique is applied to a simple quantum model utilizing the single-thread Monte Carlo algorithm to estimate the ground state energy or dominant eigenvalue of a Hamiltonian matrix.

Default probabilities of a holding company, with complete and partial information

This paper studies the valuation of credit risk for firms that own several subsidiaries or business lines. We provide simple analytical approximating expressions for probabilities of default, and for equity–debt market values, both in the case when the information is available in continuous time as well as in the case that it is not instantaneously available. The total firm’s asset value being modeled as a sum of lognormal random variables, we use convex upper and lower approximations to infer these analytical approximating expressions. We extend the model to firms financed by multiple stochastic liabilities and conclude by numerical illustrations.

Approximate Default Probabilities of a Holding Company, with Complete and Partial Information

This paper studies the valuation of credit risk for firms that own several subsidiaries or business lines. We provide simple analytical approximating expressions for probabilities of default, and for equity-debt market values, both in the case when the information is available in continuous time as well as in the case that it is not instantaneously available. The total firm's asset value being modeled as a sum of lognormal random variables, we use convex upper and lower approximations to infer these analytical approximating expressions. We extend the model to firms financed by multiple stochastic liabilities and conclude by numerical illustrations.

SIR distribution analysis in cellular networks considering the joint impact of path-loss, shadowing and fast fading

Abstract In this paper, we propose an analysis of the joint impact of path-loss, shadowing and fast fading on cellular networks. Two analytical methods are developed to express the outage probability. The first one based on the Fenton-Wilkinson approach, approximates a sum of log-normal random variables by a log-normal random variable and approximates fast fading coefficients in interference terms by their average value. We denote it FWBM for Fenton-Wilkinson based method. The second one is based on the central limit theorem for causal functions. It allows to approximate a sum of positive random variables by a Gamma distribution. We denote it CLCFM for central limit theorem for causal functions method. Each method allows to establish a simple and easily computable outage probability formula, which jointly takes into account path-loss, shadowing and fast fading. We compute the outage probability, for mobile stations located at any distance from their serving BS, by using a fluid model network that considers the cellular network as a continuum of BS. We validate our approach by comparing all results to extensive Monte Carlo simulations performed in a traditional hexagonal network and we provide the limits of the two methods in terms of system parameters. The proposed framework is a powerful tool to study performances of cellular networks, e.g., OFDMA systems (WiMAX, LTE).

A Low-Complexity Approximation to Lognormal Sum Distributions via Transformed Log Skew Normal Distribution

Sums of lognormal random variables (RVs) occur in many important problems in wireless communication. The lognormal sum distribution is known to have no closed form and is difficult to numerically compute. Several methods have been proposed to approximate the lognormal sum distribution. In this paper, we first propose a low-complexity approximation method called log skew normal (LSN) approximation to model and approximate the lognormal sum distributed RVs. For typical lognormal sum cases in wireless communication, the proposed LSN method has high accuracy in most of the region of the cumulative distribution function (cdf), particularly in the lower region. The closed-form probability density function (pdf) and cdf of the resulting LSN RV are presented, and its parameters are derived from those of the individual lognormal RVs by using a moment-matching technique. However, the LSN approximation has a restriction for the skewness of samples in the logarithm domain. To overcome this drawback, a transformed LSN (TLSN) approximation method is proposed, which uses another parameter to control the skewness of samples in the transform logarithm domain. Simulation results on the pdf and cdf of lognormal sum RVs confirm the effectiveness of the TLSN approximation method.

New Quadrature-Based Approximations for the Characteristic Function and the Distribution Function of Sums of Lognormal Random Variables

Characterizing the distribution of the sum of lognormal random variables (RVs) is still an open issue. This paper proposes simple and new quadrature-based approximations of the characteristic function (CF) and the cumulative distribution function (cdf) of the sum of independent and correlated lognormal RVs. Exploiting the recent Hermite-Gauss quadrature-based approximation, which is provided for the CF of a single lognormal RV, this paper proposes expressions that are given in terms of quadrature nodes and weights, as well as in terms of the parameters of individual lognormal RVs and the covariance matrix for the correlated RV case. More importantly, the developed expression for the cdf does not require prior knowledge of the actual cdf or the employment of specialized numerical integration methods. Numerical examples and comparisons with approximation techniques for the cdf of the sum found in the literature are provided. The examples show that while most of the known approximation techniques are valid for either small or large values of the abscissa, but not both, the proposed cdf formula provides reasonable approximation over a wide range of the abscissa. Furthermore, the study shows that as the actual cdf of the sum departs from the straight-line shape and its concavity increases, when plotted on a normal probability scale, even techniques that specialize in approximating the low-end tail of the cdf start to produce higher errors. The proposed formula continues to accurately approximate the actual cdf.

Lower convex order bound approximations for sums of log-skew normal random variables

When it comes to modeling dependent random variables, not surprisingly, the multivariate normal distribution has received the most attention because of its many appealing properties. However, when it comes to practical implementation, the same family of distribution is often rejected for modeling financial and insurance data because they do not apparently behave in the multivariate normal sense. In this paper, we consider the construction of lower convex order bounds, in the sense of Kaas et al. (Insur. Math. Econ. 2000; 27:151–168), to approximate sums of dependent log-skew normal random variables. The dependence structure of these random variables is based on the class of multivariate closed skew-normal (CSN) distribution that appears in Gonzalez-Farias et al. (Skew-Elliptical Distributions and their Applications. A Journey Beyond Normality, Genton MG (ed.), Chapter 6. Chapman & Hall/CRC: Boca Raton, FL, 2004; 25–42) and which carries several interesting properties of the normal distribution apart from allowing additional parameters to regulate skewness. The bounds that we present in this paper are therefore natural extensions to the results presented in Dhaene et al. (Insurance: Mathematics and Economics 2002; 31(2): 133–161; Insurance: Mathematics and Economics 2002b; 31(1):3–33), where bounds for sums of log-normal random variables have been derived. These lower bound approximations are constructed based on the additional information provided by a conditioning variable which when optimally chosen can provide an accurate approximation. We exploit inherent properties of this family of skew-normal distributions in order to choose the optimal conditioning variable. Results of our simulations provide an indication of the performance of these approximations. Copyright © 2010 John Wiley & Sons, Ltd.

Accurate computation of the MGF of the lognormal distribution and its application to sum of lognormals

Sums of lognormal random variables (RVs) are of wide interest in wireless communications and other areas of science and engineering. Since the distribution of lognormal sums is not log-normal and does not have a closed-form analytical expression, many approximations and bounds have been developed. This paper develops two computational methods for the moment generating function (MGF) or the characteristic function (CHF) of a single lognormal RV. The first method uses classical complex integration techniques based on steepest-descent integration. The saddle point of the integrand is explicitly expressed by the Lambert function. The steepest-descent (optimal) contour and two closely-related closed-form contours are derived. A simple integration rule (e.g., the midpoint rule) along any of these contours computes the MGF/CHF with high accuracy. The second approach uses a variation on the trapezoidal rule due to Ooura and Mori. Importantly, the cumulative distribution function of lognormal sums is derived as an alternating series and convergence acceleration via the Epsilon algorithm is used to reduce, in some cases, the computational load by a factor of 106! Overall, accuracy levels of 13 to 15 significant digits are readily achievable.

On the capacity of log-normal fading channels

In this letter we provide an analytical expression for the moments of the capacity for the log-normal fading channel. Since the developed expression involves infinite series, we show that the error that results from the truncation of these series is insignificant. We also analyze in more details the ergodic capacity by giving a simpler expression for the remainder of the truncated series. Relying on the fact that the sum of log-normal random variables (RV) is well approximated by another lognormal RV, we further utilize the obtained results to approximate the capacity of diversity combining techniques in correlated lognormal fading channels. The results that we provide in this letter are an important tool for measuring the performance of communication links in a log-normal environment.

Further results on the approximation of log-normal power sum via pearson type IV distribution: a general formula for log-moments computation

In, we have recently proposed a general approach for approximating the power sum of Log-Normal Random Variables (RVs) by using the Pearson system of distributions. Therein, we have also highlighted the main advantages of using Pearson approximation instead of the usual Log-Normal one, and compared the proposed method with other approaches available in the open technical literature. However, despite being very accurate, the proposed method may be, in some circumstances, computational demanding since a non-linear least-squares problem needs to be solved numerically to get an accurate approximation. Motivated by the above consideration, the aim of this Letter is to provide an alternative approach for computing the parameters of the approximating Pearson distribution. The proposed solution is based on the Method of Moments (MoMs) in the logarithmic domain. In particular, by using some known properties of the Laplace transform, we will show that the MGF of the Log-Normal power sum in the logarithmic domain (denoted as log-MGF) can be obtained from the Mellin transform of the MGF of the Log-Normal power sum in the linear domain. From the estimated log-MGF, we will then compute the desired log-moments required for Pearson approximation. Numerical results will be also shown in order to substantiate the accuracy of the proposed method.