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Journal articles on the topic 'Product of independent random variables'

Author: Grafiati

Published: 4 June 2021

Last updated: 15 February 2022

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1

Dettmann, Carl P., and Orestis Georgiou. "Product of independent uniform random variables." Statistics & Probability Letters 79, no. 24 (December 2009): 2501–3. http://dx.doi.org/10.1016/j.spl.2009.09.004.

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2

Cline, D. B. H., and G. Samorodnitsky. "Subexponentiality of the product of independent random variables." Stochastic Processes and their Applications 49, no. 1 (January 1994): 75–98. http://dx.doi.org/10.1016/0304-4149(94)90113-9.

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3

Jovanovic Dolecek, Gordana. "Teaching Random Variables." U.Porto Journal of Engineering 7, no. 1 (February 19, 2021): 10–15. http://dx.doi.org/10.24840/2183-6493_007.001_0003.

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This paper presents demo program for teaching random variables dedicated to demonstration and visualization of behaviour of the product of m independent uniform random variables. The demo is used as a complementary tool in the teaching Random signals and processes curriculum at the graduate level. The demo is made in MATLAB and utilizes all benefits provided by MATLAB, such as simplicity, and good graphic environment, interactive use, among others. Additionally, this demo is used with other demo programs, also made in MATLAB, in teaching/learning process of random variables. Students were very satisfied with this demo program expressing that helped them understanding better the behaviour of the product of uniform random variables.

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4

Salo, J., H. M. El-Sallabi, and P. Vainikainen. "The Distribution of the Product of Independent Rayleigh Random Variables." IEEE Transactions on Antennas and Propagation 54, no. 2 (February 2006): 639–43. http://dx.doi.org/10.1109/tap.2005.863087.

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5

Sato, Hiroshi. "On the convergence of the product of independent random variables." Journal of Mathematics of Kyoto University 27, no. 2 (1987): 381–85. http://dx.doi.org/10.1215/kjm/1250520722.

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6

Behboodian, J. "Symmetric sum and symmetric product of two independent random variables." Journal of Theoretical Probability 2, no. 2 (April 1989): 267–70. http://dx.doi.org/10.1007/bf01053413.

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7

Su, Chun, and Yu Chen. "On the behavior of the product of independent random variables." Science in China Series A 49, no. 3 (January 2006): 342–59. http://dx.doi.org/10.1007/s11425-006-0342-z.

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8

Nadarajah, Saralees, and Arjun K. Gupta. "On the product and ratio of Bessel random variables." International Journal of Mathematics and Mathematical Sciences 2005, no. 18 (2005): 2977–89. http://dx.doi.org/10.1155/ijmms.2005.2977.

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The distributions of products and ratios of random variables are of interest in many areas of the sciences. In this paper, the exact distributions of the product|XY|and the ratio|X/Y|are derived whenXandYare independent Bessel function random variables. An application of the results is provided by tabulating the associated percentage points.

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9

Hall, Peter, and Eugene Seneta. "Products of independent, normally attracted random variables." Probability Theory and Related Fields 78, no. 1 (1988): 135–42. http://dx.doi.org/10.1007/bf00718041.

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10

Abramov, V. A. "Convergence of products of independent random variables." Journal of Soviet Mathematics 35, no. 2 (October 1986): 2293–301. http://dx.doi.org/10.1007/bf01105645.

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11

Shimura, Takaaki. "The product of independent random variables with slowly varying truncated moments." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 62, no. 2 (April 1997): 186–97. http://dx.doi.org/10.1017/s1446788700000756.

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AbstractThe Mellin-Stieltjes convolution and related decomposition of distributions in M(α) (the class of distributions μ on (0, ∞) with slowly varying αth truncated moments ) are investigated. Maller shows that if X and Y are independent non-negative random variables with distributions μ and v, respectively, and both μ and v are in D2, the domain attraction of Gaussian distribution, then the distribution of the product XY (that is, the Mellin-Stieltjes convolution μ ^ v of μ and v) also belongs to it. He conjectures that, conversely, if μ ∘ v belongs to D2, then both μ and v are in it. It is shown that this conjecture is not true: there exist distributions μ ∈ D2 and v μ ∈ D2 such that μ ^ v belongs to D2. Some subclasses of D2 are given with the property that if μ ^ v belongs to it, then both μ and v are in D2.

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12

Garg, Mridula, Ajay Sharma, and Pratibha Manohar. "The distribution of the product of two independent generalized trapezoidal random variables." Communications in Statistics - Theory and Methods 45, no. 21 (July 13, 2015): 6369–84. http://dx.doi.org/10.1080/03610926.2014.882954.

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13

Grigorevskii, N. V. "Ideal metrics and products of independent random variables." Journal of Soviet Mathematics 32, no. 1 (January 1986): 13–24. http://dx.doi.org/10.1007/bf01084494.

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14

Ilic, Ivana. "Weak convergence of product of sums of independent variables with missing values." Filomat 24, no. 3 (2010): 73–81. http://dx.doi.org/10.2298/fil1003073i.

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Let (Xn) be a sequence of independent and non-identically distributed random variables. We assume that only observations of (Xn) at certain points are available. We study limit properties in the sense of weak convergence in the space D[0,1] of certain processes based on an incomplete sample from {X1, X2 ...,Xn }. This is an extension of the results of Matula and Stepien [2009. Weak convergence of products of sums of independent and non-identically distributed random variables. J. Math. Anal. Appl. 353, 49-54].

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15

Krstic, Dragana, Petar Nikolic, Danijela Aleksic, Sinisa Minic, Dragan Vuckovic, and Mihajlo Stefanovic. "Product of Three Random Variables and its Application in Relay Telecommunication Systems in the Presence of Multipath Fading." Journal of Telecommunications and Information Technology 1 (March 29, 2019): 83–92. http://dx.doi.org/10.26636/jtit.2019.130018.

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In this paper, the product of three random variables (RVs) will be considered. Distribution of the product of independent random variables is very important in many applied problems, including wireless relay telecommunication systems. A few of such products of three random variables are observed in this work: the level crossing rate (LCR) of the product of a Nakagami-m random variable, a Rician random variable and a Rayleigh random variable, and of the products of two Rician RVs and one Nakagami-m RV is calculated in closed forms and presented graphically. The LCR formula may be later used for derivation of average fade duration (AFD) of a wireless relay communication radio system with three sections, working in the multipath fading channel. The impact of fading parameters and multipath fading power on the LCR is analyzed based on the graphs presented.

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16

Szczepański, Jerzy. "A remark on the distribution of products of independent normal random variables." Science, Technology and Innovation 10, no. 3 (March 10, 2021): 30–37. http://dx.doi.org/10.5604/01.3001.0014.7861.

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We present a proof of the explicit formula of the probability density function of the product of normally distributed independent random variables using the multiplicative convolution formula for Meijer G functions.

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17

Kasahara, Yuji. "A note on the product of independent random variables with regularly varying tails." Tsukuba Journal of Mathematics 42, no. 2 (December 2018): 295–308. http://dx.doi.org/10.21099/tkbjm/1554170426.

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18

Nadarajah, S., and S. Kotz. "Comments on "On the Distribution of the Product of Independent Rayleigh Random Variables." IEEE Transactions on Antennas and Propagation 54, no. 11 (November 2006): 3570–71. http://dx.doi.org/10.1109/tap.2006.884313.

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19

Hao Lu, Yunfei Chen, and Ning Cao. "Accurate Approximation to the PDF of the Product of Independent Rayleigh Random Variables." IEEE Antennas and Wireless Propagation Letters 10 (2011): 1019–22. http://dx.doi.org/10.1109/lawp.2011.2168938.

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20

Marques, Filipe J., and Florence Loingeville. "Improved near-exact distributions for the product of independent Generalized Gamma random variables." Computational Statistics & Data Analysis 102 (October 2016): 55–66. http://dx.doi.org/10.1016/j.csda.2016.04.004.

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21

Hassan, N. J., A. Hawad Nasar, and J. Mahdi Hadad. "Distributions of the Ratio and Product of Two Independent Weibull and Lindley Random Variables." Journal of Probability and Statistics 2020 (May 1, 2020): 1–8. http://dx.doi.org/10.1155/2020/5693129.

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In this paper, we derive the cumulative distribution functions (CDF) and probability density functions (PDF) of the ratio and product of two independent Weibull and Lindley random variables. The moment generating functions (MGF) and the k-moment are driven from the ratio and product cases. In these derivations, we use some special functions, for instance, generalized hypergeometric functions, confluent hypergeometric functions, and the parabolic cylinder functions. Finally, we draw the PDF and CDF in many values of the parameters.

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22

Mingo, James A., and Alexandru Nica. "Crossings of Set-Partitions and Addition Crossings of Graded-Independent Random Variables." International Journal of Mathematics 08, no. 05 (August 1997): 645–64. http://dx.doi.org/10.1142/s0129167x97000342.

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We consider a q-deformation, introduced in [7], of the cumulants associated with a linear functional on polynomials; combinatorially, the deformation is defined using crossing numbers of set-partitions. The paper is concerned with the case q = -1. We show that if the linear functional μ : C[X] → C is symmetric (i.e.μ(Xn) = 0 for n odd), then the exponential generating function of the even (-1)-cumulants of μ is equal to [Formula: see text], (as a formal power series in z). We discuss the connection of this fact with the form of independence for (non-commutative) random variables which corresponds to the operation of Z2-graded tensor product.

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23

Bercovici, Hari, and Vittorino Pata. "Limit laws for products of free and independent random variables." Studia Mathematica 141, no. 1 (2000): 43–52. http://dx.doi.org/10.4064/sm-141-1-43-52.

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24

Simonelli, Italo. "Convergence and symmetry of infinite products of independent random variables." Statistics & Probability Letters 55, no. 1 (November 2001): 45–52. http://dx.doi.org/10.1016/s0167-7152(01)00126-2.

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25

Simonelli, Italo. "“Convergence and symmetry of infinite products of independent random variables”." Statistics & Probability Letters 62, no. 3 (April 2003): 323. http://dx.doi.org/10.1016/s0167-7152(03)00049-x.

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26

Shanchao, Yang. "Moment inequalities for sums of products of independent random variables." Statistics & Probability Letters 76, no. 18 (December 2006): 1994–2000. http://dx.doi.org/10.1016/j.spl.2006.05.004.

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27

Matsak, I. K., and A. N. Plichko. "Khinchin's inequality for k-fold products of independent random variables." Mathematical Notes of the Academy of Sciences of the USSR 44, no. 3 (September 1988): 690–94. http://dx.doi.org/10.1007/bf01159131.

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28

WANG, YANLING, SUXIA YAO, and HONGXIA DU. "Asymptotic distribution of products of sums of independent random variables." Proceedings - Mathematical Sciences 123, no. 2 (April 12, 2013): 283–92. http://dx.doi.org/10.1007/s12044-013-0130-y.

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29

Khemissi, Eliza Anna. "Operations on Risk Variables." Folia Oeconomica Stetinensia 15, no. 2 (December 1, 2015): 42–52. http://dx.doi.org/10.1515/foli-2015-0034.

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Abstract In the article the author considers and analyzes operations and functions on risk variables. She takes into account the following variables: the sum of risk variables, its product, multiplication by a constant, division, maximum, minimum and median of a sum of random variables. She receives the formulas for probability distribution and basic distribution parameters. She conducts the analysis for dependent and independent random variables. She proposes the examples of the situations in the economy and production management of risk modelled by these operations. The analysis is conducted with the way of mathematical proving. Some of the formulas presented are taken from the literature but others are the permanent results of the author.

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30

Salo, J., H. M. El-Sallabi, and P. Vainikainen. "Reply to "Comments on 'On the Distribution of the Product of Independent Rayleigh Random Variables'"." IEEE Transactions on Antennas and Propagation 54, no. 11 (November 2006): 3571–72. http://dx.doi.org/10.1109/tap.2006.884317.

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31

Matula, Przemyslaw, and Iwona Stepien. "Weak and Almost Sure Convergence for Products of Sums of Associated Random Variables." ISRN Probability and Statistics 2012 (June 14, 2012): 1–14. http://dx.doi.org/10.5402/2012/107096.

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We study weak convergence of product of sums of stationary sequences of associated random variables to the log-normal law. The almost sure version of this result is also presented. The obtained theorems extend and generalize some of the results known so far for independent or associated random variables.

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32

Ly, Sel, Kim-Hung Pho, Sal Ly, and Wing-Keung Wong. "Determining Distribution for the Product of Random Variables by Using Copulas." Risks 7, no. 1 (February 25, 2019): 23. http://dx.doi.org/10.3390/risks7010023.

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Determining distributions of the functions of random variables is one of the most important problems in statistics and applied mathematics because distributions of functions have wide range of applications in numerous areas in economics, finance, risk management, science, and others. However, most studies only focus on the distribution of independent variables or focus on some common distributions such as multivariate normal joint distributions for the functions of dependent random variables. To bridge the gap in the literature, in this paper, we first derive the general formulas to determine both density and distribution of the product for two or more random variables via copulas to capture the dependence structures among the variables. We then propose an approach combining Monte Carlo algorithm, graphical approach, and numerical analysis to efficiently estimate both density and distribution. We illustrate our approach by examining the shapes and behaviors of both density and distribution of the product for two log-normal random variables on several different copulas, including Gaussian, Student-t, Clayton, Gumbel, Frank, and Joe Copulas, and estimate some common measures including Kendall’s coefficient, mean, median, standard deviation, skewness, and kurtosis for the distributions. We found that different types of copulas affect the behavior of distributions differently. In addition, we also discuss the behaviors via all copulas above with the same Kendall’s coefficient. Our results are the foundation of any further study that relies on the density and cumulative probability functions of product for two or more random variables. Thus, the theory developed in this paper is useful for academics, practitioners, and policy makers.

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33

Beylkin, Gregory, Lucas Monzón, and Ignas Satkauskas. "On computing distributions of products of non-negative independent random variables." Applied and Computational Harmonic Analysis 46, no. 2 (March 2019): 400–416. http://dx.doi.org/10.1016/j.acha.2018.01.002.

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34

Latała, Rafał. "L 1-norm of combinations of products of independent random variables." Israel Journal of Mathematics 203, no. 1 (October 2014): 295–308. http://dx.doi.org/10.1007/s11856-014-1076-1.

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35

DI NUNNO, GIULIA. "RANDOM FIELDS: NON-ANTICIPATING DERIVATIVE AND DIFFERENTIATION FORMULAS." Infinite Dimensional Analysis, Quantum Probability and Related Topics 10, no. 03 (September 2007): 465–81. http://dx.doi.org/10.1142/s0219025707002828.

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The non-anticipating stochastic derivative represents the integrand in the best L2-approximation for random variables by Itô non-anticipating integrals with respect to a general stochastic measure with independent values on a space–time product. In this paper some explicit formulas for this derivative are obtained.

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36

Zheng, Fa-mei. "A Limit Theorem for Random Products of Trimmed Sums of i.i.d. Random Variables." Journal of Probability and Statistics 2011 (2011): 1–13. http://dx.doi.org/10.1155/2011/181409.

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Let be a sequence of independent and identically distributed positive random variables with a continuous distribution function , and has a medium tail. Denote and , where , , and is a fixed constant. Under some suitable conditions, we show that , as , where is the trimmed sum and is a standard Wiener process.

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37

Dai, Deliang, and Yuli Liang. "High-Dimensional Mahalanobis Distances of Complex Random Vectors." Mathematics 9, no. 16 (August 7, 2021): 1877. http://dx.doi.org/10.3390/math9161877.

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In this paper, we investigate the asymptotic distributions of two types of Mahalanobis distance (MD): leave-one-out MD and classical MD with both Gaussian- and non-Gaussian-distributed complex random vectors, when the sample size n and the dimension of variables p increase under a fixed ratio c=p/n→∞. We investigate the distributional properties of complex MD when the random samples are independent, but not necessarily identically distributed. Some results regarding the F-matrix F=S2−1S1—the product of a sample covariance matrix S1 (from the independent variable array (be(Zi)1×n) with the inverse of another covariance matrix S2 (from the independent variable array (Zj≠i)p×n)—are used to develop the asymptotic distributions of MDs. We generalize the F-matrix results so that the independence between the two components S1 and S2 of the F-matrix is not required.

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38

Talagrand, M. "A new isoperimetric inequality for product measure and the tails of sums of independent random variables." Geometric and Functional Analysis 1, no. 2 (June 1991): 211–23. http://dx.doi.org/10.1007/bf01896379.

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39

Pederzoli, Giorgio. "Some properties of beta functions and the distribution for the product of independent beta random variables." Trabajos de Estadistica y de Investigacion Operativa 36, no. 1 (February 1985): 122–28. http://dx.doi.org/10.1007/bf02888658.

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40

Nevzorov, Valery B. "Representations of ordered random variables in the terms of sums or products of independent variables." Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy 3(61), no. 4 (2016): 627–40. http://dx.doi.org/10.21638/11701/spbu01.2016.412.

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41

Götze, Friedrich, Alexey Naumov, and Alexander Tikhomirov. "Local laws for non-Hermitian random matrices and their products." Random Matrices: Theory and Applications 09, no. 04 (November 15, 2019): 2150004. http://dx.doi.org/10.1142/s2010326321500040.

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We consider products of independent [Formula: see text] non-Hermitian random matrices [Formula: see text]. Assume that their entries, [Formula: see text], are independent identically distributed random variables with zero mean, unit variance. Götze and Tikhomirov [On the asymptotic spectrum of products of independent random matrices, preprint (2010), arXiv:1012.2710] and O’Rourke and Sochnikov [Products of independent non-Hermitian random matrices, Electron. J. Probab. 16 (2011) 2219–2245] proved that under these assumptions the empirical spectral distribution (ESD) of [Formula: see text] converges to the limiting distribution which coincides with the distribution of the [Formula: see text]th power of random variable uniformly distributed in the unit circle. In this paper, we provide a local version of this result. More precisely, assuming additionally that [Formula: see text] for some [Formula: see text], we prove that ESD of [Formula: see text] converges to the limiting distribution on the optimal scale up to [Formula: see text] (up to some logarithmic factor). Our results generalize the recent results of Bourgade et al. [Local circular law for random matrices, Probab. Theory Related Fields 159 (2014) 545–595], Tao and Vu [Random matrices: Universality of local spectral statistics of non-Hermitian matrices, Ann. Probab. 43 (2015) 782–874] and Nemish [Local law for the product of independent non-hermitian random matrices with independent entries, Electron. J. Probab. 22 (2017) 1–35]. We also give further development of Stein’s type approach to estimate the Stieltjes transform of ESD.

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42

Siregar, Dahrul, Ahmad Harun Daulay, and Saparuddin Siregar. "Increasing Customer's Saving Interest through Religiusity, Product Perception and Knowledge." Budapest International Research and Critics Institute (BIRCI-Journal): Humanities and Social Sciences 4, no. 1 (February 4, 2021): 918–25. http://dx.doi.org/10.33258/birci.v4i1.1693.

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The number of Islamic banks is not significant with the Muslim population in this country. The efforts to increase the number of customers and use of Islamic/ sharia products need to be increased. The aim of this study is to analyze the effect of religiosity, product perception and knowledge on people’s saving interest in Islamic banking products. The target population and sample in this study are 135 Moslems by using purposive random sampling technique. Data collection using a questionnaire. The results showed that all independent variables of religiosity, product perception and knowledge have a positive and significant effect on saving interest in Islamic banking products. The most dominant variable in influencing consumers' interest in saving is product perception.

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43

Nevzorov, V. B. "Representations of ordered random variables in the terms of the sums or products of independent variables." Vestnik St. Petersburg University: Mathematics 49, no. 4 (October 2016): 361–70. http://dx.doi.org/10.3103/s1063454116040117.

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44

LIU, YAN, and QIHE TANG. "THE SUBEXPONENTIAL PRODUCT CONVOLUTION OF TWO WEIBULL-TYPE DISTRIBUTIONS." Journal of the Australian Mathematical Society 89, no. 2 (May 18, 2010): 277–88. http://dx.doi.org/10.1017/s1446788710000182.

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AbstractLet X1 and X2 be two independent and nonnegative random variables with distributions F1 and F2, respectively. This paper proves that if both F1 and F2 are of Weibull type and fulfill certain easily verifiable conditions, then the distribution of the product X1X2, called the product convolution of F1 and F2, belongs to the class 𝒮* and, hence, is subexponential.

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45

Romdhoni, Abdul Haris, and Dita Ratna Sari. "Pengaruh Pengetahuan, Kualitas Pelayanan, Produk, dan Religiusitas terhadap Minat Nasabah untuk Menggunakan Produk Simpanan pada Lembaga Keuangan Mikro Syariah." Jurnal Ilmiah Ekonomi Islam 4, no. 02 (July 31, 2018): 136. http://dx.doi.org/10.29040/jiei.v4i02.307.

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This study aims to examine the influence of knowledge, service quality, product, and religiosity on customers' interest in using savings products at BMT Amanah Ummah Gumpang Kartasura, Sukoharjo. In this study for independent variables are knowledge, service quality, product, and religiosity, while for the dependent variable in this study is the interest of customers using deposit products. This study uses a sample of 100 people with sampling using a random sampling method. This study uses quantitative methods and primary data using a questionnaire that must be answered by the respondent. Data analysis in this study used Multiple Linear Regression. The results of this study can be concluded based on the t test, the variables of knowledge and religiosity have an influence on the interest of customers using savings products. While service and product quality variables do not have an influence on customer interest by using deposit products. Based on the F test shows that the knowledge, quality of service, products, and religiosity simultaneously influence the interest of customers to use savings products at BMT Amanah Ummah Gumpang Kartasura, Sukoharjo.

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46

Yamashita Rios de Sousa, Arthur Matsuo, Hideki Takayasu, Didier Sornette, and Misako Takayasu. "Sigma-Pi Structure with Bernoulli Random Variables: Power-Law Bounds for Probability Distributions and Growth Models with Interdependent Entities." Entropy 23, no. 2 (February 19, 2021): 241. http://dx.doi.org/10.3390/e23020241.

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The Sigma-Pi structure investigated in this work consists of the sum of products of an increasing number of identically distributed random variables. It appears in stochastic processes with random coefficients and also in models of growth of entities such as business firms and cities. We study the Sigma-Pi structure with Bernoulli random variables and find that its probability distribution is always bounded from below by a power-law function regardless of whether the random variables are mutually independent or duplicated. In particular, we investigate the case in which the asymptotic probability distribution has always upper and lower power-law bounds with the same tail-index, which depends on the parameters of the distribution of the random variables. We illustrate the Sigma-Pi structure in the context of a simple growth model with successively born entities growing according to a stochastic proportional growth law, taking both Bernoulli, confirming the theoretical results, and half-normal random variables, for which the numerical results can be rationalized using insights from the Bernoulli case. We analyze the interdependence among entities represented by the product terms within the Sigma-Pi structure, the possible presence of memory in growth factors, and the contribution of each product term to the whole Sigma-Pi structure. We highlight the influence of the degree of interdependence among entities in the number of terms that effectively contribute to the total sum of sizes, reaching the limiting case of a single term dominating extreme values of the Sigma-Pi structure when all entities grow independently.

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47

Ivanoff, B. Gail, and N. C. Weber. "Tail probabilities for weighted sums of products of normal random variables." Bulletin of the Australian Mathematical Society 58, no. 2 (October 1998): 239–44. http://dx.doi.org/10.1017/s0004972700032214.

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Weighted sums of products of independent normal random variables arise naturally as distributional limits for various statistics. This note investigates the rate at which the tail probability of these sums approaches zero.

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48

Denisov, Denis, and Bert Zwart. "On a Theorem of Breiman and a Class of Random Difference Equations." Journal of Applied Probability 44, no. 04 (December 2007): 1031–46. http://dx.doi.org/10.1017/s0021900200003715.

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We consider the tail behavior of the product of two independent nonnegative random variables X and Y. Breiman (1965) has considered this problem, assuming that X is regularly varying with index α and that E{Yα+ε} < ∞ for some ε > 0. We investigate when the condition on Y can be weakened and apply our findings to analyze a class of random difference equations.

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49

Denisov, Denis, and Bert Zwart. "On a Theorem of Breiman and a Class of Random Difference Equations." Journal of Applied Probability 44, no. 4 (December 2007): 1031–46. http://dx.doi.org/10.1239/jap/1197908822.

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We consider the tail behavior of the product of two independent nonnegative random variables X and Y. Breiman (1965) has considered this problem, assuming that X is regularly varying with index α and that E{Yα+ε} < ∞ for some ε > 0. We investigate when the condition on Y can be weakened and apply our findings to analyze a class of random difference equations.

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Leipnik, Roy B. "On lognormal random variables: I-the characteristic function." Journal of the Australian Mathematical Society. Series B. Applied Mathematics 32, no. 3 (January 1991): 327–47. http://dx.doi.org/10.1017/s0334270000006901.

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Abstract:

AbstractThe characteristic function of a lognormal random variable is calculated in closed form as a rapidly convergent series of Hermite functions in a logarithmic variable. The series coefficients are Nielsen numbers, defined recursively in terms of Riemann zeta functions. Divergence problems are avoided by deriving a functional differential equation, solving the equation by a de Bruijn integral transform, expanding the resulting reciprocal Gamma function kernel in a series, and then invoking a convergent termwise integration. Applications of the results and methods to the distribution of a sum of independent, not necessarily identical lognormal variables are discussed. The result is that a sum of lognormals is distributed as a sum of products of lognormal distributions. The case of two lognormal variables is outlined in some detail.

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