scholarly journals A reciprocal relation for Hermite polynomials

10.53733/88 ◽  
2021 ◽  
Vol 51 ◽  
pp. 109-114
Author(s):  
Saralees Nadarajah ◽  
C Withers
Keyword(s):  
Hermite Polynomial ◽  
Hermite Polynomials ◽  
Random Variable ◽  
Normal Random Variable ◽  
Reciprocal Relation ◽  
Unit Normal

For $x\in \mathbb{R}$, the ordinary Hermite polynomial $H_k(x)$ can be written\begin{eqnarray*}\displaystyleH_k(x)= \mathbb{E} \left[ (x + {\rm i} N)^k \right] =\sum_{j=0}^k {k\choose j} x^{k-j} {\rm i}^j \mathbb{E} \left[ N^j \right],\end{eqnarray*}where ${\rm i} = \sqrt{-1}$ and $N$ is a unit normal random variable.  We prove the reciprocal relation\begin{eqnarray*}\displaystylex^k=\sum_{j=0}^k {k\choose j} H_{k-j}(x)\ \mathbb{E} \left[ N^j \right].\end{eqnarray*}A similar result is given for the multivariate Hermite polynomial.  

Generation of Correlated Logistic-Normal Random Variates for Medical Decision Trees

1998 ◽  
Vol 37 (03) ◽  
pp. 235-238 ◽  
Author(s):  
M. El-Taha ◽  
D. E. Clark
Keyword(s):  
Monte Carlo ◽  
Decision Trees ◽  
Computer Algebra ◽  
Taylor Series ◽  
Computer Algebra System ◽  
Random Variable ◽  
Monte Carlo Analysis ◽  
Normal Random Variable ◽  
Medical Decision ◽  
Theoretical Results

AbstractA Logistic-Normal random variable (Y) is obtained from a Normal random variable (X) by the relation Y = (ex)/(1 + ex). In Monte-Carlo analysis of decision trees, Logistic-Normal random variates may be used to model the branching probabilities. In some cases, the probabilities to be modeled may not be independent, and a method for generating correlated Logistic-Normal random variates would be useful. A technique for generating correlated Normal random variates has been previously described. Using Taylor Series approximations and the algebraic definitions of variance and covariance, we describe methods for estimating the means, variances, and covariances of Normal random variates which, after translation using the above formula, will result in Logistic-Normal random variates having approximately the desired means, variances, and covariances. Multiple simulations of the method using the Mathematica computer algebra system show satisfactory agreement with the theoretical results.

scholarly journals Hermite polynomial normal transformation for structural reliability analysis

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Jinsheng Wang ◽  
Muhannad Aldosary ◽  
Song Cen ◽  
Chenfeng Li
Keyword(s):  
Reliability Analysis ◽  
Hermite Polynomial ◽  
Structural Reliability ◽  
Hermite Polynomials ◽  
Random Variables ◽  
Transformation Method ◽  
Statistical Moments ◽  
Content Type ◽  
Structural Reliability Analysis ◽  
Normal Transformation

Purpose Normal transformation is often required in structural reliability analysis to convert the non-normal random variables into independent standard normal variables. The existing normal transformation techniques, for example, Rosenblatt transformation and Nataf transformation, usually require the joint probability density function (PDF) and/or marginal PDFs of non-normal random variables. In practical problems, however, the joint PDF and marginal PDFs are often unknown due to the lack of data while the statistical information is much easier to be expressed in terms of statistical moments and correlation coefficients. This study aims to address this issue, by presenting an alternative normal transformation method that does not require PDFs of the input random variables. Design/methodology/approach The new approach, namely, the Hermite polynomial normal transformation, expresses the normal transformation function in terms of Hermite polynomials and it works with both uncorrelated and correlated random variables. Its application in structural reliability analysis using different methods is thoroughly investigated via a number of carefully designed comparison studies. Findings Comprehensive comparisons are conducted to examine the performance of the proposed Hermite polynomial normal transformation scheme. The results show that the presented approach has comparable accuracy to previous methods and can be obtained in closed-form. Moreover, the new scheme only requires the first four statistical moments and/or the correlation coefficients between random variables, which greatly widen the applicability of normal transformations in practical problems. Originality/value This study interprets the classical polynomial normal transformation method in terms of Hermite polynomials, namely, Hermite polynomial normal transformation, to convert uncorrelated/correlated random variables into standard normal random variables. The new scheme only requires the first four statistical moments to operate, making it particularly suitable for problems that are constraint by limited data. Besides, the extension to correlated cases can easily be achieved with the introducing of the Hermite polynomials. Compared to existing methods, the new scheme is cheap to compute and delivers comparable accuracy.

Improved approximate confidence intervals for the mean of a log-normal random variable

2002 ◽  
Vol 21 (10) ◽  
pp. 1443-1459 ◽  
Author(s):  
Douglas J. Taylor ◽  
Lawrence L. Kupper ◽  
Keith E. Muller
Keyword(s):  
Confidence Intervals ◽  
Random Variable ◽  
Normal Random Variable ◽  
The Mean ◽  
Approximate Confidence Intervals ◽  
Log Normal

scholarly journals Occupation times of alternating renewal processes with Lévy applications

2018 ◽  
Vol 55 (4) ◽  
pp. 1287-1308 ◽  
Author(s):  
Nicos Starreveld ◽  
Réne Bekker ◽  
Michel Mandjes
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Random Variable ◽  
Levy Processes ◽  
Normal Random Variable ◽  
Occupation Time ◽  
Renewal Processes ◽  
Tail Asymptotics ◽  
Occupation Times

AbstractIn this paper we present a set of results relating to the occupation time α(t) of a processX(·). The first set of results concerns exact characterizations of α(t), e.g. in terms of its transform up to an exponentially distributed epoch. In addition, we establish a central limit theorem (entailing that a centered and normalized version of α(t)∕tconverges to a zero-mean normal random variable ast→∞) and the tail asymptotics of ℙ(α(t)∕t≥q). We apply our findings to spectrally positive Lévy processes reflected at the infimum and establish various new occupation time results for the corresponding model.

On the normal convergence of a class of simple batch epidemics

1981 ◽  
Vol 18 (01) ◽  
pp. 31-41
Author(s):  
Naftali A. Langberg
Keyword(s):  
Stochastic Process ◽  
Mild Condition ◽  
Random Variable ◽  
Normal Random Variable ◽  
Contagious Disease ◽  
Normal Convergence

A group of n susceptible individuals exposed to a contagious disease is considered. It is assumed that at each instant in time one or more susceptible individuals can contract the disease. The progress of this epidemic is modeled by a stochastic process Xn (t), t in [0,∞) representing the number of infective individuals at time t. It is shown that Xn (t), with the suitable standardization and under a mild condition, converges in distribution as n → ∞to a normal random variable for all t in (0, t 0), where t 0 is an identifiable number.

scholarly journals Approximating Multivariate Tempered Stable Processes

2012 ◽  
Vol 49 (1) ◽  
pp. 167-183 ◽  
Author(s):  
Boris Baeumer ◽  
Mihály Kovács
Keyword(s):  
Particle Tracking ◽  
Random Variable ◽  
Normal Random Variable ◽  
Stable Processes ◽  
Lévy Measure ◽  
Simple Method ◽  
Linear Rate ◽  
Skorokhod Topology ◽  
Lower Dimensional ◽  
Random Angle

We give a simple method to approximate multidimensional exponentially tempered stable processes and show that the approximating process converges in the Skorokhod topology to the tempered process. The approximation is based on the generation of a random angle and a random variable with a lower-dimensional Lévy measure. We then show that if an arbitrarily small normal random variable is added, the marginal densities converge uniformly at an almost linear rate, providing a critical tool to assess the performance of existing particle tracking codes.

scholarly journals A CLT FOR THE THIRD INTEGRATED MOMENT OF BROWNIAN LOCAL TIME INCREMENTS

2011 ◽  
Vol 11 (01) ◽  
pp. 5-48
Author(s):  
JAY ROSEN
Keyword(s):  
Brownian Motion ◽  
Local Time ◽  
Random Variable ◽  
Normal Random Variable ◽  
Higher Moments ◽  
Second Moment ◽  
The Third ◽  
Brownian Local Time

Let [Formula: see text] denote the local time of Brownian motion. Our main result is to show that for each fixed t[Formula: see text] as h → 0, where η is a normal random variable with mean zero and variance one, that is independent of [Formula: see text]. This generalizes our previous result for the second moment. We also explain why our approach will not work for higher moments.

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