Truncated random variables with application to random surfaces

There has been considerable interest in obtaining discrete results for random surfaces. Standard results have been published in journals of physics or engineering which have emphasised the applications. This paper gives a detailed background of the mathematical methods needed so that the central connection, namely truncated random variables, between these standard results can be understood. Distributions of discrete peak measures are obtained from the distributions of discrete profile measures of a random Gaussian surface by applying results for the distributions of truncated random variables. This enable the moments to be obtained from known results for the truncated distributions.     

On simulating truncated stable random variables

Soltani and Shirvani (Comput Stat 25:155–161, 2010) proposed a scheme for simulating truncated stable random variables. That involves solving a nonlinear transformation in each realization. Here, we propose alternative schemes to generate truncated stable random variables. Our schemes are more general (for example, incorporates one-sided and two-sided truncations) and are shown to be more efficient.     

Imprecise random variables,random sets,and Monte Carlo simulation

The paper addresses the evaluation of upper and lower probabilities induced by functions of an imprecise random variable. Given a function g and a family $X_{\lambda }$ of random variables, where the parameter λ ranges in an index set Λ, one may ask for the upper/lower probability that $g(X_{\lambda })$ belongs to some Borel set B. Two interpretations are investigated. In the first case, the upper probability is computed as the supremum of the probabilities that $g(X_{\lambda })$ lies in B. In the second case, one considers the random set generated by all $g(X_{\lambda })$, $\lambda \in \mathrm{\Lambda }$, e.g. by transforming $X_{\lambda }$ to standard normal as a common probability space, and computes the corresponding upper probability. The two results are different, in general. We analyze this situation and highlight the implications for Monte Carlo simulation. Attention is given to efficient simulation procedures and an engineering application is presented.     

On simulation of tempered stable random variates

Various simulation methods for tempered stable random variates with stability index greater than one are investigated with a view towards practical implementation, in particular cases of very small scale parameter, which correspond to increments of a tempered stable Lévy process with a very short stepsize. Methods under consideration are based on acceptance-rejection sampling, a Gaussian approximation of a small jump component, and infinite shot noise series representations. Numerical results are presented to discuss advantages, limitations and trade-off issues between approximation error and required computing effort. With a given computing budget, an approximative acceptance-rejection sampling technique Baeumer and Meerschaert (2009) [11] is both most efficient and handiest in the case of very small scale parameter and moreover, any desired level of accuracy may be attained with a small amount of additional computing effort.     

Generating correlated random variables for a simulation model

In most simulation textbooks, a great deal of attention is given to generating independent random variables. The topic of generating correlated random variables is either omitted or given only a cursory analysis. The purpose of this note is to illustrate how correlated random variables were generated in a simulation model for analysing a firm's ability to meet the demand for its product.     

On a characterization question for symmetric random variables

If X₁, X₂ are independent with common density g symmetric about zero, then $\text{P(X}{}_1\text{+ αX}{}_2\text{> 0) =}\text{1}\text{2}$ for all real α. We provide a counter example to show that the converse is false and thus settle a question posed by Burdick (1972).     

Extending simulation uses of antithetic variables: partially monotone functions, random permutations, and random subsets

We show how to effectively use antithetic variables to evaluate the expected value of (a) functions of independent random variables, when the functions are monotonic in only some of their variables, (b) Schur functions of random permutations, and (c) monotone functions of random subsets.     

On the simulation of shot noise and some other random variables

It is shown that the random voltage V_t resulting from pulses with independent random amplitude Y_i Poisson arrivals, and exponential decay, can be asymptotically represented, in the stationary case, by the following random variable; namely a sum of products of random variables:

$\text{W=}\text{∑}\text{i=1}\text{∞}\text{U}{}_{\mathrm{i}}\text{Y}{}_{\mathrm{i}}\text{,}$

where

$\text{U}{}_{\mathrm{i}}\text{=}\text{∏}\text{j=1}\text{i}\text{X}{}_{\mathrm{j}}{}^{\mathrm{\beta /\lambda }}\text{.}$

Here X_j are independent uniform random variables, β>0 is the decay parameter, λ>0 is the rate of the Poisson process.     

On characterization of multivariate stable distributions via random linear statistics

LetX ₁,X ₂, ...,X _n be independent and identically distributed random vectors inR ^d , and letY=(Y ₁,Y ₂, ...,Y _n )′ be a random coefficient vector inR ⁿ , independent ofX _j ^/′ . We characterize the multivariate stable distributions by considering the independence of the random linear statistic $$U = Y_1 X_1 + Y_2 X_2 + \cdot \cdot \cdot + Y_n X_n $$ and the random coefficient vectorY.     

A characterization for truncated cauchy random variables with nonzero skewness parameter

Soltani and Shirvani (Comput Stat 25:155–161, 2010) provided a characterization and a simulation method for truncated stable random variables when the characteristic exponent $\alpha \ne 1 $ , and left the case $\alpha =1$ open. The case of $\alpha =1$ is treated in this article.