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A new system of multivariate distributions with fixed marginal distributions is introduced via the consideration of random variates that are randomly chosen pairs of order statistics of the marginal distributions. The distributions allow arbitrary positive or negative Pearson correlations between pairs of random variates and generalise the Farlie-Gumbel-Morgenstern distribution. It is shown that the copulas of these distributions are special cases of the Bernstein copula. Generation of random numbers from the distributions is described, and formulas for the Kendall and grade (Spearman) correlations are given. Procedures for data fitting are described and illustrated with examples.

We propose new control variates for variance reduction in estimation of mean values using the Metropolis–Hastings algorithm. Traditionally, states that are rejected in the Metropolis–Hastings algorithm are simply ignored, which intuitively seems to be a waste of information. We present a setting for construction of zero mean control variates for general target and proposal distributions and develop ideas for the standard Metropolis–Hastings and reversible jump algorithms. We give results for three simulation examples. We get best results for variates that are functions of the current state x and the proposal y, but we also consider variates that in addition are functions of the Metropolis–Hastings acceptance/rejection decision. The variance reduction achieved varies depending on the target distribution and proposal mechanisms used. In simulation experiments, we typically achieve relative variance reductions between 15% and 35%.

A novel, simple and efficient method for the generation of Tikhonov (a.k.a. von Mises) random variates is proposed. In the proposed method, circular variates of a prescribed Tikhonov distribution pT(x;alpha,xi) are generated via the transformation of variates selected randomly, on a one-for-one basis, from a bank of K distinct Cauchy and Gaussian generators. The mutually exclusive probabilities of sampling from each of the Cauchy or Gaussian generators, as well as the variance and half-width parameters that specify the latter, are derived directly from the Cauchy, Gaussian and Tikhonov characteristic functions, all of which are either known or given in closed form. The proposed random mixture technique is extremely efficient in that a single pair of uniform random numbers is consumed in the generation of each Tikhonov (or von Mises) sample, regardless of the prescribed concentration and centrality parameters (alpha, xi), all requiring neither the rejection of samples, nor the repetitive evaluation of computationally demanding functions. Additional attractive features of the method are as follows. By construction, the first (dominant) N circular moments of Tikhonov variates generated with the proposed random mixture technique are the ones that best approximate their corresponding theoretical values, with errors measured exactly. The exact distribution of generated Tikhonov variates is determined analytically, and its (Kullback-Leibler) divergence to the exact Tikhonov PDF is shown also analytically to be negligible. Finally, the technique establishes a connection between Tikhonov and Gaussian variates which can be exploited, e.g., in the generation of piecewise-continuous pseudo-random functions with Tikhonov-distributed outcomes.

This paper proposes an envelope rejection method for computer sampling from the negative binomial distribution, with index parameter @a and probability parameter p, based on a simple probability distribution inequality. Comparative timings with existing methods of generating NB variates show that the proposed method is comparable or faster when @a is not too large (@a 0.7.

We describe how dependencies between random numbers generated with some popular pseudo-random number generators can be detected using general purpose machine-learning techniques. This is a novel approach, since usually pseudo-random number generators are evaluated using tests specifically designed for this purpose. Such specific tests are more sensitive. Hence, detecting the dependence using machine-learning methods implies that the dependence is indeed very strong. The most important example of a generator, where dependencies may easily be found using our approach, is MATLAB's function rand if the method state is used. This method was the default in MATLAB versions between 5 (1995) and 7.3 (2006b), i.e., for more than 10 years. In order to evaluate the strength of the dependence in it, we used the same machine-learning tools to detect dependencies in some other random number generators, which are known to be bad or insufficient for large simulations: the infamous RANDU, ANSIC, the oldest generator in C library, minimal standard generator, suggested by Park and Miller (1988), and the rand function in Microsoft C compiler.

Random number generators are widely used in different applications. However, it is difficult to obtain a good random number generator in low precision and resource constrained system, such as an eight-bit micro-controller system which is still commonly used in industrial and consumer markets. This paper provides a practical solution for this problem based on chaotic systems. By the use of a modified Chua's circuit, it is demonstrated that the sampled state, after post-processing by a high-dimensional chaotic map, can be used as a random source even in an eight-bit environment. The randomness of the generated sequence is testified and confirmed by different statistical tests and the up-to-date statistical suite.

For decades, stochastic modellers have used computerized random number generators to produce random numeric sequences fitting a specified statistical distribution. Unfortunately, none of the random number generators we tested satisfactorily produced the target distribution. The result is generated distributions whose mean even diverges from the mean used to generate them, regardless of the length of run. Non-uniform distributions from short sequences of random numbers are a major problem in stochastic climate generation, because truly uniform distributions are required to produce the intended climate parameter distributions. In order to ensure generation of a representative climate with the stochastic weather generator CLIGEN within a 30-year run, we tested the climate output resulting from various random number generators. The resulting distributions of climate parameters showed significant departures from the target distributions in all cases. We traced this failure back to the uniform random number generators themselves. This paper proposes a quality control approach to select only those numbers that conform to the expected distribution being retained for subsequent use. The approach is based on goodness-of-fit analysis applied to the random numbers generated. Normally distributed deviates are further tested with confidence interval tests on their means and standard deviations. The positive effect of the new approach on the climate characteristics generated and the subsequent deterministic process-based hydrology and soil erosion modelling are illustrated for four climatologically diverse sites.

Statistical distributions generated from any J- or U-shaped random variables are cumbersome to derive if not completely indefinable and thus are unavailable analytically because of the singularities at the tails of the basic random variable. This paper presents a computational method for providing a numerical convolution derived from a basic U-shaped random variable composed of a continuous part mixed with (or contaminated by) a discrete part at the tails. The J-shaped sampling distribution case is implied as a special case. Though the computations are based on a background Normal Distribution, it can be generalized on any other distribution. Such distributions will open up an area of sampling distributions of mixed random variables that are not elaborately covered in textbooks dealing with the theory of distributions.

A new algorithm for generating random variates from the t distribution is presented. It is based on a generalization of the ratio of uniforms method for the Cauchy distribution. It is simple, short and uniformly fast for all degrees of freedom.

A serious gap in the Proof of Pakes’s paper on the convolution equivalence of infinitely divisible distributions on the line is completely closed. It completes the real analytic approach to Sgibnev’s theorem. Then the convolution equivalence of random sums of IID random variables is discussed. Some of the results are applied to random walks and Levy processes. In particular, results of Bertoin and Doney and of Korshunov on the distribution tail of the supremum of a random walk are improved. Finally, an extension of Rogozin’s theorem is proved.

The importance of random number generators has increased over the years. This follows from the fact that contemporary research methods rely more and more on simulation and the increased importance of encryption technology. The output of a random number generator is created by either an algorithm or a physical device. The most popular method for random number generation is through the use of an algorithm. This article presents a new category of physical random bit generator that is packaged by several manufacturers. A statistical analysis of the output from the generators is given.

A program for identifying the form of the distribution law of a random quantity is described. The dependence of the optimum number of intervals when constructing histograms from a volume of the sample (10–100,000) and of the kurtosis (in the limits 2–9.65) is determined by computer modeling. Using 50 different functions, the program can also model random quantities with unimodal and bimodal distribution laws.

In early fall 2004, the Air Force Studies and Analyses Agency (AFSAA) forwarded a proposed research topic, endorsed by the Air Force Chief of Staff, to the Department of Operational Sciences at the Air Force Institute of Technology. This topic requested an examination of potential variance reduction techniques (VRTs) for use with the THUNDER combat model. THUNDER is a large-scale, highly aggregated, discrete-event simulation of campaign-level military operations used to examine issues involving the utility and effectiveness of air and space power in a theater-level, joint warfare context. A new, state of the art combined multiple recursive generator was incorporated into THUNDER to synchronize the random inputs for common random numbers (CRN) and antithetic variates (AV). This work discusses the application of CRN, AV, and control variates (CV), within THUNDER, and the resulting variance reductions obtained for an example scenario.

Random numbers generators are widely used in different fields like cryptography, gaming development, artificial intelligence, etc. Being the engine of some security (cryptographic) protocols, the development of good and secure Random Number Generators aroused the attention of the research communities worldwide. Random Number Generators split into two categories: True Random Number Generators (based on physical non-deterministic processes, like the jitter of an oscillator) and Pseudo Random Number Generators (based on mathematical properties, where the output at the time t is based somehow on the output on time t-1). It is well known that the output distribution of a Random Number Generator (either True or Pseudo) output tends toward 50%. There are some particular cases where this probability must be different (some of these situations will be described here) and this paper will present a way of obtaining this. The novelty of this paper is based on a new approach on the classic Gollmann Cascade PRNG and starting from this, the development of a FPGA implementation to increase its security.

In the paper the method based on order statistics for processing of the random observations with so called Flatten — Gaussian distribution that is a convolution of rectangular and normal distributions, is proposed. This method gives a smaller standard uncertainty of a location parameter compared to a standard uncertainty of a mean value. The efficiency of the method increases with the increase content of the uniform component. The implementation of the proposed method does not require complicated calculations and based on an analytical expression of a normalized Flatten — Gaussian density distribution. The research results of method, which are the standard uncertainty depending on the relative content of components and the number of observations, are presented.

A convolution regression model with random design is considered. We investigate the estimation of the derivatives of an unknown function, element of the convolution product. We introduce new estimators based on wavelet methods and provide theoretical guarantees on their good performances.

Generalized simulated annealing is an important method for finding the global minimum of a function. However, it requires the generation of Tsallis random numbers. Here we show that a D-dimensional Tsallis random variable is a multivariate t or Student distributed random variable. Using this finding we design an exact Tsallis random number generator and, finally, present some applications of this generator to generalized simulated annealing.

The Gibbs sampler is a popular Markov chain Monte Carlo routine for generating random variates from distributions otherwise difficult to sample. A number of implementations are available for running a Gibbs sampler varying in the order through which the full conditional distributions used by the Gibbs sampler are cycled or visited. A common, and in fact the original, implementation is the random scan strategy, whereby the full conditional distributions are updated in a randomly selected order each iteration. In this paper, we introduce a random scan Gibbs sampler which adaptively updates the selection probabilities or “learns” from all previous random variates generated during the Gibbs sampling. In the process, we outline a number of variations on the random scan Gibbs sampler which allows the practitioner many choices for setting the selection probabilities and prove convergence of the induced (Markov) chain to the stationary distribution of interest. Though we emphasize flexibility in user choice and specification of these random scan algorithms, we present a minimax random scan which determines the selection probabilities through decision theoretic considerations on the precision of estimators of interest. We illustrate and apply the results presented by using the adaptive random scan Gibbs sampler developed to sample from multivariate Gaussian target distributions, to automate samplers for posterior simulation under Dirichlet process mixture models, and to fit mixtures of distributions.

This paper extends the sum-of-uniforms method to generate correlated random variables with certain marginal distributions. We first use the transformation method to derive the joint probability density function of the correlated uniform random variables. We also demonstrate that the sum-of-uniforms method can be extended to generate correlated random variables with certain marginal distributions including uniform, exponential, Erlang, Bernoulli, binomial, geometric, and negative binomial. Finally, this paper presents the exact correlation coefficients of such correlated random variables.

The acceptance/rejection approach is widely used in universal nonuniform random number generators. Its key part is an accurate approximation of a given probability density from above by a hat function. This article uses a piecewise constant hat function, whose values are overestimates of the density on the elements of the partition of the domain. It uses a sawtooth overestimate of Lipschitz continuous densities, and then examines all local maximizers of such an overestimate. The method is applicable to multivariate multimodal distributions. It exhibits relatively short preprocessing time and fast generation of random variates from a very large class of distributions.

Markov chain Monte Carlo (MCMC) routines have revolutionized the application of Monte Carlo methods in statistical application and statistical computing methodology. The Hastings sampler, encompassing both the Gibbs and Metropolis samplers as special cases, is the most commonly applied MCMC algorithm. The performance of the Hastings sampler relies heavily on the choice of sweep strategy, that is, the method by which the components or blocks of the random variable X of interest are visited and updated, and the choice of proposal distribution, that is the distribution from which candidate variates are drawn for the accept–reject rule in each iteration of the algorithm. We focus on the random sweep strategy, where the components of X are updated in a random order, and random proposal distributions, where the proposal distribution is characterized by a randomly generated parameter. We develop an adaptive Hastings sampler which learns from and adapts to random variates generated during the algorithm towards choosing the optimal random sweep strategy and proposal distribution for the problem at hand. As part of the development, we prove convergence of the random variates to the distribution of interest and discuss practical implementations of the methods. We illustrate the results presented by applying the adaptive componentwise Hastings samplers developed to sample multivariate Gaussian target distributions and Bayesian frailty models. c 2004 Elsevier B.V. All rights reserved.

Knuth [The Art of Computer Programming, vol. 2: Semi-Numerical Algorithms, 3rd ed., Addison-Wesley, Reading, MA, 1997] shows that iterations of a random function perform poorly on average as a random number generator. He proposes a generalization in which the next value depends on two or more previous values. This note demonstrates, via an analysis of the cycle length of a random k-ary map, the equally poor performance of a random instance in Knuth's more general model.

Pseudo Random Number Generator (PRNG) uses a deterministic system and an initial seed to generate random numbers. In order for the output sequence to be truly random, a truly random input seed is used. Most True Random Number Generators (TRNG), use noise in the form nuclear decay, atmospheric noise, electrical noise or Brownian motion as their initial seed. In order to reduce the computational complexity, we use a simple setup of a fish tank as the variable environment, capturing its images over time. The image data is then applied to a reduction algorithm and hash function to generate the initial seed. We propose a cost efficient method of extracting the true seed from the image data and applying it to a pseudo random generator, a Linear Congruential Generator (LCG) in our case to give true random numbers.

An error in the Gaussian random variate generator by Kinderman and Ramage is described that results in the generation of random variates with an incorrect distribution. An additional statement that corrects the original algorithm is given.

A one-way coupled chaotic map lattice is used for generating pseudo-random numbers. It is shown that with suitable cooperative applications of both chaotic and conventional approaches, the output of the spatiotemporally chaotic system can easily meet the practical requirements of random numbers, i.e., excellent random statistical properties, long periodicity of computer realizations, and fast speed of random number generations. This pseudo-random number generator system can be used as ideal synchronous and self-synchronizing stream cipher systems for secure communications.

Summary This paper investigates a new random contraction scheme which complements the length-biasing and convolution contraction schemes considered in the literature. A random power contraction is used with order statistics, leading to new and elegant characterizations of the power distribution. In view of Rossberg's counter-example of a non-exponential law with exponentially distributed spacings of order statistics, possibly the most appealing consequence of the result is a characterization of the exponential distribution via an independent exponential shift of order statistics.

In this thesis, Bayesian parameter estimation procedure is discussed for the mean change model of multivariate normal random variates under the assumption of noninformative priors for all the parameters. Parameters are estimated by Gibbs sampling method. In Gibbs sampler, the change point parameter is generated by Metropolis-Hastings algorithm. We apply our methodology to numerical data to examine it.

The most efficient and easiest way for implementing a multiple recursive random number generator is the approximate factoring (AF) method. In the literature, the AF method can be applied in the three types of restriction on multiplier. This paper considers the effectiveness and efficiency of these restrictions from the theoretical and empirical points of view. The most effective and efficient restriction is presented. The number of arithmetic operations required by the AF method is also discussed.

A random number generator based on the addition of points on an elliptic curve over finite fields is proposed. By using the proposed generator together with the elliptic curve cryptography (ECC) algorithm, we can save hardware and software components. For hardware implementation, the proposed generator can be implemented using the existing ECC arithmetic processor. Up to 29% of gate counts can be saved when compared to the case of implementing a random number generator separately. Theoretical analyses show that periods of the proposed random number generator are sufficiently long. Moreover, the generated sequences have passed the FIPS 140-2 statistical tests. As a result, the proposed generator is suitable to be a reliable and efficient random number generator in ECC systems.

A wide variety of random number generators is discussed based on truncating functional outcomes and considering the fractional remainders as random digits in the interval, ]0,1[. These generators do not require seeding in the traditional sense, moreover offer an infinite number of outcomes, apparently without periodicity. These generators are trivial in their software implementation. Those that are based on logarithms perform best in tests of randomness. When applied for spatial simulation, though, quality of the random number generator seems unimportant to the outcome.

For continuous random variables, their distribution data can be approximated to generalized exponential-sum distribution functions by optimization tools like TOMLAB. Generalized exponential-sum distribution functions have larger representations and wider applications than pure exponential-sum distribution functions. On the bases of the n-fold convolution of pure exponential-sum distribution functions, the n-fold convolution of generalized exponential-sum distribution functions has been developed in this paper. Compared with the n-fold convolution of pure exponential-sum distribution functions, the results presented in this paper for the calculation of the n-fold convolution of generalized exponential-sum distribution functions are more general and applicable. As a matter of fact, the solution for computing the n-fold convolution of pure exponential-sum distribution functions is contained in the solution for the calculation of the n-fold convolution of generalized exponential-sum distribution functions. Limited testing results show that the formulae presented in this paper are correct.

We use inequalities to design short universal algorithms that can be used to generate random variates from large classes of univariate continuous or discrete distributions (including all log-concave distributions). The expected time is uniformly bounded over all these distributions for a particular generator. The algorithms can be implemented in a few lines of high level language code.

27.1 The RandomGen class, and the StdGen generator 23627.2 The Random class 23927.3 The global random number generator 240

Monte Carlo computations are considered easy to parallelize. However, the results can be adversely affected by defects in the parallel pseudorandom number generator used. A parallel pseudorandom number generator must be tested for two types of correlations--(i) intra-stream correlation, as for any sequential generator, and (ii) inter-stream correlation for correlations between random number streams on different processes. Since bounds on these correlations are difficult to prove mathematically, large and thorough empirical tests are necessary. Many of the popular pseudorandom number generators in use today were tested when computational power was much lower, and hence they were evaluated with much smaller test sizes.This paper describes several tests of pseudorandom number generators, both statistical and application-based. We show defects in several popular generators. We describe the implementation of these tests in the SPRING [ACM Trans. Math. Software 26 (2000) 436; SPRNG--scalable parallel random number generators. SPRNG 1.0--http://www.ncsa.uiuc.edu/ Apps/SPRNG; SPRNG 2.0--http://sprng.cs.fsu.edu] test suite and also present results for the tests conducted on the SPRNG generators. These generators have passed some of the largest empirical random number tests.

The lack of a well-defined and meaningful measure of the quality of computer-generated correlated random variates complicates the design and comparison of generation algorithms for communications system simulation applications. This paper presents quantitative quality measures for computer-generated random variates having a multivariate Gaussian distribution or having a distribution that derives from the multivariate Gaussian distribution. These measures are, in particular, useful for digital communication system simulation applications. A theoretical basis for the definition of the measures is given and it is shown that the measures also possess useful intuitive interpretations. Examples of the use of these quality measures are provided.

In this paper, a method called the vertical strip (VS) method is proposed for generating non-uniform random variates with a given density. It can be considered as an improvement of the grid method as the VS method avoids setting up a directory to store information on big rectangles. Unlike the horizontal strip method that is based on the Riemann integral, the VS method is based on the Lebesgue integral and can be applied to unbounded densities or densities with infinite support. Applications of the VS method for generating random variates, which follow the exponential distribution and normal density, are also given.

Cryptography techniques rely on chains of random numbers used to generate safe encryption keys. Since random number generator algorithms are in fact pseudo-random their behavior can be predicted if the generation method is known and as such they cannot be used for perfectly safe communications. In this article, we present a perfectly random generator based on quantum measurement processes. The main advantage of such a generator is that using quantum mechanics, its behavior cannot be predicted in any way. We verify the randomness of our generator and compare it to commonly used pseudo-random generators.

Adding random disturbances to Modelica models is necessary to represent stochastic fluctuations like sensor noise, air gusts and road irregularities. In this paper, we present a library to specify a pseudo random noise for continuous-time simulations. The random number generator, a probability density function and a frequency spectrum can be defined independently. A new random number generator is proposed to generate a continuous random signal, which is proven to be highly suitable for continuous models. The performance of the noise models is tested in two benchmarks using an academic as well as a realistic model both showing a remarkable increase in simulation speed.

Power-law distributions are typical macroscopic features occurring in almost all complex systems observable in nature. As a result, researchers in quantitative analyses must often generate random synthetic variates obeying power-law distributions. The task is usually performed through standard methods that map uniform random variates into the desired probability space. Whereas all these algorithms are theoretically solid, in this paper we show that they are subject to severe machine-dependent limitations. As a result, two dramatic consequences arise: (i) the sampling in the tail of the distribution is not random but deterministic; (ii) the moments of the sample distribution, which are theoretically expected to diverge as functions of the sample sizes, converge instead to finite values. We provide quantitative indications for the range of distribution parameters that can be safely handled by standard libraries used in computational analyses. Whereas our findings indicate possible reinterpretations of numerical results obtained through flawed sampling methodologies, they also pave the way for the search for a concrete solution to this central issue shared by all quantitative sciences dealing with complexity., 8 pages, 4 figures

Random unique number generator can be used for generating a series of unpredictable and unrepeatable numbers within limited ranges of data and numbers. These numbers are usually distributed equally, random, independent, unpredictable and unrepeatable. A good random number generator has to be effective for a long period and has good statistical distribution and efficient generating performance. This study proposes a computational methodology to generate pseudo-random numbers based on random base polynomial, which uses less memory but generates a great deal of unrepeated pseudo-random numbers. Then this method adopts the multi-thread parallelization to effectively get the benefits of multi-core processors to accelerate the generation of a huge amount of pseudo-random numbers.

A random number generation method based on a cross coupled chaotic oscillator is introduced. Numerical model for the proposed design has been developed where bootstrap method is utilized which allows us to estimate the statistical characteristics of underlying chaotic signal. Numerical results verifying the feasibility and correct operation of the random number generator are given such that numerically generated binary sequences fulfill FIPS-140-2 statistical test suites for randomness without post-processing. Presented random number generator features much higher and constant throughput rates and allows offset compensation.

Random number is widely used in cryptographic applications, which is mainly used as key. Because the security of key totally depends on the amount and randomness of itself, it's very important to produce random numbers for cryptographic applications. This paper presents a method to generate random numbers for cryptographic applications. NIST Statistical Test Suite which provides 15 statistical methods is used to test the randomness of the random number generated by this method. Because the tests focus on a variety of different types of non-randomness, not all tests are needed. The chosen statistical tests are Frequency (Monobit) Test, Frequency Test within a Block, The Cumulative Sums (Cusums) Test, The Runs Test, Test for the Longest Run of Ones in a Block, Discrete Fourier Transform (Specral) Test, Approximate Entropy Test and Serial Test. The result of tests shows that the random number generated by the random number generator is random. Therefore the conclusion is the random number generated is random enough for cryptographic applications.

Limit theorems are proved establishing criteria of convergence of the distributions of random sums and statistics constructed from samples with random sizes to exponential power laws. INTRODUCTION. EXPONENTIAL POWER DISTRIBUTIONS Let 0 < α 6 2. Exponential power distribution is the absolutely continuous distribution defined by its Lebesgue probability density

Random number generation is still an important research field in many scientific applications today. Cryptography, Monte Carlo simulations and commertial applications all rely on reference random data. Randomness tests and basic statistics share the same history. Randomness can be summarized as the unpredictability of future samples of a random number generator even in the presence of known all past values. Various randomness tests are developed and due to their individual contributions, usually a battery of tests are applied to verify a random number generator. In signal processing however, the error of a specific observed sample set to a given distribution could be much more important when it is used as the input for a system model. Recently, this distance of finite samples set to a given distribution is studied and a quantitative measure for quality is proposed. Multi run computations like Monte Carlo simulations, often rely on accurate statistical data for high repetibility. Otherwise when the data is not accurate, the results could often rely on the source of random data generator. Many runs are often required to gain a confidence in the presence of those variances. In this work, recently proposed quasi-random number generator utilizing method of uniform sampling (MUS) is tested using standard goodness-of-fitness tests. MUS-QRNG numbers are shown to have exact statistics and also their randomness test results are observed to be similar to well known reference generator of Matlab. MUS-QRNG is proposed for high quality random data generation.

Random sampling is a technique for signal acquisition which is gaining popularity in practical signal processing systems. Nowadays, event-driven analog-to-digital converters make random sampling feasible in practical applications. A process of random sampling is defined by a sampling pattern, which indicates signal sampling points in time. Practical random sampling patterns are constrained by ADC characteristics and application requirements. In this paper authors introduce statistical methods which evaluate random sampling pattern generators with emphasis on practical applications. Furthermore, the authors propose a new random pattern generator which copes with strict practical limitations imposed on patterns, with possibly minimal loss in randomness of sampling. The proposed generator is compared with existing sampling pattern generators using the introduced statistical methods. It is shown that the proposed algorithm generates random sampling patterns dedicated for event-driven-ADCs better than existed sampling pattern generators. Finally, implementation issues of random sampling patterns are discussed., Comment: 29 pages, 12 figures, submitted to Circuits, Systems and Signal Processing journal