Large Number Of Random Variables Research Articles

Optimal testing of VLSI analog circuits

The high cost of capital equipment for production testing coupled with the time that an analog circuit spends on a tester has made it imperative to minimize average chip testing time during production. Testing time can be reduced by decreasing the number of tests that need to be performed on a circuit and by optimizing the order of the tests. This can be done by studying performance data for a sample of chips or using simulation data. The advantage of simulation data is that a large number of circuits do not have to be nonoptimally tested in order to generate the data set. Generating simulation data for very large scale integration (VLSI) analog circuits has been considered to be very difficult because of the computational cost of simulating a large system, the very large number of random variables needed to model the manufacturing process, and the need to model and simulate not only random parameter variations but also spot defects. These problems are overcome by decomposing a circuit into several blocks, which are linked together by a behavioral model of the system. In order for this to work, the blocks must not depend on a large set of significant random variables, and the blocks should not be tightly coupled. In this paper we demonstrate our statistical simulation methodology for a VLSI analog circuit. The proposed approach to defect simulation is applied to a parallel filter bank which has a total of 32 channels, is composed of 67 blocks, and has over 4040 transistors and 750 capacitors. Simulation results are used to optimize the test set for this circuit, by showing that the frequency response tests at many frequencies are redundant.

Summing received signal powers with arbitrary probability density functions on a logarithmic scale

A technique is presented to calculate the probability density function (pdf) for a sum of random variables that have pdf's on a logarithmic scale. In mobile radio it is often necessary to calculate the pdf of the total received signal power, which is the “power sum” of a number of simultaneously received signal powers. When the signal powers are given on a linear scale (e.g. Watts) probability density functions (pdf's) of the individual signals can be convolved to give the pdf for the received power of all the signals together. When, as is usual, the signal powers are given on a logarithmic scale (e.g. dBs) this is not possible. The simple convolution for the linear domain must now be replaced by a convolution for the logarithmic domain, which is not straightforward to compute. In this paper, a method is presented to compute the pdf of a power sum of two random variables, the logarithmic convolution. The results are not in closed form, numerical integration is necessary to find the resulting pdf. The method can be applied recursively to give results for power sums of more than two random variables. Although methods exist that give solutions in a closed form, they mainly use approximations and are valid only for specific distributions. The method presented in this paper yields exact results for arbitrary distributions. The results of the logarithmic convolution are verified by Monte-Carlo simulations. Even for large numbers of random variables the power sum results are shown to be correct.

Efficient Algorithm for Second‐Order Reliability Analysis

In the second‐order reliability method the principal curvatures of the limit‐state surface at the design point are used to construct a paraboloid approximation of the surface, which is then used to compute a second‐order estimate of the failure probability. The principal curvatures are the eigenvalues of the Hessian of the surface. In this paper an efficient algorithm is developed to determine the principal curvatures without computing the Hessian. The curvatures are computed in an iterative manner using the gradient of the limit‐state function, and are obtained in the decreasing order of their absolute magnitudes, which is also the order of their importance in reliability analysis. The computation can be terminated when the last curvature obtained is sufficiently small. The method is efficient for problems with large numbers of random variables, especially when an efficient algorithm for computing the gradient is available. Several numerical examples, including a finite‐element application involving 99 r...

A strange pigeon-hole principle

Using Ramsey theory, we establish the following pigeon-hole type principle: From a large number of random variables (functions, vectors, etc.) one can always select two, X and Y, such that P(X < Y) ∼ 1/2. We apply the principle for a poset problem.

Relationships Among Water Storage Variables

Among a large number of random variables related to water storage for given sample size n, the most studied are the range Rn, deficit Dn, and negative run‐sum Vn. A simulation of 5,000 samples for various n was performed for three assumed population processes: normal independent, normal dependent, and gamma independent. It was found by simulation, and verified by an analytical derivation, that the range and deficit are equal (Rn=Dn) in 50% of generated samples, while in the other 50% the range is greater that the deficit (Rn>Dn). Probabilities of Dn=Vn range from zero to more than 0.60; they decrease as n and skewness Cs increase, but increase as the lag‐1 autocorrelation coefficient increases. The case of Rn=Dn=Vn has similar patterns. The ratios of sample means for these three storage‐related variables, either for 5,000 samples or only the samples for which the values of these variables are not equal, exhibit these general patterns: (1) Ratios Dn/Rn increase with n converging to an asymptotic value of 0.7854 in case of 5,000 samples; (2) for Rn>Dn samples, the ratios Dn/Rn are much smaller than for the entire set of 5,000 samples with Rn⩾Dn; (3) ratios Vn/Dn and Vn/Rn are much smaller than Dn/Rn. The pairwise lag‐0 correlation coefficients of Rn,Dn, and Vn for 5,000 sample estimates show that corr (Rn,Dn) changes around 0.45 as n increases in the case Cs=0; this correlation is small (0-0.25), but increase with n for Cs=3. For Rn>Dn samples, corr (Rn,Dn) becomes negative, oscillating around -0.07. corr (Dn,Vn) shows a decrease with an increase of n for Cs=3, but increases with an increase of autocorrelation, corr (Rn,Vn) decreases with an increase of n, changes slightly with autocorrelation and with skewness.

Marcinkiewicz Laws and Convergence Rates in the Law of Large Numbers for Random Variables with Multidimensional Indices

Consider a set of independent identically distributed random variables indexed by $Z^d_+$, the positive integer $d$-dimensional lattice points, $d \geqq 2$. The classical Kolmogorov-Marcinkiewicz strong law of large numbers is generalized to this case. Also, convergence rates in the law of large numbers are derived, i.e., the rate of convergence to zero of, for example, the tail probabilities of the sample sums is determined.

Convergence rates in the law of large numbers for random variables on partially ordered sets

Let (A, ≤) be a partially ordered set, {Xα} a collection of i. i. d. random variables, indexed byA. Let\(S_\alpha = \sum _{\beta \leqslant \alpha } X_\beta \), |α|=card {β∈A, β∈α}. We study the convergence rates ofSα/|α|. We derive for a large class of partially ordered sets theorems, like the following one: For suitabler, t with 1/2< <r/t≤1:E|X|tM (|X|t/r)<∞ andEX=μ if and only if $$S_\alpha = \sum _{\beta \leqslant \alpha } X_\beta $$ for all e>0, where\(M(x) = \sum _{j< x} d(j)\) withd(j)=card {α∈A, |α|=j}.

LAWS OF LARGE NUMBERS FOR NORMED LINEAR SPACES AND CERTAIN FRECHET SPACES

General introduction.- Mathematical preliminaries.- Random elements in separable metric spaces.- Laws of large numbers for random variables and separable Hilbert spaces.- Strong laws of large numbers for normed linear spaces.- Weak laws of large numbers for normed linear spaces.- Laws of large numbers for Frechet spaces.- Some applications.

A class of “central limit theorems” for convolution products of generalized functions

Introduction and summary. From the naive viewpoint of a nonprobabilist, the remarkable fact about the central limit theorem is that (under suitable conditions) the average of a large number of random variables converges (in a suitable sense) to a limit which is independenit of the things being averaged, but is completely determined by the averaging process itself. In the present paper, this situation is abstracted from its connection with probability. As we shall see, the restriction to positive functionals which characterizes probability is in no way necessary for validity of the limit theorems. Moreover, we are able to dispense completely with measure theory, in both definitions and proofs. In a purely algebraic and function-space context, we find a whole sequence of averaging processes, each with its own limit function; the first two of them generalize the weak law of large numbers and the central limit theorem. We prove convergence in three different senses, under three different sets of assumptions: first convergence of moments, then uniform convergence of characteristic functions, and finally weak convergence of distribution functions. Our first theorem is purely combinatorial; our next, essentially a calculus theorem; in our third we use function-space techniques borrowed from the modern theory of partial differential equations. By doing so, we obtain rather refined and precise results on the strongest norm in which we can assert convergence. In this respect our results may be of interest even in the classical probabilistic cases. In particular (see Corollary 2 below) we show that if a very mild extra regularity condition is imposed on the summands in the classical central limit theorem, then not only do the partial sums converge in law to the Gaussian, but every derivative of the c.d.f. of the partial sum exists and converges to the corresponding derivative of the Gaussian. Theorem 3 possesses an interpretation in terms of electrical engineering, which is given in the Remark following Corollary 2. It also has a surprising application to numerical analysis, which is described in the Postscript. We begin with an abstract algebra A over the real numbers, equipped with a linear functional Y. (If a E A were a random variable, Y(a`) would be its kth