This paper reviews the Fourier-series method for calculating cumulative distribution functions (cdf's) and probability mass functions (pmf's) by numerically inverting characteristic functions, Laplace transforms and generating functions. Some variants of the Fourier-series method are remarkably easy to use, requiring programs of less than fifty lines. The Fourier-series method can be interpreted as numerically integrating a standard inversion integral by means of the trapezoidal rule. The same formula is obtained by using the Fourier series of an associated periodic function constructed by aliasing; this explains the name of the method. This Fourier analysis applies to the inversion problem because the Fourier coefficients are just values of the transform. The mathematical centerpiece of the Fourier-series method is the Poisson summation formula, which identifies the discretization error associated with the trapezoidal rule and thus helps bound it. The greatest difficulty is approximately calculating the infinite series obtained from the inversion integral. Within this framework, lattice cdf's can be calculated from generating functions by finite sums without truncation. For other cdf's, an appropriate truncation of the infinite series can be determined from the transform based on estimates or bounds. For Laplace transforms, the numerical integration can be made to produce a nearly alternating series, so that the convergence can be accelerated by techniques such as Euler summation. Alternatively, the cdf can be perturbed slightly by convolution smoothing or windowing to produce a truncation error bound independent of the original cdf. Although error bounds can be determined, an effective approach is to use two different methods without elaborate error analysis. For this...

: When computing the infimal convolution of a convex function f with the squared norm, one obtains the so-called Moreau-Yosida regularization of f . Among other things, this function has a Lipschitzian gradient. We investigate some more of its properties, relevant for optimization. Our main result concerns second-order differentiability and is as follows. Under assumptions that are quite reasonable in optimization, the Moreau-Yosida is twice diffferentiable if and only if f is twice differentiable as well. In the course of our development, we give some results of general interest in convex analysis. In particular, we establish primaldual relationship between the remainder terms in the first-order development of a convex function and its conjugate. Key-words: Convex optimization, mathematical programming, proximal point, secondorder differentiability. (R'esum'e : tsvp) Unite de recherche INRIA Rocquencourt Domaine de Voluceau, Rocquencourt, BP 105, 78153 LE CHESNAY Cedex (France) Telep...

Stochastic Petri nets have been used to analyze the performance and reliability of complex systems comprising concurrency and synchronization. Various extensions have been proposed in literature in order to broaden their field of application to an increasingly larger range of real situations. In this paper we extend the class of Markov Regenerative Stochastic Petri Nets* (MRSPN*s), removing the restriction that at most one generally distributed timed transition can be enabled in any marking. This new class of Petri Nets, which we call Concurrent Generalized Petri Nets (CGPNs) allows simultaneous enabling of immediate, exponentially and generally distributed timed transitions, under the hypothesis that the latter are all enabled at the same instant. The stochastic process underlying a CGPN is shown to be still an MRGP. We evaluate the kernel distribution of the underlying MRGP and define the steps required to generate it automatically. The methodology described is used to assess the beh...

this dissertation isdev eloping a ranking and selection technique fore#ectiv ely comparing tracking algorithms that maintain target tracks after they hav e been initialized

Variance estimation and ranking methods are developed for stochastic processes modeled by Gaussian mbcture distributions. It is shown that the variance estimate from a Gaussian mixture distribution has the same properties as a variance estimate from a single Gaussian distribution based on a reduced number of samples. Hence, well known tools of variance estimation and ranking of single Gaussian distributions can be applied to Gaussian mixture distributions. As an application example, optimization of sensor processing order in the sequential multi-target multi-sensor joint probabilistic data associ- ation (MSJPDA) algorithm is presented.

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. We consider the integrodifferential equation u(t; x) = R t 0 a(t \Gamma s)uxx (s; x)ds with initial and boundary conditions corresponding to the Rayleigh problem. The kernel has the form a(t) = a 0 + a1 t + R t 0 a 1 (s)ds, where a 0 0, a1 0, and a 1 2 L 1 loc (R+ ) is of positive type and satisfies the condition R 1 0 e \Gammafflt ja 1 (t)jdt ! 1 for every ffl ? 0. Solving the equation numerically and performing a careful error analysis we show that the solution u(t; x) need not be nondecreasing in t 0 for fixed x ? 0, if a 1 is nonnegative, nonincreasing, and convex. The same result is shown to hold under the assumption that a 1 is completely positive. This answers a question that remained unsolved in [J. Pruß, Math. Ann., 279 (1987), p. 330]. In the case where a1 is convex, piecewise linear, the solution is shown to be almost everywhere equal to a function which is discontinuous across infinitely many parallel lines. Key words. viscoelasticity, integrodifferentia...

a linear system is much more valuable if it is a accompanied by an estimate of the error. The most often used approach in the error analysis of linear systems is inverse error analysis, which has been developed mainly by Wilkinson. Another approach is interval analysis, as developed by Hansen [ 5, 6 ] and Moore [ 7 ]. In the latter method, the solution consists of an interval for each of the element of x, and the algorithm is designed so that the interval is guaranteed to contain the true value of that element. In this Appendix the solution of an ill - conditioned systems will be treated also as a problem of interval estimation and incorporate the a priori information about the solution into the problem as constraints. Then, it will be shown how the resulting constrained estimation problem can be reduced to a problem in mathematical programming which, in turn, can be solved to give confidence intervals for the true solution of the system. A.3.2 Numerical solutions An excellent surve