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...

The limiting distribution of the normalized number of key comparisons required by the Quicksort sorting algorithm is known to be the unique fixed point of a certain distributional transformation T ---unique, that is, subject to the constraints of zero mean and finite variance. We show that a distribution is a fixed point of T if and only if it is the convolution of with a Cauchy distribution of arbitrary center and scale. In particular, therefore, is the unique fixed point of T having zero mean. 1

We examine the conventional wisdom that commends the use of direct search methods in the presence of random noise. To do so, we introduce new formulations of stochastic optimization and direct search. These formulations suggest a natural strategy for constructing globally convergent direct search algorithms for stochastic optimization by controlling the error rates of the ordering decisions on which direct search depends. This strategy is successfully applied to the class of generalized pattern search methods. However, a great deal of sampling is required to guarantee convergence with probability one. Contents 1 Introduction 2 2 Stochastic Optimization 2 3 Direct Search 5 3.1 The Deterministic Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3.2 The Stochastic Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 4 Convergence Theory 7 5 Pattern Search 9 5.1 Numerical Optimization . . . . . . . . . . . . . . . . . . . . . . . ....

Abstract. We consider the problem of maximizing expected utility from consumption in a constrained incomplete semimartingale market with a random endowment process, and establish a general existence and uniqueness result using techniques from convex duality. The notion of asymptotic elasticity of Kramkov and Schachermayer is extended to the time-dependent case. By imposing no smoothness requirements on the utility function in the temporal argument, we can treat both pure consumption and combined consumption/terminal wealth problems, in a common framework. To make the duality approach possible, we provide a detailed characterization of the enlarged dual domain which is reminiscent of the enlargement of L1 to its topological bidual (L∞) ∗ , a space of finitely-additive measures. As an application, we treat the case of a constrained Itô-process market-model. 1.

Motivated by the desire to model complex features of network traffic revealed in traffic measurements, such as heavy-tail probability distributions, long-range dependence, self similarity and nonstationarity, we propose a nonstationary offered-load model, in which connections of multiple types arrive according to independent nonhomogeneous Poisson processes, and general bandwidth stochastic processes describe the individual user bandwidth requirements at multiple links of a communication network during their connections. For example, an individual bandwidth process may be an on-off process where the on and off times have general (possibly heavy-tail) distributions. We obtain expressions for the moment generating function, mean and variance of the total required bandwidth of all customers on each link at any designated time. We suggest making decisions based on the probability that demand will exceed supply, or other designated target level, at each time of interest, using (i) numerical...

Abstract. Let Tn(x) denote the time of first visit of a point x on the lattice torus Z 2 n = Z 2 /nZ 2 by the simple random walk. The size of the set of α, n-late points Ln(α) = {x ∈ Z 2 n: Tn(x) ≥ α 4 π (nlog n)2} is approximately n 2(1−α) , for α ∈ (0,1) (Ln(α) is empty if α> 1 and n is large enough). These sets have interesting clustering and fractal properties: we show that for β ∈ (0,1) a disc of radius n β centered at non-random x typically contains about n 2β(1−α/β2) points from Ln(α) (and is empty if β < √ α), whereas choosing the center x of the disc uniformly in Ln(α) boosts the typical number α, n-late points in it to n 2β(1−α). We also estimate the typical number of pairs of α, n-late points within distance n β of each other; this typical number can be significantly smaller than the expected number of such pairs, calculated by Brummelhuis and Hilhorst (1991). On the other hand, our results show that the number of ordered pairs of late points within distance n β of each other, is larger than what one might predict by multiplying the total number of late points by the number of late points in a disc of radius n β centered at a typical late point. 1.

This paper considers the Poisson equation associated with time-homogeneous Markov chains on a countable state space. The discussion emphasizes probabilistic arguments and focuses on three separate issues, namely (i) the existence and uniqueness of solutions to the Poisson equation, (ii) growth estimates and bounds on these solutions and (iii) their parametric dependence. Answers to these questions are obtained under a variety of recurrence conditions, and extensions to noncountable state spaces are outlined. Motivating applications can be found in the theory of Markov decision processes in both its adaptive and non-adaptive formulations, and in the theory of Stochastic Approximations. The results complement available results from Potential Theory for Markov chains, and are therefore of independent interest. Keywords : Markov Chains, Poisson Equation, smoothness of solutions. September 21, 1994. 2 Electrical Engineering Department and Systems Research Center, University of Maryland, C...

Suppose that “uncertainty ” about labor market conditions has increased. Does this change induce an unemployed worker to search longer, or shorter? This paper shows that the answer is drastically different depending on whether an increase in “uncertainty ” is an increase in risk or that in true uncertainty in the sense of Frank Knight. We show in a general framework that, while an increase in risk (the mean-preserving spread of the wage distribution that the worker thinks she faces) increases the reservation wage, an increase in the Knightian uncertainty (a decrease in her confidence about the wage distribution) reduces it. We are grateful to seminar participants at Western Ontario and SUNY-Buffalo for their helpful comments. The

Suppose the true density generating data e x n = (x 1 ; : : : ; x n ) is in a parametric family denoted n ( e x n j 1 ), where is a real parameter, but that n ( e x n j 1 ) is not known in detail. One may try to model the data by using a dierent conditional distribution q n ( e x n j 1 ) = Q n i=1 q i (x i j 1 ), which assumes independence even when this is not valid. Here we take the q i 's to be the marginals from n since this is an optimal choice. The independence density q n can be used to obtain a q n -based MLE, ^ q , and a q n -based posterior, w q . We examine the performance of ^ q and w q under n in two situations. The rst situation assumes no extra structure on n , only that it satises some laws of large numbers. The second situation assumes that n may be realized as a mixture over nuisance parameters of some underlying higher dimensional conditional independence model. Under our conditions, none of the parameters in this conditional independence model need \fade out" as n increases to get consistency of estimators based on q n . Assessing convergence in n ( e x n j 1 ), we nd that w q is consistent and asymptotically normal in both cases, with asymptotic variance unchanged from what one would expect if the data were generated by an independence model. The asymptotic distribution of ^ q need not be normal nor be scaled by the Fisher information. Consequently, posterior inference based on the product of marginals is dierent from MLE-type inference derived from the product of marginals. This analysis is distinct from the usual \estimation in the presence of nuisance parameters" analysis in that we are interested in estimators based on the product of marginals q n , not estimato...

We establish the limiting distribution of the number T n of random functions on a set of size n which must be composed before a constant function results. In more detail, let f 1 , f 2 , . . . be independent draws from the uniform distribution over all functions from . . . , n} into itself. For t = 1, 2, . . . let g t := f t f 1 denote the composition of the first t random maps, and let T n be the smallest t such that g t is constant. Then T n /n converges in distribution, with convergence of moment generating functions (and hence of all moments), to the infinite convolution of exponential distributions with rates , j = 2, 3, . . . .