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21
The FourierSeries Method For Inverting Transforms Of Probability Distributions
, 1991
"... This paper reviews the Fourierseries 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 Fourierseries method are remarkably easy ..."
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Cited by 149 (51 self)
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This paper reviews the Fourierseries 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 Fourierseries method are remarkably easy to use, requiring programs of less than fifty lines. The Fourierseries 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 Fourierseries 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...
A Characterization Of The Set Of Fixed Points Of The Quicksort Transformation
, 2000
"... 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 dist ..."
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Cited by 18 (10 self)
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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
Zitković, “Optimal consumption from investment and random endowment in incomplete semimartingale markets
 Ann. Probab
, 2003
"... 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 Kr ..."
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Cited by 16 (0 self)
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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 timedependent 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 finitelyadditive measures. As an application, we treat the case of a constrained Itôprocess marketmodel. 1.
A Nonstationary OfferedLoad Model for Packet Networks
, 1998
"... Motivated by the desire to model complex features of network traffic revealed in traffic measurements, such as heavytail probability distributions, longrange dependence, self similarity and nonstationarity, we propose a nonstationary offeredload model, in which connections of multiple types arriv ..."
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Cited by 12 (2 self)
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Motivated by the desire to model complex features of network traffic revealed in traffic measurements, such as heavytail probability distributions, longrange dependence, self similarity and nonstationarity, we propose a nonstationary offeredload 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 onoff process where the on and off times have general (possibly heavytail) 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...
On the Use of Direct Search Methods for Stochastic Optimization
 Rice University, Department of
, 2000
"... 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 ..."
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Cited by 10 (0 self)
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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 . . . . . . . . . . . . . . . . . . . . . . . ....
Search and Knightian uncertainty
, 2001
"... 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 unce ..."
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Cited by 7 (0 self)
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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 meanpreserving 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 SUNYBuffalo for their helpful comments. The
Late points for random walks in two dimensions
, 2005
"... 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, nlate 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 eno ..."
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Cited by 5 (3 self)
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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, nlate 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 nonrandom 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, nlate points in it to n 2β(1−α). We also estimate the typical number of pairs of α, nlate 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.
On Compositions of Random Functions on a Finite Set
, 2002
"... 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 ..."
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Cited by 4 (0 self)
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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, . . . .
On The Poisson Equation For Markov Chains: Existence Of Solutions And Parameter Dependence By Probabilistic Methods
, 1994
"... This paper considers the Poisson equation associated with timehomogeneous 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 estim ..."
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Cited by 3 (0 self)
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This paper considers the Poisson equation associated with timehomogeneous 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 nonadaptive 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...
An isomorphism theorem for random interlacements
 Electron. Commun. Probab
, 2012
"... We consider continuoustime random interlacements on a transient weighted graph. We prove an identity in law relating the field of occupation times of random interlacements at level u to the Gaussian free field on the weighted graph. This identity is closely linked to the generalized second RayKnig ..."
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Cited by 2 (1 self)
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We consider continuoustime random interlacements on a transient weighted graph. We prove an identity in law relating the field of occupation times of random interlacements at level u to the Gaussian free field on the weighted graph. This identity is closely linked to the generalized second RayKnight theorem of [2], [4], and uniquely determines the law of occupation times of random interlacements at level u.