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...

We present analytical expressions for the probability density function (PDF) of the random mutual information between transmitted and received vector signals of a random space-time independent and identically distributed (i.i.d.) multipleinput multiple-output (MIMO) channel, assuming that the transmitted signals from the multiple antennas are Gaussian i.i.d.. We show that this PDF can be well approximated by a Gaussian distribution, and such a Gaussian approximation is based on expressions for the given PDF's mean and variance that we derive. We prove that at high SNR, every 3 dB increase in signal to noise ratio (SNR) leads to an increase in outage rate approximately equal to min(M,N ), where M and N denote the number of transmit- and receive-antennas, respectively. A simple expression for the moment generating function of the mutual information PDF is also provided, based on which we establish normality of the PDF, when both M and N are large, and the SNR is large.

This paper studies the impact of long-range-dependent (LRD) traffic on the performance of reassembly and multiplexing queueing. A queueing model characterizing the general reassembly and multiplexing operations performed in packet networks is developed and analyzed. The buffer overflow probabilities for both reassembly and multiplexing queues are derived by extending renewal analysis and Beneˇs fluid queue analysis, respectively. Tight upper and lower bounds of the frame loss probabilities are also analyzed and obtained. Our analysis is not based on existing asymptotic methods, and it provides new insights regarding the practical impact of LRD traffic. For the reassembly queue, the results show that LRD traffic and conventional Markov traffic yield similar queueing behavior. For the multiplexing queue, the results show that the LRD traffic has a significant impact on the buffer requirement when the target loss probability is small, including for practical ranges of buffer size or maximum delay.

A flexible efficient and accurate inverse Laplace transform algorithm is developed. Based on the quotient-difference methods the algorithm computes the coefficients of the continued fractions needed for the inversion process. By combining diagonalwise operations and the recursion relations in the quotient-difference schemes, the algorithm controls the dimension of the inverse Laplace transform approximation automatically. Application of the algorithm to the solute transport equations in porous media is explained in a general setting. Also, a numerical simulation is performed to show the accuracy and efficiency of the developed algorithm. Key words. Inverse Laplace transform, time-integration, transport equation, porous media. AMS subject classfications. 65M60, 65Y20. 1

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Abstract. The Ornstein-Uhlenbeck process has been proposed as a model for the spontaneous activity of a neuron. In this model, the firing of the neuron corresponds to the first passage of the process to a constant boundary, or threshold. While the Laplace transform of the first-passage time distribution is available, the probability distribution function has not been obtained in any tractable form. We address the problem of estimating the parameters of the process when the only available data from a neuron are the interspike intervals, or the times between firings. In particular, we give an algorithm for computing maximum likelihood estimates (MLEs) and their corresponding confidence regions for the three identifiable (of the five model) parameters by numerically inverting the Laplace transform. A comparison of the two-parameter algorithm (where the time constant τ is known a priori) to the threeparameter algorithm shows that significantly more data is required in the latter case to achieve comparable parameter resolution as measured by 95 % confidence intervals widths. The computational methods described here are an efficient, biophysicallyrevealing alternative to the well-known moment-based estimations for OU integrate and fire (IF) models. Moreover, it could serve as a template for performing parameter inference on more complex IF neuronal models. Keywords: Ornstein-Uhlenbeck process, parameter inference, inverse Laplace transform 1.

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