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Showing papers on "Gaussian process published in 1983"


Journal ArticleDOI
TL;DR: In this article, it was shown that the central limit theorem holds for some non-linear functionals of stationary Gaussian fields if the correlation function of the underlying field tends fast enough to zero.

412 citations


Journal ArticleDOI
TL;DR: The problem of transmitting a sequence of identically distributed independent Gaussian random variables through a Gaussian memoryless channel with a given input power constraint, in the presence of an intelligent jammer, is considered and the optimal policy of the transmitter is to amplify the input sequence to the given power level by a linear transformation, and the receiver is to use a Bayes estimator.
Abstract: The problem of transmitting a sequence of identically distributed independent Gaussian random variables through a Gaussian memoryless channel with a given input power constraint, in the presence of an intelligent jammer, is considered. The jammer taps the channel and feeds back a signal, at a given energy level, for the purpose of jamming the transmitting sequence. Under a square-difference distortion measure which is to be maximized by the jammer and to be minimized by the transmitter and the receiver, this correspondence obtains the complete set of optimal (saddle-point) policies. The solution is essentially unique, and it is structurally different in three different regions in the parameter space, which are determined by the signal-to-noise ratios and relative magnitudes of the noise variances. The best (maximin) policy of the jammer is either to choose a linear function of the measurement he receives through channel-tapping, or to choose, in addition (and additively), an independent Gaussian noise sequence, depending on the region where the parameters lie. The optimal (minimax) policy of the transmitter is to amplify the input sequence to the given power level by a linear transformation, and that of the receiver is to use a Bayes estimator.

213 citations


Journal ArticleDOI
TL;DR: New outer bounds are demonstrated for the capacity regions of discrete memoryless interference channels and Gaussian interference channels, which improves previous knowledge when the interference is of medium strength.
Abstract: New outer bounds are demonstrated for the capacity regions of discrete memoryless interference channels and Gaussian interference channels. The bound for discrete channels coincides with the capacity region in special cases. The bound for Gaussian channels improves previous knowledge when the interference is of medium strength.

110 citations


Journal ArticleDOI
TL;DR: In this article, the mixing and dependence conditions commonly used in extreme value theory for stationary sequences were established for the normalized partial sums under the mixing conditions. But the mixing condition was not considered in this paper.
Abstract: Let $\{X_n\}$ be a stationary sequence of random variables whose marginal distribution $F$ belongs to a stable domain of attraction with index $\alpha, 0 < \alpha < 2$. Under the mixing and dependence conditions commonly used in extreme value theory for stationary sequences, nonnormal stable limits are established for the normalized partial sums. The method of proof relies heavily on a recent paper by LePage, Woodroofe, and Zinn which makes the relationship between the asymptotic behavior of extreme values and partial sums exceedingly clear. Also, an example of a process which is an instantaneous function of a stationary Gaussian process with covariance function $r_n$ behaving like $r_n \log n \rightarrow 0$ as $n \rightarrow \infty$ is shown to satisfy these conditions.

86 citations


Journal ArticleDOI
TL;DR: In this article, a class of empirical processes having the structure of $U$-statistics is considered, and the weak convergence of the processes to a continuous Gaussian process is proved in weighted sup-norm metrics stronger than the uniform topology.
Abstract: A class of empirical processes having the structure of $U$-statistics is considered. The weak convergence of the processes to a continuous Gaussian process is proved in weighted sup-norm metrics stronger than the uniform topology. As an application, a central limit theorem is derived for a very general class of non-parametric statistics.

74 citations


Journal ArticleDOI
TL;DR: In this article, the authors studied the Wiener process in terms of the zeros of the Airy function and the positive stable density of exponent 2/3 and found the distribution of the supremum of a certain stationary, mean zero, periodic Gaussian process, which corresponds to the limiting distribution of an optimal test statistic for the isotropy of a set of directions.
Abstract: Let $W(t), 0 \leq t \leq 1$, be the Wiener process tied down at $t = 0, t = 1; W(0) = W(1) = 0$. We find the distribution of $\sup_{0 \leq t \leq 1} W(t) - \int^1_0 W(t) dt$ in terms of the zeros of the Airy function and the positive stable density of exponent 2/3. This corresponds to the distribution of the supremum of a certain stationary, mean zero, periodic Gaussian process. It is also the limiting distribution of an optimal test statistic for the isotropy of a set of directions, proposed by G. S. Watson.

66 citations


Journal ArticleDOI
TL;DR: In this article, the Lagrangian Monte Carlo model is applied to an instantaneous one-dimensional cloud source (which is approximately equivalent to a continuous line source) and it is shown that at large time the Gaussian model accounts only for the meandering contribution to fluctuations, and by smoothing out all internal structure of the doud eliminates relative fluctuations.
Abstract: The consequences for the theory of concentration fluctuations of different proposals by L. F. Richardson and G. K. Batchelor about the rate of separation of pairs of marked particles (giving rise to non-Gaussian and Gaussian particle separation probability density functions respectively) are explored via two Lagrangian Monte Carlo models. The models are applied to an instantaneous one-dimensional cloud source (which is approximately equivalent to a continuous line source) and in many respects give similar results for concentration fluctuations. The crucial difference is that whereas the non-Gaussian model predicts, in agreement with observation, that at large time the fluctuations remain the same order of magnitude as the mean field (the ratio depending only on the source size), the Gaussian model incorrectly predicts that fluctuations ultimately vanish compared with the mean field. The reason for the failure of the Gaussian model is explored by partitioning the total fluctuations into contributions due to the variation of the distribution of material within the cloud (i.e. in coordinates relative to the centre-of-mass) and due to motions of the cloud as a whole (meandering). It is shown that at large time the Gaussian model accounts only for the meandering contribution to fluctuations, and by smoothing out all internal structure of the doud eliminates relative fluctuations.

61 citations



Journal ArticleDOI
TL;DR: By simulation, the possibility of generating a process with a specified spectral density and a specified first-order probability distribution by passing a Gaussian process with anappropriately chosen spectral density through an appropriately chosen zero-memory nonlinearity is explored.
Abstract: The procedure for generating a Gaussian process with a specified spectral density is well known. It is harder to generate a process with a specified spectral density and a specified first-order probability distribution. In this paper we explore, by simulation, the possibility of generating a process with such a dual specification by passing a Gaussian process with an appropriately chosen spectral density through an appropriately chosen zero-memory nonlinearity. Several applications are cited where such a dual specification is desirable.

56 citations


Journal ArticleDOI
TL;DR: In this paper, conditions for a separable Gaussian process to have sample paths of finite $p$-variation are given in terms of the mean function and the covariance function.
Abstract: For $p \geq 1$, conditions for a separable Gaussian process to have sample paths of finite $p$-variation are given in terms of the mean function and the covariance function. A process with paths of finite $p$-variation may or may not induce a tight measure on the nonseparable Banach space $B_p$. Consequences of tightness and conditions for tightness are given.

37 citations


Journal ArticleDOI
TL;DR: First and second-order approximations to the first passage time conditional probability density function of a stationary Gaussian process, with differentiable sample paths crossing a time-dependent boundary, are provided.
Abstract: First and second-order approximations to the first passage time conditional probability density function of a stationary Gaussian process, with differentiable sample paths crossing a time-dependent boundary, are explicitly provided. In the limit of infinitely large time, our results coincide with the well-known results of Kac and Rice for the constant level-crossing problem.

Journal ArticleDOI
TL;DR: In this paper, the power spectral density function is discretized by parameters and piecewise polynomials and the free parameters are determined iteratively by a least square fit.
Abstract: A method is presented to determine the power spectral density functions from given smooth response spectra. It is assumed that the underlying excitations constitute a stationary Gaussian random process. The relationship between the response spectrum and the power spectral density function is established by the probability distribution of the extreme values. The power spectral density function is discretized by parameters and piecewise polynomials and the free parameters are determined iteratively by a least square fit. Due to this discretisation all required integrations can be carried out analytically. It is also possible to obtain at least square fit value of the duration of the stationary output process. The numerical examples include the calculation of the power spectral density functions for selected response spectra.

Journal ArticleDOI
TL;DR: In this article, the problem of global estimation of the mean function of a quite arbitrary Gaussian process is considered, where the loss function in estimating the function is assumed to be of the form L ( θ, a ) = ∫ [ θ ( t ) − a ( t )] 2 μ ( dt ), and estimators are evaluated in terms of their risk function (expected loss).

Journal ArticleDOI
TL;DR: In this article, the authors prove the same result for a gaussian semimartingale and give some applications to the innovation problem, where the paths of the gaussians are decompositions of the paths into a martingale, and a predictable process of integrable variation.
Abstract: Recently N.C. Jain and D. Monrad [8] have obtained a decomposition of the paths of a gaussian quasimartingale into a martingale and a predictable process of integrable variation such that these components are jointly gaussian. In the first part of this paper we prove the same result for a gaussian semimartingale. In the second part we give some applications to the innovation problem.

Journal ArticleDOI
TL;DR: In this paper, the authors give a characterization theorem for Gaussian processes based on a limit theorem for sums of dependent random variables and prove aconverse implication for the covariance function characteristic of Gaussian process.
Abstract: The aim of this paper isto give a characterization theorem for Gaussian processes.It is wellknown that for Gaussian processes the conditional expectation is alinear function of the states of the process and the conditionalvariance is a deterministic function. In the presentpaper we show aconverse implication. We prove that these two conditions and Lipschitz condition for the covariance function characteristicGaussian processes. The proof is based on a limit theorem for sums ofdependent random variables.

Journal ArticleDOI
TL;DR: Almost sure convergence for the parameter estimate and the filtering error will be established and an almost-supermartingale convergence lemma that allows a stochastic Lyapunov-like approach is considered.

Journal ArticleDOI
TL;DR: In this paper, second-order statistics were examined for the following three correlation estimators: 1) the direct estimator, 2) a hybrid-sign estimator and 3) the polarity coincidence estimator.
Abstract: Using independent samples from two stationary Gaussian processes, second-order statistics are examined for the following three correlation estimators: 1) the direct estimator, 2) a "hybrid-sign" estimator, and 3) the polarity coincidence estimator In most cases, increasing the number of samples by a factor of two or three allows one to use the simple polarity coincidence estimator with the same accuracy as the other two estimators

Journal ArticleDOI
TL;DR: In this paper, the stability of a column subjected to a time varying axial load is considered and the stability conditions for certain class of nonstationary random forcing are also discussed.
Abstract: Dynamic stability of a Kelvin-viscoelastic column subjected to a time varying axial load is considered. Deterministic loads with sinusoidal time variation is studied which reduces to a Mathieu equation. Special attention is given to stationary random excitations. Sufficiency stability criteria for columns subjected to white noise, Gaussian and general stationary random loadings are established. The stability conditions for certain class of nonstationary random forcing are also discussed.

Journal ArticleDOI
V. Seshadri1
TL;DR: In this paper, structural properties of the inverse Gaussian distribution, together with several new characterizations based on constancy of regression of suitable functions on the sum of n independent identically distributed random variables, are presented.
Abstract: This article presents some structural properties of the inverse Gaussian distribution, together with several new characterizations based on constancy of regression of suitable functions on the sum of n independent identically distributed random variables. A decomposition of the statistic λσ (X−1i−X−1) into n - 1 independent chi-squared random variables, each with one degree of freedom, is given when n is of the form 2r.

Journal ArticleDOI
TL;DR: In this paper, the authors compared the conventional least square procedure with a robust/resistant modification in which the weight of each reflection is multiplied by a function of the ratio of its residual to a resistant measure of the width of the residual distribution on the previous cycle.
Abstract: The conventional crystallographic least-squares procedure has been compared with a robust/resistant modification in which the weight of each reflection is multiplied by a function of the ratio of its residual to a resistant measure of the width of the residual distribution on the previous cycle. Three synthetic data sets were created by adding random errors, according to various probability distributions, to the calculated structure factors for a known crystal structure. A set with a Gaussian error distribution was refined with two sets of weights: one assigned correctly in proportion to the reciprocals of the variances of the data points, the other using unit weights throughout. The second error distribution was Gaussian contaminated by 10% drawn from another Gaussian distribution with its variance nine times greater. The third distribution was a long-tailed distribution derived by dividing a random variable with a Gaussian distribution by an independent random variable with a uniform distribution. Each of the first three cases was refined to convergence using both conventional and robust/resistant procedures, with the modified procedure leading to a result at least as close to the known structure as the conventional procedure. In the fourth case, the conventional procedure gave a poor fit, but the robust/ resistant procedure converged to a reasonable approximation to the correct structure.

Journal ArticleDOI
TL;DR: In this paper, a classification is given of all Gaussian measures that have the conditional independence property and such that restricted to a subspace, they coincide with a given Gaussian measure.

Journal ArticleDOI
TL;DR: How to calculate bounds on the mean upcrossing rate of any level by a sum of several clipped normal processes is demonstrated and makes it possible to calculate a lower bound on the exceedance probability giving an evaluation of the upper bound as an approximation to the exact probability.
Abstract: Clipped normal processes are considered from the point of view of modeling transient loads on structures. Sample curves of these types of processes are pulse like, each pulse corresponding to an exceedance of the zero level by a Gaussian process. The pulses are the more narrow the larger negative is the mean of the Gaussian process. In contrast to current transient load models the pulses are of random shape. How to calculate bounds on the mean upcrossing rate of any level by a sum of several clipped normal processes is demonstrated. For high levels these founds are close. Thus, they may be used to obtain an accurate evaluation of a well-known upper bound on the probability of the sum exceeding the level within a given time period. The particular example of the sum of several mutually independent, stationary clipped normal processes permits an exact calculation of the upcrossing rate of any constant level. The upcrossing problem is, in fact, solved by observing the equivalence of the problem with a Gaussian vector process outcrossing problem from a certain convex polyhedral region in the n-dimensional space. Reliability theoretical methods for this case have been reported earlier. These methods also make it possible to calculate a lower bound on the exceedance probability giving an evaluation of the upper bound as an approximation to the exact probability.


Journal ArticleDOI
TL;DR: For stationary discrete-time Gaussian sources and the squared-error distortion measure, a trellis source code is constructed that is implementable and applicable at any nonzero rate to a stationary Gaussian source with a bounded and continuous power spectrum.
Abstract: For stationary discrete-time Gaussian sources and the squared-error distortion measure, a trellis source code is constructed. The encoder consists of a Karhunen-Loeve transform on the source output followed by a search on a trellis structured code, where the decoder is a time-variant nonlinear filter. The corresponding code theorem is proved using the random coding argument. The proof technique follows that of Viterbi and Omura, who proved the trellis coding theorem for memoryless sources. The resultant coding scheme is implementable and applicable at any nonzero rate to a stationary Gaussian source with a bounded and continuous power spectrum. Therefore. for stationary sources, it is more general than Berger's tree coding scheme, which is restricted to autoregressive Gaussian sources in a region of high rate (low distortion).


Journal ArticleDOI
TL;DR: In this article, the authors present a survey devoted to works appearing in the last 3-5 years and pertaining mainly to local properties of the trajectories of Gaussian processes, the behavior of trajectories in the uniform metric, and properties of level sets.
Abstract: The survey is devoted to works appearing in the last 3–5 years and pertaining mainly to local properties of the trajectories of Gaussian processes, the behavior of trajectories in the uniform metric, and properties of level sets. Some new results are also presented.

Journal Article
TL;DR: In this article, a new approximation to the maximum likelihood equations for a Gaussian first-order conditional scheme is proposed, which includes edge correction terms and contain errors only from the four corners.
Abstract: A new approximation to the maximum likelihood equations for a Gaussian first order conditional scheme is proposed. The approximated equations include edge correction terms and contain errors only from the four corners. By simulation, the new estimates are compared with those due to Besag (1974), which are based on a considerably coarser approximation, but the improved approximation seems to improve the estimator only if the interaction is strong and then only slightly.

Journal ArticleDOI
TL;DR: By a systematic use of thc Slepian model processes it is shown that the normalized shape of a positive FM-click approaches a certain rational function with simple random coefficients as the carrier-to-noise ratio tends to infinity.
Abstract: The stochastic behavior of FM-clicks in an unmodulated carrier plus noise is studied in terms of the conditional behavior of stationary Gaussian processes after level crossings. By a systematic use of thc Slepian model processes it is shown that the normalized shape of a positive FM-click approaches a certain rational function with simple random coefficients as the carrier-to-noise ratio tends to infinity. The outcomes of this random function illustrate the typical click shapes that have been observed experimentally. The distribution of the duration of a click is studied by means of similar model processes, and its exact asymptotic density is derived. It is seen to be very similar to a Maxwell density, although of smaller order for large durations.

Journal ArticleDOI
TL;DR: By smearing with a Gaussian distribution a family of single- spin weight functions is constructed which interpolates between an arbitrary single-spin distribution and a pure Gaussian.
Abstract: By smearing with a Gaussian distribution a family of single-spin weight functions is constructed which interpolates between an arbitrary single-spin distribution and a pure Gaussian. The observation of Baker and Bishop (1982) concerning the factorisability of the partition function of the double Gaussian model remains valid for all Gaussian-smeared models. The effects of smearing Gaussian and spherical weight functions are studied in further detail.