Limiting Gaussian Process Research Articles

An Extension of Clarke's Model With Stochastic Amplitude Flip Processes

Stochastic modeling is an essential tool for studying statistical properties of wireless channels. In multipath fading channel (MFC) models, the signal reception is modeled by a sum of wave path contributions, and Clarke's model is an important example of such which has been widely accepted in many wireless applications. However, since Clarke's model is temporally deterministic, Feng and Field noted that it does not model real wireless channels with time-varying randomness well. Here, we extend Clarke's model to a novel time-varying stochastic MFC model with scatterers randomly flipping on and off. Statistical properties of the MFC model are analyzed and shown to fit well with real signal measurements, and a limit Gaussian process is derived from the model when the number of active wave paths tends to infinity. A second focus of this work is a comparison study of the error and computational cost of generating signal realizations from the MFC model and from its limit Gaussian process. By rigorous analysis and numerical studies, we show that in many settings, signal realizations are generated more efficiently by Gaussian process algorithms than by the MFC model's algorithm. Numerical examples that strengthen these observations are also presented.

CLT for Spectra of Submatrices of Wigner Random Matrices

We prove a CLT for spectra of submatrices of real symmetric and Hermitian Wigner matrices. We show that if in the standard normalization the fourth moment of the off-digonal entries is GOE/GUE-like then the limiting Gaussian process can be viewed as a collection of simply yet nontrivially correlated two-dimensional Gaussian Free Fields.

A Gaussian limit process for optimal FIND algorithms

We consider versions of the FIND algorithm where the pivot element used is the median of a subset chosen uniformly at random from the data. For the median selection we assume that subsamples of size asymptotic to $c \cdot n^\alpha$ are chosen, where $0 < \alpha \leq \frac{1}{2}$, $c > 0$ and $n$ is the size of the data set to be split. We consider the complexity of FIND as a process in the rank to be selected and measured by the number of key comparisons required. After normalization we show weak convergence of the complexity to a centered Gaussian process as $n \to \infty$, which depends on $\alpha$. The proof relies on a contraction argument for probability distributions on càdlàg functions. We also identify the covariance function of the Gaussian limit process and discuss path and tail properties.

A large deviation principle for networks of rate neurons with correlated synaptic weights

One of the major goals of mathematical neuroscience is to rigorously justify macroscopic continuum neural-field equations by deriving them from microscopic equations governing the interactions of individual neurons. Since the macroscopic variables are continuous, their value at a particular point is usually interpreted as a mean taken over the neurons in a small neighborhood of the point. In order that the mean is an accurate approximation, it is normally assumed / proved that the neurons are approximately uncorrelated, so that the law of large numbers implies that their average behavior is close to the mean. We develop a model of neural networks with inhomogeneous weights between the neurons, and analyze its behavior as the number of neurons asymptotes to infinity. It will be seen that the inhomogeneity of the weights ensures that the neurons in the limit system are not uncorrelated. Our results thus suggest that the mean-field approximation is insufficient. We study the asymptotic behavior of a network of N firing rate neurons as the number N grows to infinity. The neurons are modeled as lying equally-spaced on a ring. The membrane potential of each neuron evolves according to a discrete time version of the Hopfield or Wilson-Cowan equations [1]. The synaptic weights J(i,j) of presynaptic neuron j and postsynaptic neuron i are modeled as Gaussian Random variables, with identical means that scale as one over N. The covariance between J(i,j) and J(k,l) also scales as one over N times C(i-k,j-l) for some fixed function C. In other words, the covariance is considered to be a function of the 'ring distance' between the presynaptic and postsynaptic. Our main result is that the behavior of the infinite size ensemble of neurons can be described by a simple nonlinear transformation of a spatially stationary (along the ring) Gaussian random process. The nonlinearity is a combination of the firing rate function and the leak. This Gaussian process is described by its mean, the same time-varying function for each neuron, and its covariance operator. The covariance operator describes the correlation between any k-tuple of neurons. It is also stationary in the sense that if we translate each neuron of the k-tuple by the same amount along the ring, the correlation does not change. We have been able to obtain explicitly the equations that describe the mean and covariance of the limit Gaussian process. They form a set of strongly coupled recursive (in time) equations. Our analysis goes beyond the identification of the asymptotic limit of the network. We also prove that in effect the probability law that describes the solutions to the network equations converges exponentially fast toward the previous limit (in a precise mathematical sense) and we have been able to compute the specific rate of convergence thanks to the use of the theory of Large Deviations [2]. This rate of convergence is given explicitly by a function (called the good rate function) defined over the set of all possible asymptotic limit probability laws. We prove that it has a unique minimum at the asymptotic limit. Most modeling of neural networks assumes / proves some sort of thermodynamic limit, whereby if one isolates a particular population of neurons in a localized area of space, they are found to fire increasingly asynchronously as the number in the population asymptotes to infinity, e.g. [3]. However our limit does not possess this property: the nontrivial covariances between the weights ensures that there are large system-wide correlations between the neurons in the asymptotic limit. A very important implication of our result is that the mean-field behavior is insufficient to characterize the behavior of a population. Our work challenges the assumption held by some that one cannot have a concise macroscopic description of a neural network without an assumption of asynchronicity at the local population level. It is a generalization of the work of Moynot and Samuelidies [4].

A simple construction of optimal estimation in multivariate marginal Cox regression

The multivariate extension of the Cox model proposed by Wei, Lin and Weissfeld in 1989 has been widely used for analyzing multivariate survival data. Under the model assumption, failure times from an individual are assumed to marginally follow their respective proportional hazards regression relation, leaving the joint distribution completely unspecified. This paper presents a simple approach to efficiency improvement through segmentation of stochastic integrals in the marginal estimating equations and incorporation of the limiting covariance structure. It is shown that when partition of the time interval is done at a suitable rate, the resulting estimator is consistent and asymptotically normal. Through the reproducing kernel Hilbert space arising from the covariance function of the limiting Gaussian process, it is also shown that the proposed estimator is asymptotically optimal within a reasonable class of estimators under marginal specification. Simulations are conducted to assess the finite-sample performance of the proposed method.

Non-uniform spacings processes

We provide a joint strong approximation of the uniform spacings empirical process and of the uniform quantile process by sequences of independent Gaussian processes. This allows us to obtain an explicit description of the limiting Gaussian process generated by the sample spacings from a non-uniform distribution. It is of the form \({B(t)+(1-\sigma_F) \{(1-t)\log(1-t)\} \int_{0}^1\{B(s)/(1-s)\}ds}\), for 0 ≤ t ≤ 1, where {B(t):0 ≤ t ≤ 1} denotes a Brownian bridge, and where \({\sigma^2_F={\rm Var}({\rm log}\,f(X))}\) is a factor depending upon the underlying distribution function \({F(\cdot)=\mathbb{P}(X\leq x)}\) through its density \({f(x)=\frac{d}{dx}F(x)}\). We provide a strong approximation of the non-uniform spacings processes by replicae of this Gaussian process, with limiting sup-norm rate \({O_{\mathbb{P}}(n^{-1/8}(\log n)^{1/2})}\). The limiting process reduces to a Brownian bridge if and only if \({\sigma^2_F=1}\), which is the case when the sample observations are exponential. For uniform spacings, we get \({\sigma^{2}_F=0}\), which is in agreement with the results of Beirlant (In: Limit theorems in probability and statistics, Proc Coll Math Soc J Bolyai, vol 36, Akademiai Kiado, Budapest, pp 77–80, 1984), and Aly et al. (Z Wahrsch Verw Gebiete 66:461–484, 1984).

A Functional Combinatorial Central Limit Theorem

The paper establishes a functional version of the Hoeffding combinatorial central limit theorem. First, a pre-limiting Gaussian process approximation is defined, and is shown to be at a distance of the order of the Lyapounov ratio from the original random process. Distance is measured by comparison of expectations of smooth functionals of the processes, and the argument is by way of Stein's method. The pre-limiting process is then shown, under weak conditions, to converge to a Gaussian limit process. The theorem is used to describe the shape of random permutation tableaux.

Dynamics of Mandelbrot cascades

Mandelbrot multiplicative cascades provide a construction of a dynamical system on a set of probability measures defined by inequalities on moments. To be more specific, beyond the first iteration, the trajectories take values in the set of fixed points of smoothing transformations (i.e., some generalized stable laws). Studying this system leads to a central limit theorem and to its functional version. The limit Gaussian process can also be obtained as limit of an “additive cascade” of independent normal variables.

Accelerated convergence for nonparametric regression with coarsened predictors

We consider nonparametric estimation of a regression function for a situation where precisely measured predictors are used to estimate the regression curve for coarsened, that is, less precise or contaminated predictors. Specifically, while one has available a sample $(W_1,Y_1),...,(W_n,Y_n)$ of independent and identically distributed data, representing observations with precisely measured predictors, where $\mathrm{E}(Y_i|W_i)=g(W_i)$, instead of the smooth regression function $g$, the target of interest is another smooth regression function $m$ that pertains to predictors $X_i$ that are noisy versions of the $W_i$. Our target is then the regression function $m(x)=E(Y|X=x)$, where $X$ is a contaminated version of $W$, that is, $X=W+\delta$. It is assumed that either the density of the errors is known, or replicated data are available resembling, but not necessarily the same as, the variables $X$. In either case, and under suitable conditions, we obtain $\sqrt{n}$-rates of convergence of the proposed estimator and its derivatives, and establish a functional limit theorem. Weak convergence to a Gaussian limit process implies pointwise and uniform confidence intervals and $\sqrt{n}$-consistent estimators of extrema and zeros of $m$. It is shown that these results are preserved under more general models in which $X$ is determined by an explanatory variable. Finite sample performance is investigated in simulations and illustrated by a real data example.

Integral curves of noisy vector fields and statistical problems in diffusion tensor imaging: Nonparametric kernel estimation and hypotheses testing

Let $v$ be a vector field in a bounded open set $G\subset {\mathbb{R}}^d$. Suppose that $v$ is observed with a random noise at random points $X_i,\ i=1,\dots, n$, that are independent and uniformly distributed in $G$. The problem is to estimate the integral curve of the differential equation \[\frac{dx(t)}{dt}=v(x(t)),\qquad t\geq 0,x(0)=x_{0}\in G,\] starting at a given point $x(0)=x_0\in G$ and to develop statistical tests for the hypothesis that the integral curve reaches a specified set $\Gamma\subset G$. We develop an estimation procedure based on a Nadaraya–Watson type kernel regression estimator, show the asymptotic normality of the estimated integral curve and derive differential and integral equations for the mean and covariance function of the limit Gaussian process. This provides a method of tracking not only the integral curve, but also the covariance matrix of its estimate. We also study the asymptotic distribution of the squared minimal distance from the integral curve to a smooth enough surface $\Gamma\subset G$. Building upon this, we develop testing procedures for the hypothesis that the integral curve reaches $\Gamma$. The problems of this nature are of interest in diffusion tensor imaging, a brain imaging technique based on measuring the diffusion tensor at discrete locations in the cerebral white matter, where the diffusion of water molecules is typically anisotropic. The diffusion tensor data is used to estimate the dominant orientations of the diffusion and to track white matter fibers from the initial location following these orientations. Our approach brings more rigorous statistical tools to the analysis of this problem providing, in particular, hypothesis testing procedures that might be useful in the study of axonal connectivity of the white matter.

Bootstrapping the Empirical Distribution Function of a Spatial Process

In this article, we consider a stationary α-mixing random field in IRd. Under a large-sample scheme that is a mixture of the so-called “infill” and “increasing domain” asymptotics, we establish a functional central limit theorem for the empirical processes of this random field. Further, we apply a blockwise bootstrap to the samples. Under the condition that the side length of the block \({\lambda_l = {\cal O}(\lambda_n^{\beta})}\) for some 0 < β < 1, where λ n is the growth rate in the increasing domain asymptotics, we show that the bootstrapped empirical process converges weakly to the same limiting Gaussian process almost surely. Extension to multivariate random fields and application to differentiable statistical functionals are also given. A spatial version of the Bernstein’s inequality is developed, which may be of some independent interest.

Asymptotic Distribution of Coordinates on High Dimensional Spheres

The coordinates $x_i$ of a point $x = (x_1, x_2, \dots, x_n)$ chosen at random according to a uniform distribution on the $\ell_2(n)$-sphere of radius $n^{1/2}$ have approximately a normal distribution when $n$ is large. The coordinates $x_i$ of points uniformly distributed on the $\ell_1(n)$-sphere of radius $n$ have approximately a double exponential distribution. In these and all the $\ell_p(n),1 \le p \le \infty,$ convergence of the distribution of coordinates as the dimension $n$ increases is at the rate $\sqrt{n}$ and is described precisely in terms of weak convergence of a normalized empirical process to a limiting Gaussian process, the sum of a Brownian bridge and a simple normal process.

Stochastic and deterministic analysis of SIS household epidemics

We analyse SIS epidemics among populations partitioned into households. The analysis considers both the stochastic and deterministic models and, unlike in previous analyses, we consider general infectious period distributions. For the deterministic model, we prove the existence of an endemic equilibrium for the epidemic if and only if the threshold parameter, R*, is greater than 1. Furthermore, by utilising Markov chains we show that the total number of infectives converges to the endemic equilibrium as t → ∞. For the stochastic model, we prove a law of large numbers result for the convergence, to the deterministic limit, of the mean number of infectives per household. This is followed by the derivation of a Gaussian limit process for the fluctuations of the stochastic model.

Asymptotic fluctuations of mutagrams

Abrupt changes in sequential data, known as change-points, can be detected by fitting differences of one-sided kernel estimators. The detection depends on both a bandwidth and a threshold, where we declare a change-point whenever the differences of the one-sided kernel estimators exceed the threshold. The joint behavior of the difference of one-sided kernel estimators and scale can be graphically represented in a mutagram [Müller, H.G., Wai, N., 2004. Change trees and mutagrams for the visualization of local changes in sequence data. J. Comput. Statist. Graph. 13, 571–585.]. Here we study the fluctuations of the mutagram values in dependency on different bandwidth choices, corresponding to scale. Study of these fluctuations is related to the SiZer approach of Chaudhuri and Marron [2000. Scale space view of curve estimation. Ann. Statist. 28, 408–428.]. We explore the asymptotic properties and convergence of a local bandwidth process for differences of one-sided kernel estimators. These processes are shown to be tight with a Gaussian limit process that represents the asymptotic fluctuations of mutagrams. These fluctuations are illustrated with a sequence of vertical ocean shear data.

The empirical distribution function and partial sum process of residuals from a stationary arch with drift process

The weak convergence of the empirical process and partial sum process of the residuals from a stationary ARCH-M model is studied. It is obtained for and\(\sqrt n \) consistent estimate of the ARCH-M parameters. We find that the limiting Gaussian processes are no longer distribution free and hence residuals cannot be treated as i.i.d. In fact the limiting Gaussian process for the empirical process is a standard Brownian bridge plus an additional term, while the one for partial sum process is a standard Brownian motion plus an additional term. In the special case of a standard ARCH process, that is an ARCH process with no drift, the additional term disappears. We also study a sub-sampling technique which yields the limiting Gaussian processes for the empirical process and partial sum process as a standard Brownian bridge and a standard Brownian motion respectively.

Partial Sum Process for Records

Suppose the upper records \({\left\{ {X_{{L_{n} }} } \right\}}\) from a sequence of i.i.d. random variables is in the domain of attraction of a normal distribution. Consider the D(0,1]-valued process {Zn(·)} constructed by usual interpolation of the partial sums of the records. We prove that under some mild conditions, {Zn} converges to a limiting Gaussian process in D(0,1]. As a consequence, the partial sums of records is asymptotically normal.

Gaussian Approximation to the Partial Sum Processes of Moving Averages

The authors study approximation to the partial sum processes which is based on the stationary sequences of random variables having the structure of the so-called moving averages of independent identically distributed observations. In particular, the rates of convergence both in Donsker's and Strassen's invariance principles are obtained in the case when the limit Gaussian process is a fractional Brownian motion with an arbitrary Hurst parameter.

The $\bm{L}_\mathbf{1}$-norm density estimator process

The notion of an $L_{1}$-norm density estimator process indexed by a class of kernels is introduced. Then a functional central limit theorem and a Glivenko--Cantelli theorem are established for this process. While assembling the necessary machinery to prove these results, a body of Poissonization techniques and restricted chaining methods is developed, which is useful for studying weak convergence of general processes indexed by a class of functions. None of the theorems imposes any condition at all on the underlying Lebesgue density $f$. Also, somewhat unexpectedly, the distribution of the limiting Gaussian process does not depend on $f$.

On Nonparametric Estimation of Reliability Indices

The paper proposes nonparametric estimates for exponential cumulative intensity functions based on the generalized model of random censorship from the right. For respectively normed estimates, the results on strong approximationof them by a sequence of square integrable orthogonal martingales and on the weak convergence to the limit Gaussian process are proved.

Asymptotics of random convex polygonal lines

The present paper deals with the limit shape of random plane convex polygonal lines whose edges are independent and identically distributed, with finite first moment. The smoothness of a limit curve depends on some properties of the distribution. The limit curve is determined by the projection of the distribution to the unit circle. This correspondence between limit curves and measures on the unit circle is proved to be a bijection. The emphasis is on limit distributions of deviations of random polygonal lines from a limit curve. Normed differences of Minkowski support functions converge to a Gaussian limit process. The covariance of this process can be found in terms of the initial distribution. In the case of uniform distribution on the unit circle, a formula for the covariance is found. The main result is that a.s. sample functions of the limit process have continuous first derivative satisfying the Holder condition of order a, for any fixed a with 0<a<1/2. Bibliography: 7 titles.