The square of a Gaussian Markov process and nonlinear prediction

An explicit formula is obtained for the nonlinear predictor of Y(t) = X(t)² − E(X(t)²), where X(t) is an N-ple Gaussian Markov process.     

A Kuznetsov Formula for Kloosterman Sums on GLn

We prove a Kuznetsov sum formula for Kloosterman sums on GL _n corresponding to the big Bruhat cell. Using this formula, a weighted sum of Kloosterman sums can be expressed in spectral decomposition. A non-trivial estimate of the spectral side might lead to a proof of cancellations in sums of Kloosterman sums on GL _n.     

Continuity of the Filter with Unbounded Observation Coefficients

In this article, we study the continuity with respect to the trajectories of the observation process for the filter associated with nonlinear filtering problems when the coefficients depend on both the signal and the observation and the observation coefficient is unbounded.

To achieve this task we define a formal unnormalized filter and prove by limiting arguments that it is related to the original filter through a generalized Bayes formula, and is locally Lipschitz continuous with respect to the uniform norm.     

Non-linear prediction of the degree n of a Gaussian N-ple Markov process

An explicit formula is obtained for the nonlinear predicion of Y(t) = Xⁿ(t), where X(t) is an N-ple Gaussian Markov process.     

Small Deviations for Gaussian Markov Processes Under the Sup-Norm

Let {X(t); 0t1} be a real-valued continuous Gaussian Markov process with mean zero and covariance (s, t) = EX(s) X(t) 0 for 0<s, t<1. It is known that we can write (s, t) = G(min(s, t)) H(max(s, t)) with G>0, H>0 and G/H nondecreasing on the interval (0, 1). We show that

Asymptotic behavior of Bayes estimators for hidden Markov models with application to ion channels

In this paper we study the asymptotic behavior of Bayes estimators for hidden Markov models as the number of observations goes to infinity. The theorem that we prove is similar to the Bernstein—von Mises theorem on the asymptotic behavior of the posterior distribution for the case of independent observations. We show that our theorem is applicable to a wide class of hidden Markov models. We also discuss the implication of the theorem’s assumptions for several models that are used in practical applications such as ion channel kinetics.      

Reliability Approximation for Markov Chain Imbeddable Systems

In the present article, a simple method is developed for approximating the reliability of Markov chain imbeddable systems. The approximating formula reduces the problem to the reliability assessment of smaller systems with structure similar to the original systems. Two specific reliability structures which have attracted considerable research interest recently (r-within-consecutive-k-out-of-n system and two dimensional r-within-k₁ × k₂-out-of-n₁ × n₂ system) are studied by the new approach and numerical calculations are carried out, which reveal the high quality of our approximations. Several possible extensions and generalizations are also presented in brief.     

Optimal filters for a hidden Markov random field model

A Markov random field (MRF) is a useful technical tool for modeling dynamics systems exhibiting some type of spatio-temporal variability. In this paper, we propose optimal filters for the states of a partially observed temporal Markov random field. We also discuss parameters estimation. This generalizes an earlier work by Elliott and Aggoun [1].     

Testing Gaussian process with applications to super-resolution

This article introduces exact testing procedures on the mean of a Gaussian process X derived from the outcomes of ${\ell }_1$-minimization over the space of complex valued measures. The process X can be thought as the sum of two terms: first, the convolution between some kernel and a target atomic measure (mean of the process); second, a random perturbation by an additive centered Gaussian process. The first testing procedure considered is based on a dense sequence of grids on the index set of X and we establish that it converges (as the grid step tends to zero) to a randomized testing procedure: the decision of the test depends on the observation X and also on an independent random variable. The second testing procedure is based on the maxima and the Hessian of X in a grid-less manner. We show that both testing procedures can be performed when the variance is unknown (and the correlation function of X is known). These testing procedures can be used for the problem of deconvolution over the space of complex valued measures, and applications in frame of the Super-Resolution theory are presented. As a byproduct, numerical investigations may demonstrate that our grid-less method is more powerful (it detects sparse alternatives) than tests based on very thin grids.     

Trace Formula for p-Adics

We give a p-adic analogue of Selberg's trace formula relating the trace of a semigroup generated by a natural elliptic operator with a sum over contributions coming from closed geodesics. The construction uses probabilistic methods to define the generator.     

Bivariate Markov chain embeddable variables of polynomial type

The primary aim of the present article is to provide a general framework for investigating the joint distribution of run length accumulating/enumerating variables by the aid of a Markov chain embedding technique. To achieve that we introduce first a class of bivariate discrete random variables whose joint distribution can be described by the aid of a Markov chain and develop formulae for their joint probability mass function, generating functions and moments. The results are then exploited for the derivation of the distribution of a bivariate run-related statistic. Finally, some interesting uses of our results in reliability theory and educational psychology are highlighted. Research supported by General Secretary of Research and Technology of Greece under grand PENED 2001.     

Large-Deviation Local Theorem for Additive Functionals of a Markov Gaussian Field. III

This paper is a continuation of [1]–[2].     

Large-Deviation Local Theorem for Additive Functionals of a Markov Gaussian Field. I

We prove a local limit theorem (LLT) on Cramer-type large deviations for sums S _V = _t V ( _t ), where _t , t Z , 1, is a Markov Gaussian random field, V Z , and is a bounded Borel function. We get an estimate from below for the variance of S _V and construct two classes of functions , for which the LLT of large deviations holds.     

Large-Deviation Local Theorem for Additive Functionals of a Markov Gaussian Field. II

This paper is a continuation of [1].     

On Dynkin's Markov property of random fields associated with symmetric processes

Let p(t, x, y) be a symmetric transition density with respect to a σ-finite measure m on (E, $\text{E}$), g(x,y)=∫p(t,x,y)dt, and $\text{M}\text{={σ-finite measures μ?0:∫g(x,y)μ(}\text{d}\text{x)μ(}\text{d}\text{y)<∞}}$. There exists a Gaussian random field $\text{Φ={?}{}_{\mathrm{\mu }}\text{:μ?}\text{M}\text{}}$ with mean 0 and covariance $\text{E}\text{?}{}_{\mathrm{\mu }}\text{?}{}_{\mathrm{\nu }}\text{=∫g(x,y)μ(}\text{d}\text{x)ν(}\text{d}\text{y)}$. Letting $\text{F}\text{(B)=σ{?}{}_{\mathrm{\mu }}\text{:μ(B}{}^{\mathrm{c}}\text{)=0}}$ we consider necessary and sufficient conditions for the Markov property (MP) on sets B, C: $\text{F}$(B), $\text{F}$(C) c.i. given $\text{F}$(B ∩ C). Of crucial importance is the following, proved by Dynkin: $\text{E}\text{{?}{}_{\mathrm{\mu }}\text{∣}\text{F}\text{(B)}=?}{}_{\mathrm{\mu B}}$, where μ_B is the hitting distribution of the process corresponding to p, m with initial law μ. Another important fact is that ?_μ=?_ν iff μ, ν have the same potential. Putting these together with an additional transience assumption, we present a potential theoretic proof of the following necessary and sufficient condition for (MP) on sets B, C: For every x?E, T_B∩C=T_B+T_C∮ θ_TB=T_C+T_B∮θ_TC a.s. P^x where, for D ? $\text{E}$, T_D is the hitting time of D for the process associated with p, m. This implies a necessary condition proved by Dynkin in a recent preprint for the case where B∪C=E and B, C are finely closed.     

Universally consistent conditionalU-statistics for absolutely regular processes and its applications for hidden Markov models

A general class of conditionalU-statistics was introduced by W. Stute as a generalization of the Nadaraya-Watson estimates of a regression function. It was shown that such statistics are universally consistent. Also, universal consistentcies of the window andk _n -nearest neighbor estimators (as two special cases of the conditionalU-statistics) were proved. In this paper, we extend these results from the independent case to dependent case. The result is applied to verify the Bayes risk consistency of the corresponding discrimination rules. Research supported by the Office of Naval Research Contract N00014-91-J-1020.     

Stochastic Detectability and Mean Bounded Error Covariance of the Recursive Kalman Filter with Markov Jump Parameters

In this article, we study the error covariance of the recursive Kalman filter when the parameters of the filter are driven by a Markov chain taking values in a countably infinite set. We do not assume ergodicity nor require the existence of limiting probabilities for the Markov chain. The error covariance matrix of the filter depends on the Markov state realizations, and hence forms a stochastic process. We show in a rather direct and comprehensive manner that this error covariance process is mean bounded under the standard stochastic detectability concept. Illustrative examples are included.     

Characterization of regular Markov chains for asymptotically independent variables

The characterization problem of a homogeneous Markov chain (with either finitely many or a countable number of states) is considered for the class of the variables satisfying the asymptotic independence. An allied result when some state distribution is given beforehand at some step is also presented.     

Image segmentation by polygonal Markov Fields

This paper advocates the use of multi-coloured polygonal Markov fields for model-based image segmentation. The formal construction of consistent multi-coloured polygonal Markov fields by Arak–Clifford–Surgailis and its dynamic representation are specialised and adapted to our context. We then formulate image segmentation as a statistical estimation problem for a Gibbsian modification of an underlying polygonal Markov field, and discuss the choice of Hamiltonian. Monte Carlo techniques, including novel Gibbs updates for the Arak model, to estimate the model parameters and find an optimal partition of the image are developed. The approach is applied to image data, the first published application of polygonal Markov fields to segmentation problems in the mathematical literature. Work carried out under project PNA4.3 ‘Stochastic Geometry’. This research was supported by the EC 6th Framework Programme Priority 2 Information Society Technology Network of Excellence MUSCLE (Multimedia Understanding through Semantics, Computation and Learning; FP6-507752), and partially by the Foundation for Polish Science (FNP) and by the Polish Minister of Scientific Research and Information Technology, grant 1 P03A 018 28 (2005-2007) [the third author].