In this paper, we study asymptotic stability properties of risk-sensitive filters with respect to their initial conditions. In particular, we consider a linear time-invariant systems with initial conditions that are not necessarily Gaussian. We show that in the case of Gaussian initial conditions, the optimal risksensitive filter asymptotically converges to any suboptimal filter initialized wit...

Lars Lindbom, Mikael Sternad and Anders Ahlén Signals and Systems, Uppsala University, PO Box 528, SE-75120, Uppsala, Sweden. [email protected], ms,aa @signal.uu.se Abstract: Adaptation laws with constant gains, that adjust parameters of linear regression models, are investigated. The class of algorithms includes LMS as its simplest member. Closed-form expressions for the tracking MSE...

We provide the solutions of linear, left-invariant, 2nd-order stochastic evolution equations on the 2D-Euclidean motion group. These solutions are given by group-convolution with the corresponding Green’s functions that we derive in explicit form in Fourier space. A particular case coincides with the hitherto unsolved forward Kolmogorov equation of the so-called direction process, the exact sol...

\begin{abstract} Let $M$ be a complete Riemannian manifold and $G$ Lie subgroup of the isometry group acting freely properly on $M.$ We study Dirichlet Problem% \[ \left\{ \begin{array} [c]{l}% \operatorname{div}\left( \frac{a\left( \left\Vert \nabla u\right\Vert \right) }{\left\Vert }\nabla u\right) =0\text{ in }\Omega u|\partial\Omega=\varphi \end{array} \right. \] where $\Omega$ is $G-$invar...

It was shown by Samelson [9] and Wang [10] that each compact Lie group K of even dimension admits left-invariant complex structures. When K has odd dimension it admits a left-invariant CR-structure of maximal dimension. This has been proved recently by Charbonnel and Khalgui [2] who have also given a complete algebraic description of these structures. In this article we present an alternative a...

We are interested in the statistical behavior of a certain continued fraction whose associated dynamical system has infinite invariant measure. show growth rate denominator $Q_n$ $n$-th convergent negative expansion $x$ and approximation: $$ \frac{\log{n}}{n}\log{\left|x-\frac{P_n}{Q_n}\right|}\rightarrow -\frac{\pi^2}{3} \quad \text{in measure} for $x$. In course proof, we reprove related resu...

We study semi-Riemannian submanifolds of arbitrary codimension in a Lie group [Formula: see text] equipped with bi-invariant metric. In particular, we show that, if the normal bundle is closed under bracket, then any Jacobi operator equals square associated invariant shape text]. This permits to understand curvature adaptedness geometrically, terms left translations. For example, case where Rie...

Approximations of slightly nonlinear systems with linear time-invariant (LTI) models are often used in applications. Here, LTI models that are optimal approximations in the mean-square error sense are studied. It is shown that these models can be very sensitive to small nonlinearities. Furthermore, the significance of the distribution of the input process is discussed. From the examples studied...

In this paper, we study asymptotic stability properties of risk-sensitive lters with respect to their initial conditions. In particular, we consider a linear time-invariant systems with initial conditions that are not necessarily Gaussian. We show that in the case of Gaussian initial conditions, the optimal risk-sensitive lter asymptotically converges to any suboptimal lter initialized with an ...

The least-mean-squares (LMS) algorithm is analysed as a feedback control system. It is shown that despite the fact that LMS is a time-variant system that in fact it behaves much as a linear time-invariant (LTI) closed-loop control system. Therefore, it is possible to treat the LMS algorithm as a control system in the classical sense and define properties such as bandwidth to determine how fast ...