For Markov chains with a finite, partially ordered state space, we show strong stationary duality under the condition of Möbius monotonicity of the chain. We give examples of dual chains in this context which have no downwards transitions. We illustrate general theory by an analysis of nonsymmetric random walks on the cube with an interpretation for unreliable networks of queues.

Abstract. Consider a (robotic) explorer starting an exploration of an unknown terrain from its base station. As the explorer has only limited communication radius, it is necessary to maintain a line of robotic relay stations following the explorer, so that consecutive stations are within the communication radius of each other. This line has to start in the base station and to end at the explorer.In the simple scenario considered here we assume an obstacle-free terrain, so that the shortest connection (the one which needs the smallest number of relay stations) is a straight line. We consider an explorer who goes an arbitrary, typically winding way, and define a very simple, intuitive, fully local, distributed strategy for the relay stationsour GO-TO-THE-MIDDLE strategy -to maintain a line from the base station to the robot as short as possible.Besides the definition of this strategy, we present an analysis of its performance under different assumptions. For the static case we prove a bound on the convergence speed, for the dynamic case we present experimental evaluations that show the quality of our strategy under different types of routes the explorer could use.

We present herein a robust algorithm for cell tracking in a sequence of time-lapse 2-D fluorescent microscopy images. Tracking is performed automatically via a multiphase active contours algorithm adapted to the segmentation of clustered nuclei with obscure boundaries. An ellipse fitting method is applied to avoid problems typically associated with clustered, overlapping, or dying cells, and to obtain more accurate segmentation and tracking results. We provide quantitative validation of results obtained with this new algorithm by comparing them to the results obtained from the established CellProfiler, MTrack2 (plugin for Fiji), and LSetCellTracker software.

We give explicit formulas for ruin probabilities in a multidimensional Generalized Gambler's ruin problem. The generalization is best interpreted as a game of one player against d other players, allowing arbitrary winning and losing probabilities (including ties) depending on the current fortune with particular player. It includes many previous other generalizations as special cases. Instead of usually utilized first-step-like analysis we involve dualities between Markov chains. We give general procedure for solving ruin-like problems utilizing Siegmund duality in Markov chains for partially ordered state spaces studied recently in context of Möbius monotonicity.

This paper studies nonparametric regression with long memory (LRD) errors and predictors. First, we formulate general conditions which guarantee the standard rate of convergence for a nonparametric kernel estimator. Second, we calculate the Mean Integrated Squared Error (MISE). In particular, we show that LRD of errors may influence MISE. On the other hand, an estimator for a shape function is typically not influenced by LRD in errors. Finally, we investigate properties of a data-driven bandwidth choice. We show that Averaged Squared Error (ASE) is a good approximation of MISE, however, this is not the case for a cross-validation criterion.