We investigate the large-sample behavior of change-point tests based on weighted two-sample U-statistics, in case short-range dependent data. Under some mild mixing conditions, we establish convergence test statistic to an extreme value distribution. A simulation study shows that are superior non-weighted versions when occurs near boundary time interval, while they loose power center.

We define weighted tree-walking automata. We show that the class of tree series recognizable by weighted tree-walking automata over a commutative semiring K is a subclass of the class of regular tree series over K. If K is not a ring, then the inclusion is strict.

We explore the connection between supports of equilibrium measures and quadrature identities, especially in case point sources added to external field Q(z)=|z|2p with p∈N. Along way, we describe some domains respect weighted area measure |z|2pdAz complex boundary |z|−2pdz.