Consider a class of skew product transformations consisting of an ergodic ora periodic transformation on a probability space (M, B, m) in the base and asemigroup of transformations on another probability space (W,F,P) in the fibre.Under suitable mixing conditions for the fibre transformation, we show that theproperties ergodicity, weakly mixing, and strongly mixing are passed on fromthe base transformation to the skew product (with respect to the productmeasure). We derive ergodic theorems with respect to the skew product on theproduct space. The main aim of this paper is to establish uniform convergencewith respect to the base variable for the series of ergodic averages of afunction F on the product of the two probability spaces along the orbits ofsuch a skew product. Assuming a certain growth condition for the couplingfunction, a strong mixing condition on the fibre transformation, and continuityandintegrability conditions for F, we prove uniform convergence in the base andL^p(P)-convergence in the fibre. Under an equicontinuity assumption on F wefurther show P-almost sure convergence in the fibre. Our work has anapplication in information theory: It implies convergence of the averages offunctions on random fields restricted to parts of stair climbing patternsdefined by a direction.
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