  We study the numerical integration problem for functions with infinitely many
variables. The function spaces of integrands we consider are weighted
reproducing kernel Hilbert spaces with norms related to the ANOVA decomposition
of the integrands. The weights model the relative importance of different
groups of variables. We investigate randomized quadrature algorithms and
measure their quality by estimating the randomized worst-case integration
error.
  In particular, we provide lower error bounds for a very general class of
randomized algorithms that includes non-linear and adaptive algorithms.
Furthermore, we propose new randomized changing dimension algorithms and
present favorable upper error bounds. For product weights and
finite-intersection weights our lower and upper error bounds match and show
that our changing dimension algorithms are optimal in the sense that they
achieve convergence rates arbitrarily close to the best possible convergence
rate. As more specific examples, we discuss unanchored Sobolev spaces of
different degrees of smoothness and randomized changing dimension algorithms
that use as building blocks scrambled polynomial lattice rules.
  Our analysis extends the analysis given in [J. Baldeaux, M. Gnewuch. Optimal
randomized multilevel algorithms for infinite-dimensional integration on
function spaces with ANOVA-type decomposition. arXiv:1209.0882v1 [math.NA],
Preprint 2012]. In contrast to the previous article we now investigate a
different cost model for algorithms. With respect to that cost model,
randomized multilevel algorithms cannot, in general, achieve optimal
convergence rates, but, as we prove for important classes of weights, changing
dimension algorithms actually can.
