  We study a randomized quadrature algorithm to approximate the integral of
periodic functions defined over the high-dimensional unit cube. Recent work by
Kritzer, Kuo, Nuyens and Ullrich (2019) shows that rank-1 lattice rules with a
randomly chosen number of points and good generating vector achieve almost the
optimal order of the randomized error in weighted Korobov spaces, and moreover,
that the error is bounded independently of the dimension if the weight
parameters, $\gamma_j$, satisfy the summability condition
$\sum_{j=1}^{\infty}\gamma_j^{1/\alpha}<\infty$, where $\alpha$ is a smoothness
parameter. The argument is based on the existence result that at least half of
the possible generating vectors yield almost the optimal order of the
worst-case error in the same function spaces.
  In this paper we provide a component-by-component construction algorithm of
such randomized rank-1 lattice rules, without any need to check whether the
constructed generating vectors satisfy a desired worst-case error bound.
Similarly to the above-mentioned work, we prove that our algorithm achieves
almost the optimal order of the randomized error and that the error bound is
independent of the dimension if the same condition
$\sum_{j=1}^{\infty}\gamma_j^{1/\alpha}<\infty$ holds. We also provide
analogous results for tent-transformed lattice rules for weighted half-period
cosine spaces and for polynomial lattice rules in weighted Walsh spaces,
respectively.
