  In this note, we study a concatenation of quasi-Monte Carlo and plain Monte
Carlo rules for high-dimensional numerical integration in weighted function
spaces. In particular, we consider approximating the integral of periodic
functions defined over the $s$-dimensional unit cube by using rank-1 lattice
point sets only for the first $d\, (<s)$ coordinates and random points for the
remaining $s-d$ coordinates. We prove that, by exploiting a decay of the
weights of function spaces, almost the optimal order of the mean squared
worst-case error is achieved by such a concatenated quadrature rule as long as
$d$ scales at most linearly with the number of points. This result might be
useful for numerical integration in extremely high dimensions, such as partial
differential equations with random coefficients for which even the standard
fast component-by-component algorithm is considered computationally expensive.
