On the weak laws of large numbers for weighted sums of dependent identically distributed random vectors in Hilbert spaces

This note establishes the weak laws of large numbers with general normalizing sequences for weighted sums of dependent identically distributed random vectors taking values in real separable Hilbert spaces. The dependent structures include pairwise and coordinatewise negative dependence and coordinatewise negative association. The sharpness of the result for the case where the random vectors are coordinatewise negatively associated is illustrated by an example.