$X$ is a discrete random variable taking values in $\{0,1,2,3,\ldots\}$ with a probability mass function $p_X(n)$. Let
$$U_k(X)=\sum_{n=k}^\infty\sqrt{np_X(n)p_X(n-k)}$$
where $k$ is a positive integer.
This came up in the course of information theory research (with $k=1$ and $k=2$ in my case). Has anyone ever seen this quantity anywhere? Can anyone see relationships to other statistics of $X$ (maybe bounds using mean/variance/etc)? I am dumbfounded and don't know what to do with this thing...