I am interested in limit of V-statistics with asymmetric kernel function. Let asymmetric $h:\mathbb{R}^r\rightarrow \mathbb{R}$. For instance when $r=2$, $h(x,y) \ne h(y,x)$ for some $y$ and $x$. I am interested in limit of $$V_n =\frac{1}{n^r}\sum_{i_1=1}^n\cdots \sum_{i_r=1}^nh(x_{i_1},\cdots,x_{i_r}), $$ where $x_i$ are drawn from the distribution $F$ identically. So sample $x_1,\cdots,x_n$ are iid.
I guess $V_n\rightarrow \int h(x_{i_1},\cdots,x_{i_r})dF(x_{i_1})\cdots dF(x_{i_r})$ in probability. But I cannot find any reference for it.