Proving Convergence of a limit related to Cross-Spectrum Estimation
A first thing to try is to reduce the unwanted variables. What happens if you set  $\ell=s=0$?

$$ U_M(r,0,0)= \left\{\begin{matrix}
0 &amp;  r&lt;-M\\
1-\frac{r}{M}&amp; -M\leq r \leq 0 \\ 
1-\frac{0}{M}&amp; 0 &lt; r \leq 0\\ 
1-\frac{r}{M}&amp; 0&lt;r \leq M\\
0 &amp;  r&gt;M 
\end{matrix}\right.$$

Luckily, this definition seems to fit with $V_M$, since the third inequality never happens. So your $V_M(r)$ is just a special case. Nothing more to do.