below.

Following the discussion at the beginning of Section \[defint\] it is natural to expect that $T_{N,M}(x)$ converges as $M,N\to \infty$ to $\xi (x)$ for every $x\in \mathbb{R}$. We will actually show that the sequence (\[TNM\]) approximates $\xi (x)$ in the norm of the Watanabe space $\mathbb{D} ^{-\alpha ,2} $ with $\alpha >\frac{1}{2}$.

\[tt2\] For every $x\in \mathbb{R}$, the sequence $T_{N,M} (x)$ given by (\[TNM\]) converges to $\xi (x)$ as $M, N\to \infty$ in the norm $\Vert \cdot \Vert _{-\alpha ,2}$ for any $\alpha >\frac{1}{2}$.

[**Proof:** ]{} Fix again $x\in \mathbb{R}$. The first step is to express $T_{N,M}$ using multiple Wiener-Itô integrals. We have, by the integration by parts formula (\[ip\]) and the symmetrization formula (\[sim2\]) $$\begin{aligned}
T_{N,M} (x)
&=& \delta ^{(2)} \left(\sum_{i=0}^{N-1} \sum_{j=0}^{M-1} \Delta (x-W_{s_{i},t_{j}})1_{[0,s_{i}]\times [t_{j}, t_{j+1}]} (u_{1},v_{1}) 1_{[s_{i}, s_{i+1}]\times [0, t_{j}]} (u_{2}, v_{2})\right)\nonumber \\
&=& \sum_{i=0}^{N-1}\sum_{j=0}^{M-1}  a_{m}^{x}(s_{i},t_{j}) \nonumber\\
&&\times \sum_{m\geq 0} I_{m+2} \left( 1_{[0,s_{i} ]\times [0,t_{j}]}^{\otimes m} ((x_{1},y_{1}),..., (x_{m},y_{m}))  1_{[0,s_{i}]\times [t_{j}, t_{j+1}]} (u_{1},v_{1}) 1_{[s_{i}, s_{i+1}]\times [0, t_{j}]} (u_{2}, v_{2}) \right)\nonumber\\
&=&\sum_{i=0}^{N-1}\sum_{j=0}^{M-1}  a_{m}^{x}(s_{i},t_{j})\sum_{m\geq 0} I_{m+2}\left( \frac{1}{(m+1)(m+2)} \right. \nonumber\\
&& \left. \sum_{k,l=1; k\not= l}^{m+2} 1_{[0, s_{i}]\times [0, t_{j}]} ^{\otimes m}((x_{1},y_{1}),.., \overline{(x_{k},y_{k})}, \overline{(x_{l},y_{l})},.., (x_{m+2},y_{m+2}))\right. \nonumber\\
&&\left. 1_{[0,s_{i}]\times [t_{j}, t_{j+1}]} (x_{k}, y_{k}))1_{[s_{i}, s_{i+1}]\times [0, t_{j}]} (x_{l}, y_{l})\right).\label{tnmsim}\end{aligned}$$ Taking into account relation (\[tnmsim\]) 