Weak convergence of the integrated number of level crossings to the local time for Wiener processes

Let $\{X_{t},\ t\in[0,1]\}$ be a standard Wiener process defined on $(\Omega,A,\mathbb P)$. We define the regularized process $X^{\varepsilon}_{t}= \varphi_{\varepsilon}*X_{t}$, with $\varphi_{\varepsilon}(t)=\varepsilon^{-1}\varphi(t/\varepsilon)$, a kernel that approaches Dirac's delta function as $\varepsilon \rightarrow 0$. We study the convergence of
$$ Z_{\varepsilon}(f) = \varepsilon^{-1/2} \int_{-\infty}^{+\infty} \biggl [ \frac{N^{X^{\varepsilon}}(x)}{c(\varepsilon)} - L_{X}(x)\biggr]f(x)\, dx, $$
when $\varepsilon$ goes to zero, with $N^{X^{\varepsilon}}(x)$ the number of crossings for $X^{\varepsilon}$ at level $x$ in $[0,1]$ and $L_{X}(x)$ the local time of $X$ in $x$ on $[0,1]$. As a by-product of our method we also obtain a weak convergence result for the increments of the process $X$.