I am currently writing my master thesis and have a problem with Lemma 2.5 in http://www.math.utah.edu/~rice/berkeshorvathrice2012.pdf . I have to show that $$\frac{1}{\sqrt{N}}\sum_{i=1}^{\lfloor Nx \rfloor}<X_{i,m},\Phi_{\ell,m}>_{L^2[0,1]} \longrightarrow \sqrt{\lambda_{\ell,m}}W_{\ell}(x) $$ holds for $N \to \infty$ and any index $\ell$, where the convergence is in the Skorokhod space $D[0,1]$. $W_{\ell}$ is a Wiener process, $(\Phi_{\ell,m})_\ell$ is a ONS in $L^2[0,1]$, $x \in [0,1]$ and $X_{i,m}$ is a $L^2[0,1]$ valued random variable. We already know that $(<X_{i,m},\Phi_{\ell,m}>)_{i}$ is m-dependent and $$<c_m(t,\cdot),\Phi_{\ell,m}>=\lambda_{\ell,m}\Phi_{\ell,m}(t)$$ with $$c_m(t,s)=\mathbb{E}\left[X_{0,m}(t)X_{0,m}(s)\right] + \sum\limits_{i=1}^m \mathbb{E}\left[X_{0,m}(t)X_{i,m}(s)\right]+ \sum\limits_{i=1}^m \mathbb{E}\left[X_{i,m}(t)X_{0,m}(s)\right]$$ holds. My approach is the following theorem from DasGupta:
Let $(X_i)_i$ be a stationary m-dependent sequence. Let $E(X_i ) = \mu$ and $Var(X_i ) = \sigma^2 < \infty$. Then $\sqrt{n}(\overline{X} − \mu) \longrightarrow N(0, \tau^2)$, where $τ^2 = σ^2 + 2\sum_{i=2}^{m+1} Cov(X_1, X_i )$.
So I can get the $\lambda$'s but I don't see where the Wiener process comes from.