Let $D$ be the set of all Dyck paths on square grid of size $n\times n$. For any particular Dyck path, let $S(t)=X_1+X_2+\ldots +X_t$ store the path, where $X_i=\pm 1$. Being a Dyck path, we have $S(0)=0$ and $S(2n)=0$.

Consider picking uniformly random Dyck paths from the set $D$. Let $P_t(x)$ denote the probability distribution of $X_t$ subject to the uniform distribution on Dyck paths. Suppose that $n$ is very large and that I pick a uniformly random Dyck path conditioned on having the first $k$ steps being fixed, where $k=o(n)$. I would like to conclude that $\mathbb{P}(X_t=x | \mbox{first $k$ steps})\approx P_t(x)$ for $t>>k$. Is there a straightforward way of doing this? In other words, far away steps become asymptotically independent from the first $k$ steps. In general, is there a way of extending this to uniformly random paths of fixed length on regular graphs?

I can see that this is probably related to asking a similar question for brownian bridges. Specifically, I am very interested in seeing how one would obtain explicit bounds for the quantity $|\mathbb{P}(X_t=x | \mbox{first $k$ steps})-P_t(x)|$ in terms of $n$ and $t$