Could anyone please give ideas or point me out references where I can find any result concerning the locally nondeterministic (LND) property (in the sense of Berman: http://www.ams.org/journals/bull/1973-79-02/S0002-9904-1973-13225-2/S0002-9904-1973-13225-2.pdf) of Brownian bridge? There are several criteria of being LND given by Berman, but none of them fit to Brownian bridge (to the best of my knowledge). Thanks.
Here is a definition of LND: Let $X(t)$, $t\ge 0$ be a real-valued Gaussian process with mean $0$ dan let $I\subset [0,\infty)$ be an open interval. Assume that $E(X(t)^2)>0$ for all $t\in I$ and there exists $\delta >0$ such that $E((X(s)-X(t))^2)>0$ for $s,t\in I$ with $0<|s-t|<\delta$. $X$ is called LND on $I$ if for every integer $n\ge 2$, $$\lim_{\varepsilon \to 0}\inf_{t_n-t_1\le \varepsilon}V_n>0,$$ where $V_n$ is the relative prediction error: $$V_n=\frac{\mathrm{Var}(X(t_n)-X(t_{n-1})|X(t_1),\ldots,X(t_{n-1}))}{\mathrm{Var}(X(t_n)-X(t_{n-1}))}$$ and the infimum is taken over all ordered points $t_1<t_2<\ldots <t_n$ in $I$ with $t_n-t_1\le \varepsilon$.
The standard Brownian bridge is the process $X(t)$, $t\in [0,T]$ for some $0<T<\infty$ with $$X(t)=B(t)-\frac{t}{T}B(T),$$ where $B(t)$ is the standard Brownian motion starting in $0$.
