Large deviations for processes on half-line

We consider a sequence of processes X-n(t) defined on the half-line 0 <= t < infinity, n = 1, 2, .... We give sufficient conditions for Large Deviation Principle (LDP) to hold in the space of continuous functions with metric rho(kappa)(f, g) = sup(t >= 0) vertical bar f(t) - g(t)vertical bar/1 + t(1+kappa), kappa >= 0. LDP is established for Random Walks and Diffusions defined on the half-line. LDP in this space is "more precise" than that with the usual metric of uniform convergence on compacts.