Large Deviations for Processes on Half-Line: Random Walk and Compound Poisson

We establish, under the Cramer exponential moment condition in a neighbourhood of zero, the Extended Large Deviation Principle for the Random Walk and the Compound Poisson processes in the metric space $\V$ of functions of finite variation on $[0,\infty)$ with the modified Borovkov metric \r(f,g)= \r_\B(\hat{f},\hat{g}) , where $\hat f(t)= f(t)/(1+t)$, $t\in \R$, and \r_\B is the Borovkov metric. LDP in this space is "more precise" than that with the usual metric of uniform convergence on compacts.