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 $\text{V}$ of functions of finite variation on $[0,\mathrm{\infty })$ 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.