Properties of a functional of trajectories which arises in studying the probabilities of large deviations of random walks

The deviation functional (or integral) describes the logarithmic asymptotics of the probabilities of large deviations of trajectories of the random walks generated by the sums of random variables (vectors) (see [1, 2] for instance). In this article we define it on a broader function space than previously and under weaker assumptions on the distributions of jumps of the random walk. The deviation integral turns out the Darboux integral $\int F(t,u)$ of a semiadditive interval function $F(t,u)$ of a particular form. We study the properties of the deviation integral and use the results elsewhere in [3] to prove some generalizations of the large deviation principle established previously under rather restrictive assumptions.