Large deviation principles for random walk trajectories. III

The present paper is a continuation of [A. A. Borovkov and A. A. Mogulskii, Theory Probab. Appl. 57, No. 1, 1–27 (2013); translation from Teor. Veroyatn. Primen. 57, No. 1, 3–34 (2012; Zbl 1279.60037)]. It consists of two sections. Section 6 presents results similar to those obtained in Sections 4 and 5, but now in the space of functions of bounded variation with metric stronger than that of $\mathbb{D}$. In Section 7 we obtain the so-called conditional large deviation principles for the trajectories of univariate random walks with a localized terminal value of the walk. As a consequence, we prove a version of Sanov’s theorem on large deviations of empirical distributions.