Large deviations for Markov processes with discontinuous statistics. II: Random walks (Q1187106)

[For part I, by the authors and \textit{A. Weiss}, see Ann. Probab. 19, No. 3, 1280-1297 (1991; Zbl 0735.60027).] Let \(\mu^ 1\) and \(\mu^ 2\) be Borel probability measures on \(\mathbb{R}^ d\) with finite moment generating functions. The main theorem proves the large deviation principle for a random walk whose transition mechanism is governed by \(\mu^ 1\) when the walk is in the left halfspace \(\Lambda^ 1= \left\{ x\in \mathbb{R}^ d :\;x_ 1 \leq 0 \right\} \), and whose transition mechanism is governed by \(\mu^ 2\) when the walk is in the right halfspace \(\Lambda^ 2= \left\{ x\in \mathbb{R}^ d :\;x_ 1 > 0 \right\}. \) When the measures \(\mu^ 1\) and \(\mu^ 2\) are equal, the main theorem reduces to Cramér's theorem.