Exit distributions for symmetric Markov processes via Gaussian techniques

Autoregressive processes and first-hit probabilities for randomized random walks

“…Remark. For comparison, we compute (22) using combinatorial methods of Feller (1968). Consider only I = 2.…”

“…(we used formulas (11.8.6) and (11.12.5) of Feller (1968)). This quantity also represents the probability of interest for a path with N = 2n + 1 steps, since a path which is below level 2 for 2n steps remains there for at least one additional step.…”

“…In Section 3 we discuss the AR(k) sequences which arise from association with randomized random walks. In Section 4 we derive explicit formulas for first-hit probabilities of random walks and compare our results with those of Feller (1968Feller ( ), (1971.…”

“…The classic work of Feller (1968Feller ( ), (1971 devotes substantial space to the topic of random walks (cf. Chapters III and XII of Feller (1968) and Chapters II, XII, XIV, XVII of Feller (1971». Problems connected with first passages, path maxima, hitting points, etc., are treated using combinatorial methods and Laplace and Fourier transforms. The study of randomized random walks, however, is more restricted and the treatment (Feller (1971), II.7 and XIV.6) is less developed.…”