Fluctuation Theory for Markov Random Walks

Two fundamental theorems by Spitzer–Erickson and Kesten–Maller on the fluctuation-type (positive divergence, negative divergence or oscillation) of a real-valued random walk $$(S_{n})_{n\ge 0}$$ ( S n ) n ≥ 0 with iid increments $$X_{1},X_{2},\ldots $$ X 1 , X 2 , … and the existence of moments of various related quantities like the first passage into $$(x,\infty )$$ ( x , ∞ ) and the last exit time from $$(-\infty ,x]$$ ( - ∞ , x ] for arbitrary $$x\ge 0$$ x ≥ 0 are studied in the Markov-modulated situation when the $$X_{n}$$ X n are governed by a positive recurrent Markov chain $$M=(M_{n})_{n\ge 0}$$ M = ( M n ) n ≥ 0 on a countable state space $$\mathcal {S}$$ S ; thus, for a Markov random walk $$(M_{n},S_{n})_{n\ge 0}$$ ( M n , S n ) n ≥ 0 . Our approach is based on the natural strategy to draw on the results in the iid case for the embedded ordinary random walks $$(S_{\tau _{n}(i)})_{n\ge 0}$$ ( S τ n ( i ) ) n ≥ 0 , where $$\tau _{1}(i),\tau _{2}(i),\ldots $$ τ 1 ( i ) , τ 2 ( i ) , … denote the successive return times of M to state i, and an analysis of the excursions of the walk between these epochs. However, due to these excursions, generalizations of the aforementioned theorems are surprisingly more complicated and require the introduction of various excursion measures so as to characterize the existence of moments of different quantities.