Renewal Theory for Markov Chains on the Real Line

Standard renewal theory is concerned with expectations related to sums of positive i.i.d. variables, Sₙ = ∑ⁿᵢ₌₁ Zᵢ. We generalize this theory to the case where ᵢ\ is a Markov chain on the real line with stationary transition probabilities satisfying a drift condition. The expectations we are concerned with satisfy generalized renewal equations, and in our main theorems, we show that these expectations are the unique solutions of the equations they satisfy.