Strong Limit Theorems of Empirical Distributions for Large Segmental Exceedances of Partial Sums of Markov Variables

Let A₁,A₂,…,Aₙ be generated governed by an r-state irreducible Markov chain and suppose (Xᵢ,Uᵢ) are real valued independently distributed given the sequence A₁,A₂,…,Aₙ, where the joint distribution of (Xᵢ,Uᵢ) depends only on the values of Aᵢ₋₁ and Aᵢ and is of bounded support. Where A₀ is started with its stationary distribution, E X₁ < 0 and the existence of a finite cycle C = ₀ = i₀,…,Aₖ = iₖ = i₀\ such that \∑ᵐᵢ₌₁Xᵢ > 0, m = 1,…,k; C\ > 0 is assumed. For the partial sum realizations where ∑ˡᵢ₌ₖXᵢ → ∞, strong laws are derived for the sums ∑ˡᵢ₌ₖUᵢ. Examples with r = 2, X ∈ \-1, 1\ and the cases of Brownian motion and Poisson process with negative drift are worked out.