Asymptotic expansions in multidimensional Markov renewal theory and first passage times for Markov random walks (Q2759814)

Let \((X_n, S_n)\) be a Markov random walk with \(X_n= X(n)\) an irreducible aperiodic Markov chain with arbitrary state space and \(S_n= S(n)= S_0+ \xi_1+\cdots+ \xi_n\) a random walk on \(\mathbb{R}^d\) such that \(P(X_n\in A,\xi_n\in B\mid V)= P(x,A\times B)\) where \(V\) is the past including \(X(n-1)= x\). Classical renewal theory studies the measure \(U(B)= \sum_n P(S_n\in B)\) when the \(\xi_n\) are i.i.d. It shows that under moment conditions when \(\mu_1= E\xi_{11}> 0\) and \(s_1\to \infty\), then \(U(.+s)\) is asymptotically equal to a measure \(\Psi_k\) that is the product of \(1/\mu_1\) times Lebesgue measure on \((0,\infty)\) for the \(1\)-coordinate and a normal probability measure for the \((2,\dots, d)\)-coordinates with covariance matrix proportional to \(s_1\), the whole multiplied by a polynomial of degree \(k\) in \(s^{-1/2}\). We put \(y= (y_1,\dots, y_d)\) when \(y\in \mathbb{R}^d\).NEWLINENEWLINENEWLINEIt is shown that similar theorems hold for \(U(A,B)= \sum_n P(X_n\in A,S_n\in B)\), i.e. \(U(A,.+s)\) is asymptotically equal to \(\pi(A)\Phi_k\) where \(\Phi_k\) is of the same type as \(\Psi_k\) above, the main term not depending on \(A\) or the Markov chain, but the higher-order terms do. There is a local and a global version and an estimate for the error of the main term. When \(d=1\) this sharpens the classical estimate. Conditions for these theorems: uniform ergodicity for \(X_n\), a moment condition for \((X_1,\xi_1)\) and \(\xi_1\) should be strongly nonlattice under distribution \(\pi\) of \(X_1\).NEWLINENEWLINENEWLINELet \(T_b= \min\{n\geq 1: S_{n1}> b\}\). The joint distribution of \(X(T_b)\), \(S_1(T_b)- b\) and centered and normed \(S_i(T_b)\), \(i= 2,\dots, d\), is shown to converge. Under conditions for the ladder Markov walk similar to those for \((X_n, S_n)\) this is extended to an asymptotic expansion. Examples are given of classes of distributions where the conditions for \((X_n, S_n)\) imply those for the ladder walk. Proofs use the Fourier transform of the Markov transition operator and Schwartz's theory of distributions.