Contributions to Doeblin's theory of Markov processes

Exploring the qualitative behavior of uncertain dynamical systems

“…Moreover, let q | H denote the transition probability q restricted to (H, H∩Z). Then, there is a nontrivial σ -finite measure μ for which μ(·) = H q(z, ·) dμ(z) holds and, up to constant multiples, μ is the unique measure with this property [29]. If μ is even a probability measure, then H is positive Harris and μ is ergodic with μ(H) = 1.…”

Bootstrapping Robust Statistics for Markovian Data Applications to Regenerative R‐Statistics and L‐Statistics

“…A set is said to be small for X if there exist , δ > 0 and a probability measure Φ supported by S such that, for all , where Π m denotes the m ‐th iterate of the transition kernel Π. Condition shall be referred to as the minorization condition As shown in Jain and Jamison (), many accessible small sets exist for ψ‐irreducible chains: any set such that ψ( B ) > 0 contains such a set. Rather than replacing the original chain X by the multi‐variate chain , we also assume that m = 1.…”

Sample and moment lyapunov exponents of discrete linear dynamical systems

“…It remains to show that - Suppose the process (h; n 2 0 ) is ~5-recurrent on W (strongly recurrent) but not necessarily stationary. Then, by Theorem 5.3 in [18], the pair process ((s,, &); n 2 0 ) is wecurrent on pd-' x W. If we assume that the invariant set C x Q is not decomposable into a cycle (see, e.g., [19]), then the process ((s,, L), n 2 0 ) w i l l converge weakly to its stationary solution ((S,, k); n 2 0 ) in the sense that, for all B E B ([pd-' x v), the Bore1 sets on the path space of the pair process, and with v denoting any initial distribution on pd-' X W, where P, and Pn denote the probabilities obtained with initial distributions v and n, respectively (see Proposition 7.12 in [lo]). Hence, by Theorem 2.1, the process ((x,, &, ) ; n 2 0 ) = ((sn l x,, I, tn); n 2 0 ) will converge weakly to ( (sneL, Cn); n 2 0) 1.…”