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This chapter considers the Moore–Penrose inversion of full matrices with quasi-separable specifications, that is, matrices that decompose into the sum of a block-lower triangular and a block-upper triangular matrix, whereby each has a state-space realization given. We show that the Moore–Penrose inverse of such a system has, again, a quasi-separable specification of the same order of complexity as the original and show how this representation can be recursively computed with three intertwined recursions. The procedure is illustrated on a 4 ? 4 (block) example.
We construct a simple example, surely known to Harry Kesten, of an R-transient Markov chain on a countable state space S ∪ {δ}, where δ is absorbing. The transition matrix K on S is irreducible and strictly substochastic. We determine the Yaglom limit, that is, the limiting conditional behavior given nonabsorption. Each starting state x ∈ S results in a different Yaglom limit. Each Yaglom limit is an R-1-invariant quasi-stationary distribution, where R is the convergence parameter of K. Yaglom limits that depend on the starting state are related to a nontrivial R-1-Martin boundary.
Generalizing the standard frailty models of survival analysis, we propose to model frailty as a weighted Lévy process. Hence, the frailty of an individual is not a fixed quantity, but develops over time. Formulae for the population hazard and survival functions are derived. The power variance function Lévy process is a prominent example. In many cases, notably for compound Poisson processes, quasi-stationary distributions of survivors may arise. Quasi-stationarity implies limiting population hazard rates that are constant, in spite of the continual increase of the individual hazards. A brief discussion is given of the biological relevance of this finding.
Let (Xt) be a one-dimensional Ornstein-Uhlenbeck process with initial density function f : ℝ+ → ℝ+, which is a regularly varying function with exponent -(1 + η), η ∊ (0,1). We prove the existence of a probability measure ν with a Lebesgue density, depending on η, such that for every A ∊ B(R+):
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