ABSTRACT
In Chapter 13 entitled Non-Homogeneous Markov Set System (NHMSS) a novel idea is introduced for a NHMS, that is, the values of the basic parameters are viewed as being contained in intervals. After the definition of the tight interval and the Lemma that a tight interval is a convex polytope then the Chapter proceeds in the formal definition of a NHMSS. In section 13.3 after a few useful Lemmas we prove that under certain conditions the set of the expected relative population structures of NHMSS is convex. We then prove that at each time step this set is getting smaller and as https://www.w3.org/1998/Math/MathML"> t → ∞ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315269986/8c02c1de-dcf4-47d7-ae03-0d40a221dc91/content/mathx1_5.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> it converges to an non empty convex set. In section 13.4 we study the asymptotic behavior of NHMSS. It is proved that a NHMSS starting with two different sets of initial structures and allocation probabilities under certain conditions easily met in practice asymptotically converge to the same limiting set geometrically fast. Then it is proved that for a NHMSS if the sets of initial structures and allocation probabilities are compact and convex then under certain easily met conditions the limiting set of the expected relative population structures exist and the convergence is geometrically fast. In section 13.5 we study important properties of the limiting set of the expected relative population structures. The Chapter is concluded with the presentation of an illustrative potential application where all the possible difficulties in applying the present results are being resolved in two novel Lemmas and an Algorithm which lead step by step to the desired results.
