On Multivalued Amarts

Abstract

Summary. In recent years, convergence results for multivalued functions have been developed and used in several areas of applied mathematics: mathematical economics, optimal control, mechanics, etc. The aim of this note is to give a criterion of almost sure convergence for multivalued asymptotic martingales (amarts). For every separable Banach space B the fact that every L 1 -bounded B-valued martingale converges a.s. in norm to an integrable B-valued random variable (r.v.) is equivalent to the Radon-Nikodym property [6]. In this paper we solve the problem of a.s. convergence of multivalued amarts by giving a topological characterization. Preliminaries.Let X S be the set of all random elements (r.e.) defined on a probability space (Ω, A, P ) with values in a Polish (separable, complete metric) space (S, ), i.e. X S = {X : Ω → S; X −1 (B) ⊂ A}, where B = B S stands for the σ-field generated by the open subsets of S.Let P(S) denote the set of all probability measures defined on (S, B). The Lévy-Prokhorov metric on P(S) is defined as follows:where P X is the probability distribution of the r.e. X, A ε = {x : d(x, A) = inf y∈A (x, y) < ε}, andIt is known [1] that the convergence of a sequence of probability measures in the Lévy-Prokhorov metric and the weak convergence of this sequence coincide.

Some observations on synthesis and electrolytic properties of nonstoichiometric calcium zirconate

“…A green compact (CZ-51C) obtained from powder synthesized by the citrate method exhibited the most intensive changes of dimensions within a temperature range 200-1250 • C. As can be seen, this pellet reached about 13% shrinkage at 1240 • C. Whilst the other sample (CZ-51S) pellets obtained via solid state reaction also start to change their diameters at 200 • C, at 1200 • C the shrinkage reached a value of 6%. This could be attributed to hard agglomeration or non-homogeneity of CaZrO 3 particles mentioned by TEM-EDX observations [28]. Stoichiometric CaZrO 3 (CZ-50) and CaZrO 3 -doped excess CaO (CZ-51) samples pressureless sintered at 1500 • C for 2 h exhibited a relative density equal to 97-98% of theoretical density.…”

“…11. [28]. It contains a sub-circuit representing the intragrain properties connected in series to the sub-circuit representing the grain-boundary properties.…”