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August, 1975 Semigroups of Conditioned Shifts and Approximation of Markov Processes
Thomas G. Kurtz
Ann. Probab. 3(4): 618-642 (August, 1975). DOI: 10.1214/aop/1176996305


Let $\mathscr{L}$ be the space of processes, progressively measurable with respect to an increasing family of $\sigma$-algebras $\{\mathscr{F}_t\}$ and having finite mean. Then $\mathscr{J}(s)f(t) = E(f(t + s) \mid \mathscr{F}_t), f \in \mathscr{L}$, defines a semigroup of linear operators on $\mathscr{L}$. Using $\mathscr{J}(s)$ and known semigroup approximation theorems, techniques are developed for proving convergence in distribution of a sequence of (possibly non-Markov) processes to a Markov process. Results are also given which are useful in proving weak convergence. In particular for a sequence of Markov process $\{X_n(t)\}$ it is shown that if the usual semigroups $(T_n(t)f(x) = E(f(X_n(t)) \mid X(0) = x))$ converge uniformly in $x$ for $f$ continuous with compact support, then the processes converge weakly.


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Thomas G. Kurtz. "Semigroups of Conditioned Shifts and Approximation of Markov Processes." Ann. Probab. 3 (4) 618 - 642, August, 1975.


Published: August, 1975
First available in Project Euclid: 19 April 2007

zbMATH: 0318.60026
MathSciNet: MR383544
Digital Object Identifier: 10.1214/aop/1176996305

Primary: 60F99
Secondary: 60G45 , 60J25 , 60J35 , 60J60

Keywords: approximation , conditional expectation , Markov process , operator semigroup , weak convergence

Rights: Copyright © 1975 Institute of Mathematical Statistics


Vol.3 • No. 4 • August, 1975
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