The Annals of Probability

Semigroups of Conditioned Shifts and Approximation of Markov Processes

Thomas G. Kurtz

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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.

Article information

Ann. Probab., Volume 3, Number 4 (1975), 618-642.

First available in Project Euclid: 19 April 2007

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Zentralblatt MATH identifier


Primary: 60F99: None of the above, but in this section
Secondary: 60J25: Continuous-time Markov processes on general state spaces 60J35: Transition functions, generators and resolvents [See also 47D03, 47D07] 60J60: Diffusion processes [See also 58J65] 60G45

Markov process approximation operator semigroup conditional expectation weak convergence


Kurtz, Thomas G. Semigroups of Conditioned Shifts and Approximation of Markov Processes. Ann. Probab. 3 (1975), no. 4, 618--642. doi:10.1214/aop/1176996305.

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