## The Annals of Probability

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

### Semigroups of Conditioned Shifts and Approximation of Markov Processes

#### Abstract

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

**Source**

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

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176996305

**Digital Object Identifier**

doi:10.1214/aop/1176996305

**Mathematical Reviews number (MathSciNet)**

MR383544

**Zentralblatt MATH identifier**

0318.60026

**JSTOR**

links.jstor.org

**Subjects**

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

**Keywords**

Markov process approximation operator semigroup conditional expectation weak convergence

#### Citation

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. https://projecteuclid.org/euclid.aop/1176996305