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A singular perturbation approach for Liptser’s functional limit theorem and some extensions. (English) Zbl 0941.60089

Theory Probab. Math. Stat. 58, 83-87 (1999) and Teor. Jmovirn. Mat. Stat. 58, 76-80 (1998).
The authors consider a sequence of Markov processes \(x_{n}(t)\), \(t\geq 0\), \(n=1,2,\ldots\), defined on a finite state space \(E=\{ 1,2,\ldots,k\}\) with transition intensity matrices \(Q_{n}=nQ.\) Let \(g\) be a finite function defined on \(E\) (the rate function). Let \(\zeta_{n}(t)=\sqrt{n}\int_0^{t} [g(x_{n}(s))-Eg(x_{n}(s))]ds\) be a sequence of functionals. Let \(\pi\) be the invariant probability measure of \(x(t)\), where \(x(t)\) is a Markov process with the transition intensity matrix \(Q\), \(\sum_{i=1}^{k}\pi Q_{ij}=0\), \(j=1,\ldots,k\), \(\sum_{i=1}^{k}\pi_{i}=1.\) Let \((B_1,\ldots,B_{k-1})\) be the solution of the equations \[ \sum_{j=1}^{k-1}{\widetilde Q}_{ij}B_{j}=g(i)-g(k), \quad i=1,\ldots,k-1, \tag{1} \] \({\widetilde Q}_{ij}=Q_{ij}-Q_{kj}\), \(i,j=1,\ldots,k-1.\) R. Sh. Liptser [in: Statistics and control of stochastic processes. Proc. Steklov. Semin., Moscow 1984, Transl. Ser. Math. Eng., 305-316 (1985; Zbl 0598.60036)] proved the following assertion: If equation (1) has unique positive solution \((\pi_1,\ldots,\pi_{k})\), then the stochastic process \(\zeta_{n}(t)\) converges weakly in the Skorokhod topology of the space \(D[0,\infty)\) to a process \(bw\), where \(w\) is a standard Wiener process and \(b\) is a constant defined by \[ b^2=\sum_{j=1}^{k-1}B_{j}^2 \sum_{i=1}^{k}|Q_{ij}|\pi_{i}- \sum_{\substack{ i,j=1\\ i\not= j}} B_{i}B_{j}(Q_{ij}\pi_{i}+Q_{ij}\pi_{j}). \] The authors give a new short proof of this theorem. Moreover, they give a generalization of this theorem in the case of general state space and perturbed rate function.

MSC:

60J25 Continuous-time Markov processes on general state spaces
60F17 Functional limit theorems; invariance principles

Citations:

Zbl 0598.60036
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