Sufficient conditions for the convergence of local-time type functionals of Markov approximations. (Ukrainian, English) Zbl 1199.60287

Kartashov, Yu. M.

Teor. Jmovirn. Mat. Stat. 77, 36-51 (2007); translation in Theory Probab. Math. Stat. 77, 39-55 (2008).

Let \(X_n(t), n \geq1\), be a sequence of stochastic processes that approximates in the Markov sense a homogeneous Markov process \(X(t)\) (see, for example, A. M. Kulik [Theory Stoch. Process. 12, No. 28, Part 1–2, 87–93 (2006; Zbl 1142.60325)]). Consider functionals \(\phi_n\) of the form \(\phi_{n}^{s,t}(X_n)=\sum_{s<t_{k,n}<t}g_n(X_n(t_{k,n}))\), \(0\leq s<t\), \(t_{k,n}=k/n\), where \(g_n\) are nonnegative Borel functions. Consider the polygonal lines constructed from these functionals \(\psi_n^{s,t}= \phi_n^{t_{j-1,n},t_{k-1,n}}-(ns-j+1)\phi_n^{t_{j-1,n},t_{j,n}}+(nt-k+1) \phi_n^{t_{k-1,n},t_{k,n}}\), \(s\in[t_{j-1,n},t_{j,n})\), \(t\in [t_{k-1,n},t_{k,n}).\) The lines \(\psi_n\) are considered as random elements in the space \(C(\mathbb T,\mathbb R^+)\), where \(\mathbb T=\{(s,t)| 0\leq s< t\}\). The problem considered is the weak convergence of \(\psi_n\) in the space \(C(\mathbb T,\mathbb R^+)\) to the element \(\{\phi^{s,t}, (s, t)\in\mathbb T\}\), where \(\phi\) is a Wiener \(W\)-functional of the limit process \(X(t)\). The main result of the paper contains sufficient conditions for such a convergence. As an application of the main result the problem of the weak convergence of local-time type functionals of random walks on the Cantor set is considered. The functionals coincide, up to a normalizing factor, with the measure of the time spent by a random polygonal line constructed from a random walk in a neighborhood of the Cantor set. The random walk is assumed to approximate an \(\alpha\)-stable process with \(\alpha\leq1\).

Reviewer: Mikhail P. Moklyachuk (Kyïv)

MSC:

60J55	Local time and additive functionals
60J45	Probabilistic potential theory
60F17	Functional limit theorems; invariance principles

Keywords:

Markov chain; Markov process; stable process; Cantor set

Citations:

Zbl 1142.60325