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

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 $$\alpha\leq1$$.

### 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

Zbl 1142.60325
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