The conditional Lindeberg-Trotter operator in the resolution of the limit theorems with rates for dependent random variables. Applications to Markovian processes (Q1114987)

This paper is concerned with weak convergence of sums of dependent random variables (r.vs.). A well-known approach to establish this type of convergence is the Trotter operator method [cf. \textit{H. F. Trotter}, Proc. Am. Math. Soc. 10, 545-551 (1959; Zbl 0099.104)]. The authors generalize this method to the situation that the r.vs. in question are dependent. For this purpose, they introduce a conditional Trotter operator \(V_{x| {\mathcal G}}f(y):\quad C_ B\to C_ B\times Z(\Omega,{\mathcal A})\) \((C_ B\) being the space of uniformly continuous and bounded functions; \(Z(\Omega,{\mathcal A})\) being the set of measurable r.vs. on \((\Omega,{\mathcal A}))\) by \[ (V_{X| {\mathcal G}}f)(y):=\inf_{x\in A_ a(y,f)}E[f(X+x)| {\mathcal G}], \] \({\mathcal G}\subset {\mathcal A}\). Here, \(A_ a(y,f):=\{x\in {\mathbb{Q}}\), \(f(x)>f(y)\), \(y\in B_{ax}\}\), and \(B_{ax}:=\{y\in {\mathbb{R}}\); \(| x-y| <a\), \(x\in {\mathbb{Q}}\}\), \(a\in {\mathbb{Q}}\). The fact that \({\mathbb{R}}\) with the natural topology (having the basis \(\bar B:=\{B_{ax}\); \(x,a\in {\mathbb{Q}}\), \(a>0\})\) is a Polish space, ensures that all properties of this new operator hold almost surely for all \(y\in {\mathbb{R}}.\) In particular, all properties of the classical Trotter operator may be transferred to the conditional one and, if X is independent of \({\mathcal G}\), then \(V_{X| {\mathcal G}}f(y)=V_ Xf(y)\) (the classical Trotter operator). The authors then establish several general convergence assertions with O- and o-rates for dependent r.vs. and, in particular, general convergence results for Markov-processes. In the meantime, the second author [Result. Math. 15, 294-323 (1989)] has simplified the definition of \(V_{X| {\mathcal G}}f\).