The Conditional Lindeberg–Trotter Operator in the Resolution of Limit Theorems with Rates for Dependent Random Variables. Applications to Markovian Processes

Abstract

Making use of the properties of conditional expectations, a conditional Lindeberg–Trotter-operator is defined which extends the properties of the classical Lindeberg–Trotter operator to the case of dependent random variables. This approach enables one to establish general limit theorems equipped with little-$o$ and large-$\mathcal{O}$ rates for sums of dependent random variables; these are applied to several versions of the central limit theorem, the weak law of large numbers, and especially to Markovian processes, not only in the case of weak convergence in distribution but partially also for strong convergence.