In this paper, we study the discretization of ergodic Functional Central Limit Theorem (CLT) established by Bhattacharya (see in Z Wahrscheinlichkeitstheorie Verwandte Geb 60:185–201, 1982) which states following: Given a stationary and Markov process $$(X_t)_{t \geqslant 0}$$ with unique invariant measure $$\nu $$ infinitesimal generator A, then, for every smooth enough function f, $$(n^{1/2} ...

In this paper we prove the central limit theorem for the following multisequence N1 ∑ n1=1 ... Nd ∑ nd=1 f(A1 1 ...A nd d x) where f is a Hölder’s continue function, A1, ..., Ad are s× s partially hyperbolic commuting integer matrices, and x is a uniformly distributed random variable in [0, 1]. Then we prove the functional central limit theorem, and the almost sure central limit theorem. The ma...

Econometricians typically make use of functional central limit theorems to prove results for I(1) processes. For example, to establish the limit distributions of unit root tests such as the Phillips–Perron and Dickey–Fuller tests, the functional central limit theorem plays a crucial role. In this paper, it is pointed out that for linear processes, minimal conditions that ensure that only a cent...