Holderian Weak Invariance Principle for Stationary Mixing Sequences

Abstract

Abstract. We provide some sufficient mixing conditions on a strictly stationary sequence in order to guarantee the weak invariance principle in Hölder spaces. Strong mixing, ρ-mixing conditions are investigated as well as τ -dependent sequences. The main tools are deviation inequalities for mixing sequences.

Weak invariance principle in some Besov spaces for stationary martingale differences

“…Since B o ∞,α matches Hölder spaces we see, that Theorems 1.2 and 1.1 complement the weak invariance principle obtained in a Hölderian framework by [12] and [4]. Concerning condition (ii) of Theorem 1.1 we prove a need of certain extra assumption by a counterexample which for any dynamical system with positive entropy constructs a function f that satisfies the condition (i) but the convergence of polygonal line processes fails.…”

“…Remark 3.4. Using deviation inequalities, similar results as those found for the Hölderian weak invariance principle for stationary mixing and τ -dependent sequences in [4] can be found for Besov spaces.…”

Hölderian weak invariance principle under the Maxwell and Woodroofe condition

“…We may observe that condition (1.4) implies by Theorem 1 of [PUW07] that the sequence (S n (f )/ √ n) n 1 is bounded in L p ; nevertheless the counter-example given in Theorem 2.6 of [Gir16a] shows that we cannot deduce the weak invariance principle from this.…”

Testing mean changes by maximal ratio statistics

“…Nearly non stationary AR(1) case is studied in (Markevičiūtė et al 2012). HFCLT for strictly stationary martingale difference sequences, stationary strong mixing, -mixing, -dependent sequences are investigated by Giraudo (2016Giraudo ( , 2017Giraudo ( , 2018. A general method to prove an HFCLT is provided by Th.…”