We investigate the invariance principle in H{\"o}lder spaces for strictly
stationary martingale difference sequences. In particular, we show that the
sufficient condition on the tail in the i.i.d. case does not extend to
stationary ergodic martingale differences. We provide a sufficient condition on
the conditional variance which guarantee the invariance principle in H{\"o}lder
spaces. We then deduce a condition in the spirit of Hannan one.Comment: in Stochastic Processes and their Applications, Elsevier, 2016, 12
We obtain a necessary and sufficient condition for the orthomartingale-coboundary decomposition. We establish a sufficient condition for the approximation of the partial sums of a strictly stationary random fields by those of stationary orthomartingale differences. This condition can be checked under multidimensional analogues of the Hannan condition and the Maxwell-Woodroofe condition.Lemma 3.6. For each integer k 1 and each integrable and measurable function f : Ω → R, the function B k (f ) can be written in the following way:
2.4)where m k • T i i∈Z d is an orthomartingale differences random field with respect to the filtration T −i F0 i∈Z d . and for each nonempty subset J of [d] such that J = [d], the random field m k,J • T i J i∈Z d is an orthomartingale differences random field with respect to the filtration T −i J F∞1 J c i∈Z d .Considering the notations of Lemma 3.6, we introduce the following notation:(3.2.5) Therefore, for any k 1, the following equality holds
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.
We provide a new projective condition for a stationary real random field
indexed by the lattice $\Z^d$ to be well approximated by an orthomartingale in
the sense of Cairoli (1969). Ourmain result can be viewed as a multidimensional
version of the martingale-coboundary decomposition method which the idea goes
back to Gordin (1969). It is a powerfull tool for proving limit theorems or
large deviations inequalities for stationary random fields when the
corresponding result is valid for orthomartingales
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