Invariance principles under the Maxwell–Woodroofe condition in Banach spaces

Abstract

We prove that, for (adapted) stationary processes, the so-called Maxwell-Woodroofe condition is sufficient for the law of the iterated logarithm and that it is optimal in some sense. That result actually holds in the context of Banach valued stationary processes, including the case of L p -valued random variables, with 1 ≤ p < ∞. In this setting we also prove the weak invariance principle, hence generalizing a result of Peligrad and Utev [45]. The proofs make use of a new maximal inequality and of approximation by martingales, for which some of our results are also new.MSC 2010 subject classification: 60F17, 60F25, 60B12; Secondary: 37A50 Key words and phrases. Banach valued processes, compact law of the iterated logarithm, invariance principles, Maxwell-Woodroofe's condition.I am thankful to Jérôme Dedecker for providing him with a copy of [58]. I wish to thank two anonymous referees for valuable remarks that yielded an improved presentation of the paper.

Limit theorems for the left random walk on $\operatorname{GL}_{d}(\mathbb{R})$

“…Condition (23) may be seen as an L p -analogue of the so-called Maxwell-Woodroofe condition [22]. As in the papers [24], [25] or [8] (see Section D.3), it can be shown that (23) is somewhat optimal for (25).…”

“…We first give a maximal inequality in the spirit of Proposition 2 of [23]. The present form is just Proposition 4.1 of [8].…”

Behavior of the Wasserstein distance between the empirical and the marginal distributions of stationary $\alpha$-dependent sequences

“…case, the condition (3.3) is necessary and sufficient for the stochastic boundedness of √ nW1(µn, µ). In the dependent context, other general criteria have been proposed by Dédé [7] and Cuny [6]. We shall discuss these conditions in Sections 4 and 5, and show that, in the α-dependent case, the condition (3.1) is weaker than the corresponding condition obtained by applying the criteria by Dédé or Cuny.…”

“…The central limit question for √ nW1(µn, µ) has been already investigated for dependent sequences in the papers by Dédé [7] and Cuny [6] (see Sections 4 and 5 for more details). This is not the case of the upper bounds for W1(µn, µ) p, even for sequences of independent and identically distributed (i.i.d.)…”

Hölderian weak invariance principle under the Maxwell and Woodroofe condition

“…Since the convergence of the finite dimensional distributions is contained in the main result of [Vol07], the only difficulty in proving Theorem 1.1 is to establish tightness. To this aim, we shall proceed as in the proof of Theorem 5.3 in [Cun14].…”