Almost sure invariance principles via martingale approximation

Merlevède, Florence; Peligrad, Costel; Peligrad, Magda

doi:10.1016/j.spa.2011.09.004

Cited by 8 publications

(11 citation statements)

References 28 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…(5.14) This condition implies in particular that n≥1 h+K h+···+K n−1 h 2,ν n 3/2 < ∞. By Lemma 2 of [19] (and its proof), it follows that ϕ(Y 0 , Y 1 ) := lim n 1 n n j=1…”

Section: Proof Of Theorem For P =mentioning

confidence: 96%

See 1 more Smart Citation

Strong Invariance Principles with Rate for “Reverse” Martingale Differences and Applications

Cuny

Merlevède

2013

J Theor Probab

Self Cite

View full text Add to dashboard Cite

In this paper, we obtain almost sure invariance principles with rate of order n 1/p log β n, 2 < p ≤ 4, for sums associated to a sequence of reverse martingale differences. Then, we apply those results to obtain similar conclusions in the context of some non-invertible dynamical systems. For instance we treat several classes of uniformly expanding maps of the interval (for possibly unbounded functions). A general result for φ-dependent sequences is obtained in the course. (2010): 37E05, 37C30, 60F15. Mathematics Subject Classification

show abstract

“…(5.14) This condition implies in particular that n≥1 h+K h+···+K n−1 h 2,ν n 3/2 < ∞. By Lemma 2 of [19] (and its proof), it follows that ϕ(Y 0 , Y 1 ) := lim n 1 n n j=1…”

Section: Proof Of Theorem For P =mentioning

confidence: 96%

“…This condition follows directly from Theorem 12 of [19] together with the fact that (d k ) k∈N is a martingale differences sequence provided that…”

Section: Proof Of Theorem For P =mentioning

confidence: 97%

Strong Invariance Principles with Rate for “Reverse” Martingale Differences and Applications

Cuny

Merlevède

2013

J Theor Probab

Self Cite

View full text Add to dashboard Cite

show abstract

“…As already mentionned, ν admits a moment of order S − τ = p + (p − 1)τ . Hence, one can prove that condition (18) holds with r = p + τ (p − 1), by using the last statement of the following lemma (taking γ = (pr…”

Section: Lipschitz Autoregressive Modelsmentioning

confidence: 99%

On the Komlós, Major and Tusnády strong approximation for some classes of random iterates

Cuny¹,

Dedecker

Merlevède³

2018

Stochastic Processes and their Applications

Self Cite

View full text Add to dashboard Cite

The famous results of Komlós, Major and Tusnády (see [15] and [17]) state that it is possible to approximate almost surely the partial sums of size n of i.i.d. centered random variables in L p (p > 2) by a Wiener process with an error term of order o(n 1/p ). Very recently, Berkes, Liu and Wu [3] extended this famous result to partial sums associated with functions of an i.i.d. sequence, provided a condition on a functional dependence measure in L p is satisfied. In this paper, we adapt the method of Berkes, Liu and Wu to partial sums of functions of random iterates. Taking advantage of the Markovian setting, we shall give new dependent conditions, expressed in terms of a natural coupling (in L ∞ or in L 1 ), under which the strong approximation result holds with rate o(n 1/p ). As we shall see our conditions are well adapted to a large variety of models, including left random walks on GL d (R), contracting iterated random functions, autoregressive Lipschitz processes, and some ergodic Markov chains. We also provide some examples showing that our L 1 -coupling condition is in some sense optimal.

show abstract

“…While (4) and (2) coincide when p = 2, our method of proof is different from the one used in [24]. In Theorem 2.3, we shall consider also the case when p ∈ ]1, 2[.…”

Section: Martingale Approximations In Lpmentioning

confidence: 99%

“…As we mentioned in the Introduction, having estimates of the approximation error of partial sums by a martingale can be useful to derive different kinds of limit theorems for the partial sums associated with a stationary process. For instance, starting from (2), Merlevède et al [24] have obtained sufficient projective conditions in order for the partial sums to satisfy either the law of the iterated logarithm or the almost sure central limit theorem. In this section, we shall use our estimate (7), either to give new projective conditions under which the partial sums associated with a stationary process satisfy a moderate deviations type results, or to analyze the rates of convergence in the CLT in terms of Wasserstein distances.…”

Section: Applicationsmentioning

confidence: 99%

On martingale approximations and the quenched weak invariance principle

Cuny¹,

Merlevède²

2014

Ann. Probab.

Self Cite

View full text Add to dashboard Cite

In this paper, we obtain sufficient conditions in terms of projective criteria under which the partial sums of a stationary process with values in ${\mathcal{H}}$ (a real and separable Hilbert space) admits an approximation, in ${\mathbb{L}}^p({\mathcal{H}})$, $p>1$, by a martingale with stationary differences, and we then estimate the error of approximation in ${\mathbb{L}}^p({\mathcal{H}})$. The results are exploited to further investigate the behavior of the partial sums. In particular we obtain new projective conditions concerning the Marcinkiewicz-Zygmund theorem, the moderate deviations principle and the rates in the central limit theorem in terms of Wasserstein distances. The conditions are well suited for a large variety of examples, including linear processes or various kinds of weak dependent or mixing processes. In addition, our approach suits well to investigate the quenched central limit theorem and its invariance principle via martingale approximation, and allows us to show that they hold under the so-called Maxwell-Woodroofe condition that is known to be optimal.Comment: Published in at http://dx.doi.org/10.1214/13-AOP856 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

show abstract

Almost sure invariance principles via martingale approximation

Cited by 8 publications

References 28 publications

Strong Invariance Principles with Rate for “Reverse” Martingale Differences and Applications

Strong Invariance Principles with Rate for “Reverse” Martingale Differences and Applications

On the Komlós, Major and Tusnády strong approximation for some classes of random iterates

On martingale approximations and the quenched weak invariance principle

Contact Info

Product

Resources

About