Recent advances in invariance principles for stationary sequences

Merlevède, Florence; Peligrad, Magda; Utev, Sergey

doi:10.1214/154957806100000202

Cited by 114 publications

(88 citation statements)

References 51 publications

(160 reference statements)

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“…Wu and Woodroofe (2004) obtained a CLT for the sums of stationary and ergodic sequences using martingale approximation method. Merlevède, Peligrad and Utev (2006) provide a further survey of some recent results on CLT and its weak invariance principle for stationary processes.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic Normality for Weighted Sums of Linear Processes

et al. 2013

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We establish asymptotic normality of weighted sums of linear processes with general triangular array weights and when the innovations in the linear process are martingale differences. The results are obtained under minimal conditions on the weights and innovations. We also obtain weak convergence of weighted partial sum processes. The results are applicable to linear processes that have short or long memory or exhibit seasonal long memory behavior. In particular, they are applicable to GARCH and ARCH(∞) models and to their squares. They are also useful in deriving asymptotic normality of kernel type estimators of a nonparametric regression function with short or long memory moving average errors.

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Section: Introductionmentioning

confidence: 99%

Asymptotic Normality for Weighted Sums of Linear Processes

et al. 2013

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“…(> 0) is some constant. This (42) follows from the same argument as in (37), whose proof is omitted. Now, we can show that P n i=1 Z ni = O p ( n ) by the previous argument.…”

Section: Resultsmentioning

confidence: 88%

“…The proof is completed if we compute the bound of P m 1 l=1 l for each case: Proof of Inequality (37). This proof proceeds in the same way as that of Lemma 1 for 2 (0; 1), and the details are omitted.…”

Section: A4 Proofs Of Auxiliary Resultsmentioning

confidence: 99%

CONVERGENCE RATES OF SUMS OF α-MIXING TRIANGULAR ARRAYS: WITH AN APPLICATION TO NONPARAMETRIC DRIFT FUNCTION ESTIMATION OF CONTINUOUS-TIME PROCESSES

Kanaya

2016

Econom. Theory

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The convergence rates of the sums of -mixing (or strongly mixing) triangular arrays of heterogeneous random variables are derived. We pay particular attention to the case where central limit theorems may fail to hold, due to relatively strong time-series dependence and/or the nonexistence of higher-order moments. Several previous studies have presented various versions of laws of large numbers for sequences/triangular arrays, but their convergence rates were not fully investigated. This study is the …rst to investigate the convergence rates of the sums of -mixing triangular arrays whose mixing coe¢ cients are permitted to decay arbitrarily slowly. We consider two kinds of asymptotic assumptions: one is that the time distance between adjacent observations is …xed for any sample size n; and the other, called the in…ll assumption, is that it shrinks to zero as n tends to in…nity. Our convergence theorems indicate that an explicit trade-o¤ exists between the rate of convergence and the degree of dependence. While the results under the in…ll assumption can be seen as a direct extension of those under the …xed-distance assumption, they are new and particularly useful for deriving sharper convergence rates of discretization biases in estimating continuous-time processes from discretely sampled observations. We also discuss some examples to which our results and techniques are useful and applicable: a moving-average process with long lasting past shocks, a continuous-time di¤usion process with weak mean reversion, and a near-unit-root process.

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“…The invariance principle in C(H) is established under (1.6) by Dedecker and Merlevède [2] as a special case of their Theorem 5 (see also in [10] Prop. 17 and the discussion p. 21).…”

Section: Proof Of Theorem 13mentioning

confidence: 96%

“…observations. A recent survey of invariance principles in C[0, 1] for stationary sequences is [10]. For invariance principles under various weak dependence conditions, let us also mention [3].…”

mentioning

confidence: 99%

Hölderian invariance principle for Hilbertian linear processes

Račkauskas

Suquet

2009

ESAIM: PS

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Abstract. Let (ξn) n≥1 be the polygonal partial sums processes built on the linear processes Xn = i≥0 ai( n−i), n ≥ 1, where ( i)i∈Z are i.i.d., centered random elements in some separable Hilbert space H and the ai's are bounded linear operators H → H, with i≥0 ai < ∞. We investigate functional central limit theorem for ξn in the Hölder spacesand L slowly varying at infinity. We obtain the H o ρ (H) weak convergence of ξn to some H valued Brownian motion under the optimal assumption that for any c > 0, tP ( 0 > ct 1/2 ρ(1/t)) = o(1) when t tends to infinity, subject to some mild restriction on L in the boundary case α = 1/2. Our result holds in particular with the weight functionsRésumé. Soit (ξn) n≥1 le processus polygonal de sommes partielles bâti sur le processus linéaire Mathematics Subject Classification. 60F17, 60B12.

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Recent advances in invariance principles for stationary sequences

Cited by 114 publications

References 51 publications

Asymptotic Normality for Weighted Sums of Linear Processes

Asymptotic Normality for Weighted Sums of Linear Processes

CONVERGENCE RATES OF SUMS OF α-MIXING TRIANGULAR ARRAYS: WITH AN APPLICATION TO NONPARAMETRIC DRIFT FUNCTION ESTIMATION OF CONTINUOUS-TIME PROCESSES

Hölderian invariance principle for Hilbertian linear processes

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