István Berkes

On discriminating between long-range dependence and changes in mean

We develop a testing procedure for distinguishing between a long-range dependent time series and a weakly dependent time series with change-points in the mean. In the simplest case, under the null hypothesis the time series is weakly dependent with one change in mean at an unknown point, and under the alternative it is long-range dependent. We compute the CUSUM statistic T n , which allows us to construct an estimator k of a change-point. We then compute the statistic T n,1 based on the observations up to time k and the statistic T n,2 2 based on the observations after time k. The statistic M n = max[T n.1 , T n,2 ] converges to a well-known distribution under the null, but diverges to infinity if the observations exhibit long-range dependence. The theory is illustrated by examples and an application to the returns of the Dow Jones index.

Split invariance principles for stationary processes

The results of Komlos, Major and Tusnady give optimal Wiener approximation of partial sums of i.i.d. random variables and provide an extremely powerful tool in probability and statistical inference. Recently Wu [Ann. Probab. 35 (2007) 2294–2320] obtained Wiener approximation of a class of dependent stationary processes with finite pth moments,

Almost sure convergence of the Bartlett estimator

2 0

GCD sums from Poisson integrals and systems of dilated functions

, and Liu and Lin [Stochastic Process. Appl. 119 (2009) 249–280] removed the logarithmic factor, reaching the Komlos–Major–Tusnady bound

Komlós–Major–Tusnády approximation under dependence

o(n^{1/p})

On the law of the iterated logarithm for permuted lacunary sequences

. No similar results exist for

On the asymptotic behavior of weakly lacunary series

p > 4

Testing for Structural Change of AR Model to Threshold AR Model

, and in fact, no existing method for dependent approximation yields an a.s. rate better than

Selection from a stable box

o(n^{1/4})

On permutations of Hardy-Littlewood-Pólya sequences

. In this paper we show that allowing a second Wiener component in the approximation, we can get rates near to