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On non-uniform Berry-Esseen bounds for time series

Jirak M.

Probability and Mathematical Statistics

2015

Vol. 35, Fasc. 1

1--14

Given a stationary sequence {Xk}k ϵ Z, non-uniform bounds for the normal approximation in the Kolmogorov metric are established. The underlying weak dependence assumption includes many popular linear and nonlinear time series from the literature, such as ARMA or GARCH models. Depending on the number of moments p, typical bounds in this context are of the size O(mp−1 n−p/2+1), where we often find that m = mn = log n. In our setup, we can essentially improve upon this rate by the factor m−p/2, yielding a bound of O(mp/2−1 n−p/2+1). Among other things, this allows us to recover a result from the literature, which is due to Ibragimov.

J1 convergence of partial sum processes with a reduced number of jumps

Krizmanić D.

Probability and Mathematical Statistics

2015

Vol. 35, Fasc. 1

107--128

Various functional limit theorems for partial sum processes of strictly stationary sequences of regularly varying random variables in the space of càdlàg functions D[0, 1] with one of the Skorokhod topologies have already been obtained. The mostly used Skorokhod J1 topology is inappropriate when clustering of large values of the partial sum processes occurs. When all extremes within each cluster of high-threshold excesses do not have the same sign, Skorokhod M1 topology also becomes inappropriate. In this paper we alter the definition of the partial sum process in order to shrink all extremes within each cluster to a single one, which allows us to obtain the functional J1 convergence. We also show that this result can be applied to some standard time series models, including the GARCH(1, 1) process and its squares, the stochastic volatility models and m-dependent sequences.

An invariance principle for weakly dependent stationary general models

Doukhan P., Wintenberger O.

Probability and Mathematical Statistics

2007

Vol. 27, Fasc. 1

45--73

The aim of this paper is to refine a weak invariance principle for stationary sequences given by Doukhan and Louhichi [10]. Since our conditions are not causal, our assumptions need to be stronger than the mixing and causal 0-weak dependence assumptions used in Dedecker and Doukhan [5]. Here, if moments of order greater than 2 exist, a weak invariance principle and convergence rates in the CLT are obtained; Doukhan and Louhichi [10] assumed the existence of moments with order greater than 4. Besides the η and к-weak dependence conditions used previously, we introduce a weaker one, λ, which fits the Bernoulli shifts with dependent inputs.