Dependent Random Processes Research Articles

Functional repeated measures analysis of variance and its application

This paper is inspired by medical studies in which the same patients with multiple sclerosis are examined at several successive visits (doctor’s appointments) and described by fractional anisotropy tract profiles, which can be represented as f unctions. Since the observations for each patient are dependent random processes, they follow a repeated measures design for functional data. To compare the results for different visits, we thus consider functional repeated measures analysis of variance. For this purpose, a pointwise test statistic is constructed by adapting the classical test statistic for one-way repeated measures analysis of variance to the functional data framework. By integrating and taking the supremum of the pointwise test statistic, we create two global test statistics. In addition to verifying the general null hypothesis of the equality of mean functions corresponding to different objects, we also propose a simple method for post hoc analysis. We illustrate the finite sample properties of permutation and bootstrap testing procedures in an extensive simulation study. Finally, we analyze a real data example in detail. All methods are implemented in the R package rmfanova, available on CRAN.

A functional central limit theorem on non-stationary random fields with nested spatial structure

In this paper, we establish a functional central limit theorem on high dimensional random fields in the context of model-based survey analysis. For strongly-mixing non-stationary random fields, we provide an upper bound for the fourth moment of the finite population total. This inequality is the generalization of a key tool for proving functional central limit theorems in Rio (Asymptotic theory of weakly dependent random processes, Springer, Berlin, 2017). Under the nested sampling strategy, we introduce assumptions on strongly-mixing coefficients and quantile functions to show that a functional stochastic process asymptotically approaches to a Gaussian process.

Palm theory, random measures and Stein couplings

We establish a general Berry–Esseen type bound which gives optimal bounds in many situations under suitable moment assumptions. By combining the general bound with Palm theory, we deduce a new error bound for assessing the accuracy of normal approximation to statistics arising from random measures, including stochastic geometry. We illustrate the use of the bound in four examples: completely random measures, excursion random measure of a locally dependent random process, and the total edge length of Ginibre–Voronoi tessellations and of Poisson–Voronoi tessellations. Moreover, we apply the general bound to Stein couplings and discuss the special cases of local dependence and additive functionals in occupancy problems.

Dependent Lindeberg central limit theorem for the fidis of empirical processes of cluster functionals

ABSTRACTDrees H. and Rootzén H. [Limit theorems for empirical processes of cluster functionals (EPCF). Ann Stat. 2010;38(4):2145–2186] have proven central limit theorems (CLTs) for EPCF built from β-mixing processes. However, this family of β-mixing processes is quite restrictive. We expand some of those results, for the finite-dimensional marginal distributions (fidis), to a more general dependent processes family, known as weakly dependent processes in the sense of Doukhan P. and Louhichi S. [A new weak dependence condition and applications to moment inequalities. Stoch. Proc. Appl. 1999;84:313–342]. In this context, the CLT for the fidis of EPCF is sufficient in some applications. For instance, we prove the convergence without mixing conditions of the extremogram estimator, including a small example with simulation of the extremogram of a weakly dependent random process but nonmixing, in order to confirm the efficacy of our result.

The Estimation of the Fourth-Order Cumulant for Dependent Data: Consistency and Asymptotic Normality

Let {X <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> } be a stationary dependent random process with finite eight-order moments. For broad classes of processes (¿-mixing and strongly mixing), we obtain the convergence in probability, with sharp rates, of the estimate of the fourth-order cumulant from <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> observations {X <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> } <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i=1</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> . We also establish the asymptotic distribution of the estimation error. The asymptotic expression of the variance is explicitly specified.

Concentration inequalities for random fields via coupling

We present a new and simple approach to concentration inequalities in the context of dependent random processes and random fields. Our method is based on coupling and does not use information inequalities. In case one has a uniform control on the coupling, one obtains exponential concentration inequalities. If such a uniform control is no more possible, then one obtains polynomial or stretched-exponential concentration inequalities. Our abstract results apply to Gibbs random fields, both at high and low temperatures and in particular to the low-temperature Ising model which is a concrete example of non-uniformity of the coupling.

Moderate deviations for some weakly dependent random processes

In this paper we compute moderate deviations for two classes of weakly dependent processes, namely, moving averages of independent identically distributed random variables and Poisson center cluster random measures.

Large deviations for some weakly dependent random processes

In this paper we compute large deviation probabilities for two classes of weakly dependent processes, moving averages of i.i.d. random variables and Poisson center cluster random measures.

Vibration Control of Flexible Multibody Aircraft During Touchdown Impacts

A method is presented for the dynamic analysis and vibration control of large scale flexible multibody aircraft during the touchdown impacts and rollover motion. The rollover motion is simulated by modeling the uneven runway profile as a stationary zero mean space dependent random process defined by its spectral shape. The prescribed motions at the two landing gears will then represent a different time dependent random process with spectral shapes changing with the change in velocity of the aircraft. Composite materials which provide higher stiffness-to-weight ratios are used as a passive control system to reduce the aircraft vibration. The results of the numerical example presented showed that the use of composites can have a significant effect on attenuating the vibration of the aircraft during the touchdown impact and rollover motion. The numerical results are obtained using the general purpose computer program DAMS (Dynamic Analysis of Multibody System).