Asymptotic normality of quadratic forms of martingale differences

We establish the asymptotic normality of a quadratic form QnQn in martingale difference random variables ηtηt when the weight matrix A of the quadratic form has an asymptotically vanishing diagonal. Such a result has numerous potential applications in time series analysis. While for i.i.d. random variables ηtηt, asymptotic normality holds under condition ||A||sp=o(||A||)||A||sp=o(||A||), where ||A||sp||A||sp and ||A|| are the spectral and Euclidean norms of the matrix A, respectively, finding corresponding sufficient conditions in the case of martingale differences ηtηt has been an important open problem. We provide such sufficient conditions in this paper

Exploring the Fundamental Dynamics of Error-Based Motor Learning Using a Stationary Predictive-Saccade Task

The maintenance of movement accuracy uses prior performance errors to correct future motor plans; this motor-learning process ensures that movements remain quick and accurate. The control of predictive saccades, in which anticipatory movements are made to future targets before visual stimulus information becomes available, serves as an ideal paradigm to analyze how the motor system utilizes prior errors to drive movements to a desired goal. Predictive saccades constitute a stationary process (the mean and to a rough approximation the variability of the data do not vary over time, unlike a typical motor adaptation paradigm). This enables us to study inter-trial correlations, both on a trial-by-trial basis and across long blocks of trials. Saccade errors are found to be corrected on a trial-by-trial basis in a direction-specific manner (the next saccade made in the same direction will reflect a correction for errors made on the current saccade). Additionally, there is evidence for a second, modulating process that exhibits long memory. That is, performance information, as measured via inter-trial correlations, is strongly retained across a large number of saccades (about 100 trials). Together, this evidence indicates that the dynamics of motor learning exhibit complexities that must be carefully considered, as they cannot be fully described with current state-space (ARMA) modeling efforts

Luxembourg Parkinson’s study -comprehensive baseline analysis of Parkinson’s disease and atypical parkinsonism

BackgroundDeep phenotyping of Parkinson’s disease (PD) is essential to investigate this fastest-growing neurodegenerative disorder. Since 2015, over 800 individuals with PD and atypical parkinsonism along with more than 800 control subjects have been recruited in the frame of the observational, monocentric, nation-wide, longitudinal-prospective Luxembourg Parkinson’s study.ObjectiveTo profile the baseline dataset and to explore risk factors, comorbidities and clinical profiles associated with PD, atypical parkinsonism and controls.MethodsEpidemiological and clinical characteristics of all 1,648 participants divided in disease and control groups were investigated. Then, a cross-sectional group comparison was performed between the three largest groups: PD, progressive supranuclear palsy (PSP) and controls. Subsequently, multiple linear and logistic regression models were fitted adjusting for confounders.ResultsThe mean (SD) age at onset (AAO) of PD was 62.3 (11.8) years with 15% early onset (AAO < 50 years), mean disease duration 4.90 (5.16) years, male sex 66.5% and mean MDS-UPDRS III 35.2 (16.3). For PSP, the respective values were: 67.6 (8.2) years, all PSP with AAO > 50 years, 2.80 (2.62) years, 62.7% and 53.3 (19.5). The highest frequency of hyposmia was detected in PD followed by PSP and controls (72.9%; 53.2%; 14.7%), challenging the use of hyposmia as discriminating feature in PD vs. PSP. Alcohol abstinence was significantly higher in PD than controls (17.6 vs. 12.9%, p = 0.003).ConclusionLuxembourg Parkinson’s study constitutes a valuable resource to strengthen the understanding of complex traits in the aforementioned neurodegenerative disorders. It corroborated several previously observed clinical profiles, and provided insight on frequency of hyposmia in PSP and dietary habits, such as alcohol abstinence in PD.Clinical trial registration: clinicaltrials.gov, NCT05266872