The Mismatch of Intrinsic Fluctuations and the Static Assumptions of Linear Statistics

Author(s):  
Mary Jean Amon ◽  
John G. Holden
2012 ◽  
Vol 01 (04) ◽  
pp. 1250013 ◽  
Author(s):  
IOANA DUMITRIU ◽  
ELLIOT PAQUETTE

We study the global fluctuations for linear statistics of the form [Formula: see text] as n → ∞, for C1 functions f, and λ1, …, λn being the eigenvalues of a (general) β-Jacobi ensemble. The fluctuation from the mean [Formula: see text] turns out to be given asymptotically by a Gaussian process. We compute the covariance matrix for the process and show that it is diagonalized by a shifted Chebyshev polynomial basis; in addition, we analyze the deviation from the predicted mean for polynomial test functions, and we obtain a law of large numbers.


2021 ◽  
Author(s):  
Kaizhong Zheng ◽  
Baojuan Li ◽  
Hongbing Lu ◽  
Huaning Wang ◽  
Baoyu Yan ◽  
...  

Abstract Accumulating evidence suggested that the brain is highly dynamic, thus investigation of brain dynamics especially in brain connectivity would provide crucial information that stationary functional connectivity could miss. This study investigated temporal expressions of spatial modes within the default mode network (DMN), salience network (SN) and cognitive control network (CCN) using a reliable data-driven co-activation pattern (CAP) analysis. We found reduced number of CAPs, as well as transitions between different CAPs of the DMN and CCN, in patients with MDD. These results suggested reduced variability and flexibility of these two brain networks in the patients. By contrast, we also found increased number of CAPs of the SN in the patients, indicating enhanced variability of the SN in individuals with MDD. In addition, the patients were characterized by prominent activation of mPFC and insula. More importantly, we showed that our findings were robust and reproducible with another independent data set. Our findings suggest that functional connectivity in the patients may not be simply attenuated or potentiated, but just alternating faster or slower among more complex patterns. The aberrant temporal-spatial complexity of intrinsic fluctuations reflects functional diaschisis of resting-state networks as characteristic of patients with MDD.


2015 ◽  
Vol 7 (3) ◽  
pp. 1-13 ◽  
Author(s):  
Heng Zhou ◽  
Shu-Wei Huang ◽  
Yixian Dong ◽  
Mingle Liao ◽  
Kun Qiu ◽  
...  

2015 ◽  
Vol 9 (2) ◽  
pp. 64-73 ◽  
Author(s):  
Christoph Zimmer ◽  
Sven Sahle ◽  
Jürgen Pahle

Author(s):  
Florence Merlevède ◽  
Magda Peligrad ◽  
Sergey Utev

Here we apply different methods to establish the Gaussian approximation to linear statistics of a stationary sequence, including stationary linear processes, near-stationary processes, and discrete Fourier transforms of a strictly stationary process. More precisely, we analyze the asymptotic behavior of the partial sums associated with a short-memory linear process and prove, in particular, that if a weak limit theorem holds for the partial sums of the innovations then a related result holds for the partial sums of the linear process itself. We then move to linear processes with long memory and obtain the CLT under various dependence structures for the innovations by analyzing the asymptotic behavior of linear statistics. We also deal with the invariance principle for causal linear processes or for linear statistics with weakly associated innovations. The last section deals with discrete Fourier transforms, proving, via martingale approximation, central limit behavior at almost all frequencies under almost no condition except a regularity assumption.


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