scholarly journals Scaling Limit for the Ant in High-Dimensional Labyrinths

2019 ◽  
Vol 72 (4) ◽  
pp. 669-763 ◽  
Author(s):  
Gérard Ben Arous ◽  
Manuel Cabezas ◽  
Alexander Fribergh
Keyword(s):  
Scaling Limit ◽  
High Dimensional

scholarly journals Estimation of Dynamic Networks for High-Dimensional Nonstationary Time Series

Entropy ◽  
2019 ◽  
Vol 22 (1) ◽  
pp. 55 ◽  
Author(s):  
Mengyu Xu ◽  
Xiaohui Chen ◽  
Wei Biao Wu
Keyword(s):  
Time Series ◽  
Structural Breaks ◽  
Dynamic Networks ◽  
Scaling Limit ◽  
Nonstationary Time Series ◽  
Change Points ◽  
High Dimensional ◽  
Time Varying ◽  
Matrix Estimation ◽  
Sample Covariance

This paper is concerned with the estimation of time-varying networks for high-dimensional nonstationary time series. Two types of dynamic behaviors are considered: structural breaks (i.e., abrupt change points) and smooth changes. To simultaneously handle these two types of time-varying features, a two-step approach is proposed: multiple change point locations are first identified on the basis of comparing the difference between the localized averages on sample covariance matrices, and then graph supports are recovered on the basis of a kernelized time-varying constrained L 1 -minimization for inverse matrix estimation (CLIME) estimator on each segment. We derive the rates of convergence for estimating the change points and precision matrices under mild moment and dependence conditions. In particular, we show that this two-step approach is consistent in estimating the change points and the piecewise smooth precision matrix function, under a certain high-dimensional scaling limit. The method is applied to the analysis of network structure of the S&P 500 index between 2003 and 2008.

Scaling limit for the ant in a simple high-dimensional labyrinth

2018 ◽  
Vol 174 (1-2) ◽  
pp. 553-646 ◽  
Author(s):  
Gérard Ben Arous ◽  
Manuel Cabezas ◽  
Alexander Fribergh
Keyword(s):  
Scaling Limit ◽  
High Dimensional

scholarly journals Optimal scaling of the random walk Metropolis algorithm under Lp mean differentiability

2017 ◽  
Vol 54 (4) ◽  
pp. 1233-1260 ◽  
Author(s):  
Alain Durmus ◽  
Sylvain Le Corff ◽  
Eric Moulines ◽  
Gareth O. Roberts
Keyword(s):  
Random Walk ◽  
Scaling Limit ◽  
Optimal Scaling ◽  
High Dimensional ◽  
Random Walk Metropolis ◽  
Langevin Diffusion ◽  
Random Walk Metropolis Algorithm ◽  
The One ◽  
Original Derivation ◽  
Practical Implications

Abstract In this paper we consider the optimal scaling of high-dimensional random walk Metropolis algorithms for densities differentiable in the Lp mean but which may be irregular at some points (such as the Laplace density, for example) and/or supported on an interval. Our main result is the weak convergence of the Markov chain (appropriately rescaled in time and space) to a Langevin diffusion process as the dimension d goes to ∞. As the log-density might be nondifferentiable, the limiting diffusion could be singular. The scaling limit is established under assumptions which are much weaker than the one used in the original derivation of Roberts et al. (1997). This result has important practical implications for the use of random walk Metropolis algorithms in Bayesian frameworks based on sparsity inducing priors.

Lattice trees and super-Brownian motion

1997 ◽  
Vol 40 (1) ◽  
pp. 19-38 ◽  
Author(s):  
Eric Derbez ◽  
Gordon Slade
Keyword(s):  
Brownian Motion ◽  
Mean Field Theory ◽  
Scaling Limit ◽  
Mean Field ◽  
High Dimensional ◽  
High Dimensions ◽  
Lace Expansion ◽  
Brownian Excursion ◽  
Lattice Trees ◽  
Super Brownian Motion

AbstractThis article discusses our recent proof that above eight dimensions the scaling limit of sufficiently spread-out lattice trees is the variant of super-Brownian motion calledintegrated super-Brownian excursion(ISE), as conjectured by Aldous. The same is true for nearest-neighbour lattice trees in sufficiently high dimensions. The proof, whose details will appear elsewhere, uses the lace expansion. Here, a related but simpler analysis is applied to show that the scaling limit of a mean-field theory is ISE, in all dimensions. A connection is drawn between ISE and certain generating functions and critical exponents, which may be useful for the study of high-dimensional percolation models at the critical point.

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