scholarly journals Central limit theorem over non-linear functionals of empirical measures with applications to the mean-field fluctuation of interacting diffusions

2021 ◽  
Vol 26 (none) ◽  
Author(s):  
Benjamin Jourdain ◽  
Alvin Tse
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Mean Field ◽  
Linear Functionals ◽  
Field Fluctuation ◽  
Empirical Measures ◽  
Interacting Diffusions ◽  
Non Linear ◽  
The Mean

scholarly journals A STOCHASTIC ANALYSIS OF BIKE-SHARING SYSTEMS

2020 ◽  
pp. 1-58
Author(s):  
Shuang Tao ◽  
Jamol Pender
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Performance Measures ◽  
Mean Field ◽  
Finite Capacity ◽  
Field Limit ◽  
Mean Field Limit ◽  
Bike Sharing ◽  
The Mean

As more people move back into densely populated cities, bike sharing is emerging as an important mode of urban mobility. In a typical bike-sharing system (BSS), riders arrive at a station and take a bike if it is available. After retrieving a bike, they ride it for a while, then return it to a station near their final destinations. Since space is limited in cities, each station has a finite capacity of docks, which cannot hold more bikes than its capacity. In this paper, we study BSSs with stations having a finite capacity. By an appropriate scaling of our stochastic model, we prove a mean-field limit and a central limit theorem for an empirical process of the number of stations with k bikes. The mean-field limit and the central limit theorem provide insight on the mean, variance, and sample path dynamics of large-scale BSSs. We also leverage our results to estimate confidence intervals for various performance measures such as the proportion of empty stations, the proportion of full stations, and the number of bikes in circulation. These performance measures have the potential to inform the operations and design of future BSSs.

scholarly journals Local Central Limit Theorem for Multi-group Curie–Weiss Models

2021 ◽  
Author(s):  
Michael Fleermann ◽  
Werner Kirsch ◽  
Gabor Toth
Keyword(s):  
High Temperature ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Temperature Regime ◽  
Mean Field ◽  
High Temperature Regime ◽  
The Mean ◽  
Local Central Limit Theorem ◽  
Group Version

AbstractWe study a multi-group version of the mean-field Ising model, also called Curie–Weiss model. It is known that, in the high-temperature regime of this model, a central limit theorem holds for the vector of suitably scaled group magnetisations, that is, for the sum of spins belonging to each group. In this article, we prove a local central limit theorem for the group magnetisations in the high-temperature regime.

Central Limit Theorem for a System of Markovian Particles with Mean Field Interactions

2002 ◽  
pp. 202-222
Author(s):  
Tokuzo Shiga ◽  
Hiroshi Tanaka
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Mean Field

scholarly journals Bounds on the constant in the mean central limit theorem

2010 ◽  
Vol 38 (4) ◽  
pp. 1672-1689 ◽  
Author(s):  
Larry Goldstein
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
The Mean

INVARIANCE PRINCIPLES WITH LOGARITHMIC AVERAGING FOR MARTINGALES

2001 ◽  
Vol 37 (1-2) ◽  
pp. 21-52 ◽  
Author(s):  
F. Chaabane
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Rate Of Convergence ◽  
Central Limit ◽  
Gaussian Measure ◽  
Empirical Measures ◽  
Invariance Principles ◽  
The Central Limit Theorem ◽  
Almost All

For discrete martingales we show that empirical measures related to the central limit theorem when appropriately weighted converge weakly toward a Gaussian measure,for almost all trajectories.Thi result enables u to derive a weighted trong law of the large number for which we pecify both weak and trong rate of convergence.

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