Central Limit Theorem for the Local Time of a Gaussian Process

1999 ◽  
pp. 25-37
Author(s):  
Samir Ben Hariz ◽  
Paul Doukhan ◽  
José Rafael León
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Gaussian Process ◽  
Local Time ◽  
Central Limit

scholarly journals Gaussian fluctuations of Young diagrams under the Plancherel measure

2007 ◽  
Vol 463 (2080) ◽  
pp. 1069-1080 ◽  
Author(s):  
Leonid V Bogachev ◽  
Zhonggen Su
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Gaussian Process ◽  
Central Limit ◽  
Point Processes ◽  
Plancherel Measure ◽  
Young Diagrams ◽  
Finite Dimensional ◽  
The Central Limit Theorem ◽  
Local Correlations

We obtain the central limit theorem for fluctuations of Young diagrams around their limit shape in the bulk of the ‘spectrum’ of partitions λ ⊢ n ∈ (under the Plancherel measure), thus settling a long-standing problem posed by Logan & Shepp. Namely, under normalization growing like , the corresponding random process in the bulk is shown to converge, in the sense of finite-dimensional distributions, to a Gaussian process with independent values, while local correlations in the vicinity of each point, measured on various power scales, possess certain self-similarity. The proofs are based on the Poissonization techniques and use Costin–Lebowitz–Soshnikov's central limit theorem for determinantal random point processes. Our results admit a striking reformulation after the rotation of Young diagrams by 45°, whereby the normalization no longer depends on the location in the spectrum. In addition, we explain heuristically the link with an earlier result by Kerov on the convergence to a generalized Gaussian process.

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