scholarly journals A Central Limit Theorem for Predictive Distributions

Mathematics ◽  
2021 ◽  
Vol 9 (24) ◽  
pp. 3211
Author(s):  
Patrizia Berti ◽  
Luca Pratelli ◽  
Pietro Rigo
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Polish Space ◽  
Borel Subset ◽  
Covariance Structure ◽  
Predictive Distribution ◽  
Gaussian Kernel ◽  
Finite Dimensional ◽  
Borel Functions ◽  
Random Probability

Let S be a Borel subset of a Polish space and F the set of bounded Borel functions f:S→R. Let an(·)=P(Xn+1∈·∣X1,…,Xn) be the n-th predictive distribution corresponding to a sequence (Xn) of S-valued random variables. If (Xn) is conditionally identically distributed, there is a random probability measure μ on S such that ∫fdan⟶a.s.∫fdμ for all f∈F. Define Dn(f)=dn∫fdan−∫fdμ for all f∈F, where dn>0 is a constant. In this note, it is shown that, under some conditions on (Xn) and with a suitable choice of dn, the finite dimensional distributions of the process Dn=Dn(f):f∈F stably converge to a Gaussian kernel with a known covariance structure. In addition, Eφ(Dn(f))∣X1,…,Xn converges in probability for all f∈F and φ∈Cb(R).

A quantum-mechanical central limit theorem for anti-commuting observables

1973 ◽  
Vol 10 (03) ◽  
pp. 502-509 ◽  
Author(s):  
R. L. Hudson
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Dimensional Space ◽  
Quantum Mechanical ◽  
Finite Dimensional Space ◽  
Commutation Relations ◽  
Finite Dimensional

A quantum-mechanical central limit theorem for sums of pairwise anti-commuting representations of the canonical anti-commutation relations over a finite-dimensional space is formulated and proved.

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.

A quantum-mechanical central limit theorem for anti-commuting observables

1973 ◽  
Vol 10 (3) ◽  
pp. 502-509 ◽  
Author(s):  
R. L. Hudson
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Dimensional Space ◽  
Quantum Mechanical ◽  
Finite Dimensional Space ◽  
Commutation Relations ◽  
Finite Dimensional

A quantum-mechanical central limit theorem for sums of pairwise anti-commuting representations of the canonical anti-commutation relations over a finite-dimensional space is formulated and proved.

scholarly journals A Dependent Lindeberg Central Limit Theorem for Cluster Functionals on Stationary Random Fields

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 212
Author(s):  
José G. Gómez-García ◽  
Christophe Chesneau
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Random Field ◽  
Central Limit ◽  
Random Fields ◽  
Empirical Processes ◽  
Index Set ◽  
Finite Dimensional ◽  
Lindeberg Condition ◽  
Stationary Random Field

In this paper, we provide a central limit theorem for the finite-dimensional marginal distributions of empirical processes (Zn(f))f∈F whose index set F is a family of cluster functionals valued on blocks of values of a stationary random field. The practicality and applicability of the result depend mainly on the usual Lindeberg condition and on a sequence Tn which summarizes the dependence between the blocks of the random field values. Finally, in application, we use the previous result in order to show the Gaussian asymptotic behavior of the proposed iso-extremogram estimator.

scholarly journals On the Markov Transition Kernels for First Passage Percolation on the Ladder

2011 ◽  
Vol 48 (02) ◽  
pp. 366-388 ◽  
Author(s):  
Eckhard Schlemm
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Asymptotic Variance ◽  
Almost Sure Convergence ◽  
First Passage Percolation ◽  
First Passage ◽  
Percolation Problem ◽  
Vertex Set ◽  
Step Transition

We consider the first passage percolation problem on the random graph with vertex set N x {0, 1}, edges joining vertices at a Euclidean distance equal to unity, and independent exponential edge weights. We provide a central limit theorem for the first passage times l n between the vertices (0, 0) and (n, 0), thus extending earlier results about the almost-sure convergence of l n / n as n → ∞. We use generating function techniques to compute the n-step transition kernels of a closely related Markov chain which can be used to explicitly calculate the asymptotic variance in the central limit theorem.

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