A CENTRAL LIMIT THEOREM FOR NON-LINEAR FUNCTIONS OF A NORMAL STATIONARY PROCESS

1963 ◽  
Author(s):  
TZE-CHIEN SUN
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Stationary Process ◽  
Linear Functions ◽  
Non Linear

EMPIRICAL INVARIANCE PRINCIPLE FOR ERGODIC TORUS AUTOMORPHISMS: GENERICITY

2008 ◽  
Vol 08 (02) ◽  
pp. 173-195 ◽  
Author(s):  
OLIVIER DURIEU ◽  
PHILIPPE JOUAN
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Invariance Principle ◽  
Morse Function ◽  
Empirical Process ◽  
Stationary Process ◽  
Sufficient Condition ◽  
Hölder Continuous ◽  
Morse Functions

We consider the dynamical system given by an algebraic ergodic automorphism T on a torus. We study a Central Limit Theorem for the empirical process associated to the stationary process (f◦Ti)i∈ℕ, where f is a given ℝ-valued function. We give a sufficient condition on f for this Central Limit Theorem to hold. In the second part, we prove that the distribution function of a Morse function is continuously differentiable if the dimension of the manifold is at least three and Hölder continuous if the dimension is one or two. As a consequence, the Morse functions satisfy the empirical invariance principle, which is therefore generically verified.

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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