A Central Limit Theorem for the Renormalized Self-Intersection Local Time of a Stationary Process

1992 ◽  
pp. 351-363 ◽  
Author(s):  
Simeon M. Berman
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Local Time ◽  
Central Limit ◽  
Stationary Process ◽  
Intersection Local Time

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.

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