scholarly journals A moment approach for the almost sure central limit theorem for martingales

2008 ◽  
Vol 45 (1) ◽  
pp. 139-159 ◽  
Author(s):  
Bernard Bercu ◽  
Jean-Claude Fort
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Branching Processes ◽  
Convergence Of Moments ◽  
Moment Approach ◽  
Original Approach

We prove the almost sure central limit theorem for martingales via an original approach which uses the Carleman moment theorem together with the convergence of moments of martingales. Several statistical applications to autoregressive and branching processes are also provided.

scholarly journals On the Almost Sure Central Limit Theorem for Vector Martingales: Convergence of Moments and Statistical Applications

2009 ◽  
Vol 46 (01) ◽  
pp. 151-169 ◽  
Author(s):  
Bernard Bercu ◽  
Peggy Cénac ◽  
Guy Fayolle
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Branching Processes ◽  
Asymptotic Properties ◽  
Regularity Conditions ◽  
Prediction Errors ◽  
Convergence Of Moments ◽  
Even Order ◽  
Martingale Transforms

We investigate the almost sure asymptotic properties of vector martingale transforms. Assuming some appropriate regularity conditions both on the increasing process and on the moments of the martingale, we prove that normalized moments of any even order converge in the almost sure central limit theorem for martingales. A conjecture about almost sure upper bounds under wider hypotheses is formulated. The theoretical results are supported by examples borrowed from statistical applications, including linear autoregressive models and branching processes with immigration, for which new asymptotic properties are established on estimation and prediction errors.

scholarly journals On the Almost Sure Central Limit Theorem for Vector Martingales: Convergence of Moments and Statistical Applications

2009 ◽  
Vol 46 (1) ◽  
pp. 151-169 ◽  
Author(s):  
Bernard Bercu ◽  
Peggy Cénac ◽  
Guy Fayolle
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Branching Processes ◽  
Asymptotic Properties ◽  
Regularity Conditions ◽  
Prediction Errors ◽  
Convergence Of Moments ◽  
Even Order ◽  
Martingale Transforms

We investigate the almost sure asymptotic properties of vector martingale transforms. Assuming some appropriate regularity conditions both on the increasing process and on the moments of the martingale, we prove that normalized moments of any even order converge in the almost sure central limit theorem for martingales. A conjecture about almost sure upper bounds under wider hypotheses is formulated. The theoretical results are supported by examples borrowed from statistical applications, including linear autoregressive models and branching processes with immigration, for which new asymptotic properties are established on estimation and prediction errors.

A Quenched Central Limit Theorem for Reversible Random Walks in a Random Environment on Z

2014 ◽  
Vol 51 (04) ◽  
pp. 1051-1064
Author(s):  
Hoang-Chuong Lam
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Random Walks ◽  
Central Limit ◽  
Random Environment ◽  
Integrability Condition ◽  
The Central Limit Theorem ◽  
The Poisson Equation ◽  
Convergence Of Moments ◽  
Quenched Central Limit Theorem

The main aim of this paper is to prove the quenched central limit theorem for reversible random walks in a stationary random environment on Z without having the integrability condition on the conductance and without using any martingale. The method shown here is particularly simple and was introduced by Depauw and Derrien [3]. More precisely, for a given realization ω of the environment, we consider the Poisson equation (P ω - I)g = f, and then use the pointwise ergodic theorem in [8] to treat the limit of solutions and then the central limit theorem will be established by the convergence of moments. In particular, there is an analogue to a Markov process with discrete space and the diffusion in a stationary random environment.

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