scholarly journals Some remarks on Banach spaces in which martingale difference sequences are unconditional

1983 ◽  
Vol 21 (1-2) ◽  
pp. 163-168 ◽  
Author(s):  
J. Bourgain
Keyword(s):  
Banach Spaces ◽  
Martingale Difference ◽  
Difference Sequences

scholarly journals Unconditional martingale difference sequences in Banach spaces

2007 ◽  
Vol 326 (2) ◽  
pp. 1291-1309 ◽  
Author(s):  
Stuart F. Cullender ◽  
Coenraad C.A. Labuschagne
Keyword(s):  
Banach Spaces ◽  
Martingale Difference ◽  
Difference Sequences

scholarly journals Davis-type theorems for martingale difference sequences

2005 ◽  
Vol 2005 (2) ◽  
pp. 159-165 ◽  
Author(s):  
George Stoica
Keyword(s):  
Rate Of Convergence ◽  
Classical Case ◽  
Moderate Deviation ◽  
Sharp Contrast ◽  
Optimal Rate ◽  
Martingale Difference ◽  
Normalization Factor ◽  
Second Moment ◽  
Optimal Rate Of Convergence ◽  
Difference Sequences

We study Davis-type theorems on the optimal rate of convergence of moderate deviation probabilities. In the case of martingale difference sequences, under the finite pth moments hypothesis (1≤p<∞), and depending on the normalization factor, our results show that Davis' theorems either hold if and only if p>2 or fail for all p≥1. This is in sharp contrast with the classical case of i.i.d. centered sequences, where both Davis' theorems hold under the finite second moment hypothesis (or less).

scholarly journals ON THE CENTRAL AND LOCAL LIMIT THEOREM FOR MARTINGALE DIFFERENCE SEQUENCES

2004 ◽  
Vol 04 (02) ◽  
pp. 153-173
Author(s):  
MOHAMED EL MACHKOURI ◽  
DALIBOR VOLNÝ
Keyword(s):  
Limit Theorem ◽  
Local Limit Theorem ◽  
Local Limit ◽  
Ergodic Measure ◽  
Difference Sequence ◽  
Martingale Difference ◽  
Lattice Distribution ◽  
Martingale Difference Sequence ◽  
The Central Limit Theorem ◽  
Difference Sequences

Let [Formula: see text] be a Lebesgue space and T: Ω→Ω an ergodic measure-preserving automorphism with positive entropy. We show that there is a bounded and strictly stationary martingale difference sequence defined on Ω with a common nondegenerate lattice distribution satisfying the central limit theorem with an arbitrarily slow rate of convergence and not satisfying the local limit theorem. A similar result is established for martingale difference sequences with densities provided the entropy is infinite. In addition, the martingale difference sequence may be chosen to be strongly mixing.

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