The random paving property for uniformly bounded matrices

The Random Paving Property for Uniformly Bounded Matrices

This note presents a new proof of an important result due to Bourgain and Tzafriri that provides a partial solution to the Kadison–Singer problem. The result shows that every unitnorm matrix whose entries are relatively small in comparison with its dimension can be paved by a partition of constant size. That is, the coordinates can be partitioned into a constant number of blocks so that the res...

The Paving Property for Uniformly Bounded Matrices

Abstract. This note presents a new proof of an important result due to Bourgain and Tzafriri that provides a partial solution to the Kadison–Singer problem. The result shows that every unitnorm matrix whose entries are relatively small in comparison with its dimension can be paved by a partition of constant size. That is, the coordinates can be partitioned into a constant number of blocks so th...

The Paving Property for Uniformly Bounded Matrices: a New Proof

Abstract. This note presents a new proof of an important result due to Bourgain and Tzafriri that provides a partial solution to the Kadison–Singer problem. The result shows that every unitnorm matrix whose entries are relatively small in comparison with its dimension can be paved by a partition of constant size. That is, the coordinates can be partitioned into a constant number of blocks so th...

Stop Rule Inequalities for Uniformly Bounded Sequences of Random Variables

If X"0, Xx_ is an arbitrarily-dependent sequence of random variables taking values in [0,1] and if V( X0,X¡,... ) is the supremum, over stop rules /, of EX,, then the set of ordered pairs {(.*, v): x V(X0, Xx,.. .,Xn) and y £(maxyS„X¡) for some X0,..., Xn] is precisely the set C„= {(x,y):x<y<x(\ + n(\ *'/")); 0 « x « l}; and the set of ordered pairs {(x, y): x V(X0, X,,...) and y £(sup„ X„) for...

Understanding the Isometry Property of Random Matrices