Disjointification of martingale differences and conditionally independent random variables with some applications

Sergey Astashkin¹ ; Fedor Sukochev² ; Chin Pin Wong²

¹Department of Mathematics Samara State University Samara, Russia
²School of Mathematics and Statistics University of New South Wales Sydney, NSW 2052, Australia

Studia Mathematica, Tome 205 (2011) no. 2, pp. 171-200

Résumé

Disjointification inequalities are proven for arbitrary martingale difference sequences and conditionally independent random variables of the form $\{f_k(s)x_k(t)\}_{k=1}^n ,$ where $f_k$'s are independent and $x_k$'s are arbitrary random variables from a symmetric space $X$ on $[0,1].$ The main results show that the form of these inequalities depends on which side of $L_2$ the space $X$ lies on. The disjointification inequalities obtained allow us to compare norms of sums of martingale differences and non-negative random variables with the norms of sums of their independent copies. The latter results can be treated as an extension of the modular inequalities proved earlier by de la Peña and Hitczenko to the setting of symmetric spaces. Moreover, using these results simplifies the proofs of some modular inequalities.

Détail

DOI : 10.4064/sm205-2-3

Keywords: disjointification inequalities proven arbitrary martingale difference sequences conditionally independent random variables form x where independent arbitrary random variables symmetric space main results form these inequalities depends which side space lies disjointification inequalities obtained allow compare norms sums martingale differences non negative random variables norms sums their independent copies latter results treated extension modular inequalities proved earlier pe hitczenko setting symmetric spaces moreover using these results simplifies proofs modular inequalities