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On symmetric spaces containing isomorphic copies of Orlicz sequence spaces

100%

Astashkin S.

Commentationes Mathematicae

2016

tom 56

nr 1

Let an Orlicz function $N$ be $(1+\varepsilon)$-convex and $(2-\varepsilon)$-concave at zero for some $\varepsilon>0.$ Then the function $1/N^{-1}(t)$, $t\in(0,1]$, belongs to a separable symmetric space $X$ with the Fatou property, which is an interpolation space with respect to the couple $(L_1,L_2)$, whenever $X$ contains a strongly embedded subspace isomorphic to the Orlicz sequence space $l_N$. On the other hand, we find necessary and sufficient conditions on such an Orlicz function $N$ under which a sequence of mean zero independent functions equimeasurable with the function ${1}/{N^{-1}}(t)$, \(0

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

51%

Astashkin S. , Sukochev F. , Wong C.

Studia Mathematica

2011

tom 205

nr 2

171-200

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 L₂ 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.