Tangent Sequences in Orlicz and Rearrangement Invariant Spaces

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Let (fₙ) and (gₙ) be two sequences of random variables adapted to an increasing sequence of σ-algebras (Fₙ) such that the conditional distributions of fₙ and gₙ given Fₙ₋₁ coincide, and such that the sequence (gₙ) is conditionally independent. Then it is known that ∑ fₖₚ ≤ C ∑ gₖₚ, 1 ≤ p ≤ ∞ where the constant C is independent of p. The aim of this paper is to extend this result to certain classes of Orlicz and rearrangement invariant spaces. This paper includes fairly general techniques for obtaining rearrangement invariant inequalities from Orlicz norm inequalities.