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Weak star convergence of martingales in a dual space

C. Castaing, F. Ezzaki, M. Lavie, M. Saadoune (2011)

Banach Center Publications

In this paper we present various weak star Kuratowski convergence results for multivalued martingales, supermartingales and multivalued mils in the dual of a separable Banach space. We establish several integral representation formulas for convex weak star compact valued multifunctions defined on a Köthe space and derive several existence results of conditional expectation for multivalued Gelfand-integrable multifunctions. Similar convergence results for Gelfand-integrable martingales in the dual...

Weak Type Inequality for the Square Function of a Nonnegative Submartingale

Adam Osękowski (2009)

Bulletin of the Polish Academy of Sciences. Mathematics

Let f be a nonnegative submartingale and S(f) denote its square function. We show that for any λ > 0, λ ℙ ( S ( f ) ≥ λ ) ≤ π / 2 ∥ f ∥ ₁ , and the constant π/2 is the best possible. The inequality is strict provided ∥f∥₁ ≠ 0.

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