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2 Probability and Mathematical Statistics

2 Geiss S.

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A general contraction principle for vector-valued martingales

Geiss S.

Probability and Mathematical Statistics

2000

Vol. 20, Fasc. 1

93--120

We prove a contraction principle for vector-valued martingales of type [formula] where X is a Banach space with elements x1, ‧‧, xn, (Δi)ni=1 ⊂ L1(Q,P) a martingale difference sequence belonging to a certain class, [formula] a sequence of independent and symmetric random variables exponential in a certain sense, and Ai operators mapping each Δi into a non-negative random variable. Moreover, special operators Ai are discussed and an application to Banach spaces of Rademacher type α (1<α ≤ 2) is given.

Operators on martingales, ф-summing operators, and the contraction principle

Geiss S.

Probability and Mathematical Statistics

1998

Vol. 18, Fasc. 1

149--171

For the absolutely Ф-summmg operators T: X→Y between Banach spaces X and Y we consider martingale inequalities of the type…[formula] where ..[formula]…is a martingale difference sequence and i is a sequence of normalized functionals on X, and we show that these inequalities are useful in different directions. For example, for a Banach space X, x1…xn ∈X, independent standard Gaussian variables gn, and 1 < r < ∞ we deduce that..[formula]… where is a scale-valued martingale difference sequence such that [formula]…is predictable ..[formula].. is a sequence of stopping times and [formula].