A general contraction principle for vector-valued martingales
The purpose of the present paper is to relate the property ``martingale difference sequence'' to the property ``independent and symmetric'' by deriving a contraction principle of the type NEWLINE\[NEWLINE\Biggl\|\sum^n_{i=1} \Delta_i x_i\Biggr\|_{L^X_p}\leq c_p\Biggl\|\sup_{1\leq i\leq n} A_i(\Delta_i)\Biggr\|_{L_p} \Biggl\|\sum^n_{i=1} H_i x_i\Biggr\|_{L^X_1}\qquad (1\leq p< \infty).NEWLINE\]NEWLINE Here, \(x_1,\dots, x_n\) are elements of a certain Banach space \(X\), \((\Delta_i)\subset L_1(\Omega, P)\) \((1\leq i\leq n)\) is a martingale difference sequence belonging to a certain class, and \((H_i)\subset L_1(M,\nu)\) \((1\leq i\leq n)\) is a sequence of independent and symmetric random variables whose tail probabilities are decreasing exponentially fast (in some specified way). Finally, \(A_1,\dots, A_n\) are operators mapping each \(\Delta_i\) into a nonnegative random variable. The just mentioned contraction principle is applied to Banach spaces of Rademacher type \(\alpha\) \((1<\alpha\leq 2)\).
scientific article; zbMATH DE number 1706584