Series of independent vector valued random variables and absolute continuity of seminorms.

Operator Valued Series and Vector Valued Multiplier Spaces

‎Let $X,Y$ be normed spaces with $L(X,Y)$ the space of continuous‎ ‎linear operators from $X$ into $Y$‎. ‎If ${T_{j}}$ is a sequence in $L(X,Y)$,‎ ‎the (bounded) multiplier space for the series $sum T_{j}$ is defined to be‎ [ ‎M^{infty}(sum T_{j})={{x_{j}}in l^{infty}(X):sum_{j=1}^{infty}%‎ ‎T_{j}x_{j}text{ }converges}‎ ‎]‎ ‎and the summing operator $S:M^{infty}(sum T_{j})rightarrow Y$ associat...

ASYMPTOTIC BEHAVIOR OF p-PREDICTIONS FOR VECTOR VALUED RANDOM VARIABLES

Let (Q,<r, p) be a probability space and let X be a B-valued /i-essentially bounded random variable, where (B,\\ ||) is a uniformly convex Banach space. Given a, a sub-<r-algebra of a, the p-prediction (1 < p < oo) of X is defined as the best Lp-approximation to X by (»-measurable random variables. The paper proves that the Pólya algorithm is successful, i.e. the p-prediction converges to an "c...

Tail sums of convergent series of independent random variables

operator valued series and vector valued multiplier spaces

‎let $x,y$ be normed spaces with $l(x,y)$ the space of continuous‎ ‎linear operators from $x$ into $y$‎. ‎if ${t_{j}}$ is a sequence in $l(x,y)$,‎ ‎the (bounded) multiplier space for the series $sum t_{j}$ is defined to be‎ [ ‎m^{infty}(sum t_{j})={{x_{j}}in l^{infty}(x):sum_{j=1}^{infty}%‎ ‎t_{j}x_{j}text{ }converges}‎ ‎]‎ ‎and the summing operator $s:m^{infty}(sum t_{j})rightarrow y$ associat...

Absolute continuity of catalytic measure-valued branching processes

Classical super-Brownian motion (SBM) is known to take values in the space of absolutely continuous measures only if d=1. For d¿2 its values are almost surely singular with respect to Lebesgue measure. This result has been generalized to more general motion laws and branching laws (yielding di erent critical dimensions) and also to catalytic SBM. In this paper we study the case of a catalytic m...