鞅型序列的变换及其收敛性

MARTINGALE-LIKE SEQUENCE TRANSFORMS AND THEIR CONVERGENCE

本文证明了(1)设 Banach 空间 B 为 P 阶光滑的(1≤P≤2),X=(X_n,(?)_n,n≥1)为B 值鞅,v=(v_n,(?)_n,n≥1)为实值可予报序列,鞅变换 Y=(sum from i=1 to n V_i(X_i-X_(i-1)),(?)_n,n≥1)在一定的条件下具有 a.e.收敛性,L^p 收敛性及强(弱)大多数定律成立。(2)Banach空间 B 具有 Radon-Nikodym 性质,X=(X_n,(?)_n,n≥1)为 B 值依概极限鞅,实值可予报序列 V=(V_n,(?)_n,n≥1)满足 sum from i=1 to ∞ E(|V_i|~p)^(1/p)<∝,1<p<∝,若(‖X_n‖~q)一致可积,其中1/p+1/q=1,则依概极限鞅变换 Y=(Y_n=sum from i=1 to n V_i(X_i-X_(i-1),(?)_n,n≥1)为拟鞅且(Y_n)a.e.收敛。

In this paper we prove (1)Let Banach space B be p-smooth space (1≤p≤2),X=(X_n,(?),n≥1) B-valued martingale and V=(V_n,(?)_n,n≥1)real-valuedpredictable sequence,a.e.convergence,L^P convergence and the strong(weak)lawof large numaers for martingale transform Y=(sum from i=1 to n V_i(X_i-X_(i-1)),(?)_n,n≥1)are held under certain conditions.(2) Assume that B has the Radon-Nikodymproperty,X=(X_n,(?)_n,n≥1) is a game fairer with time and (‖X_n‖~q) is unifo-rmly integrable (1<q<∞).The real valued predictable sequence V=(V_n,(?)_n,n≥1) satisfies sum from i=1 to ∞ (E|V_i|~P)^(1/P)<∞where 1/p+1/q=1.Thenin a quasi-martingale and (Y_n)is almost surely convergent.