SOME DISCUSSIONS ON MOMENTS OF THE MAXIMUM OF NORMED PARTIAL SUMS OF B-VALUED RANDOM ELEMENTS

In this paper, we disouss problem on moments of the maximum of normed partial sums of B-valued random elements. For 1 \leqslant p< 2 and r>p, we prove the following three propositions are equivalent: (1) There is a constant 0< c< \infty subh that for every sequenoe \left\X_n n \geqslant 1\right\ of i.i.d. B-valued random elements with mean zero and E\left\|X_1\right\|^*< \infty the inequality \left(E\left\|\sum_k=1^n X_k\right\|^r / p\right)^s / r \leqslant O_n^1 / p\left(E\left\|X_1\right\|^r\right)^1 / r \quad(n \geqslant 1) holds. (2) For every sequence \left\X_n, n \geqslant 1\right\ of i.i.d. B-valued random elements with mean zero and E \|\left. X_1\right|^\prime< \infty the relation \lim _k \rightarrow \infty E\left(\sup _n>k n^-1 / p\left\|\sum_i=1^n X_i\right\|\right)^r-0 holds. (3) For every sequence \left\X_n, n \geqslant 1\right\ of i.i.d. B-valued random elements with mean rero and E\left|X_1\right|^r< \infty the relation \left.E\left(\sup n^-1 / p \|\sum_k=1^n X_n\right)\right)^r< \infty nolds.