Given: a regular summation method \((a_{n,m})_{n,m}\) such that \(\sum^{\infty}_{k=m}| a_{n,m+1}-a_{n,m}| \to 0\) uniformly in n and a sequence \((T_ n)_ n\) of bounded operators, chosen independently on a Banach space X. The author investigates conditions under which \(\lim_{n\to \infty}\sum^{\infty}_{m=1}a_{n,m}T_ m,...,T_ 1(x)\) (x\(\in X)\) exists strongly. From his results, the author deduces various random ergodic theorems for Banach spaces.