In this paper we prove maximal ergodic theorem and a pointwise convergence theorem. Our result is to prove the convergence of B_n(T, \alpha, f)=\frac 1n\sum^n-1_j=0\alpha_jT^j f for all $f\in L^1(\Omega, X)=L^1(X)$, where $n$ tends to infinity, $\Omega$ is a $\sigma$-finite measure space, $X$ is a reflexive Banach space, $\alpha_j$ is a bounded Besicovich sequence and $T$ is a linear operator on $L^1(X)$ which is contracting in both $L^1(X)$ and $L^\infty(X)$. Our result has the additional advantage as it is sufficiently general in order to extend the Beck and Schwartz random theorem. We can also generalize this result to a multidimensional case.