  Read-$k$ oblivious algebraic branching programs are a natural generalization
of the well-studied model of read-once oblivious algebraic branching program
(ROABPs). In this work, we give an exponential lower bound of
$\exp(n/k^{O(k)})$ on the width of any read-$k$ oblivious ABP computing some
explicit multilinear polynomial $f$ that is computed by a polynomial size
depth-$3$ circuit. We also study the polynomial identity testing (PIT) problem
for this model and obtain a white-box subexponential-time PIT algorithm. The
algorithm runs in time $2^{\tilde{O}(n^{1-1/2^{k-1}})}$ and needs white box
access only to know the order in which the variables appear in the ABP.
