  Let $\alpha$ be an irrational number, let $X_1, X_2, \ldots$ be independent,
identically distributed, integer-valued random variables, and put
$S_k=\sum_{j=1}^k X_j$. Assuming that $X_1$ has finite variance or heavy tails
$P (|X_1|>t)\sim ct^{-\beta}$, $0<\beta<2$, in Part I of this paper we proved
that, up to logarithmic factors, the order of magnitude of the discrepancy $D_N
(S_k \alpha)$ of the first $N$ terms of the sequence $\{S_k \alpha\}$ is
$O(N^{-\tau})$, where $\tau= \min (1/(\beta \gamma), 1/2)$ (with $\beta=2$ in
the case of finite variances) and $\gamma$ is the strong Diophantine type of
$\alpha$. This shows a change of behavior of the discrepancy at
$\beta\gamma=2$. In this paper we determine the exact order of magnitude of
$D_N (S_k \alpha)$ for $\beta\gamma<1$, and determine the limit distribution of
$N^{-1/2} D_N (S_k \alpha)$. We also prove a functional version of these
results describing the asymptotic behavior of a wide class of functionals of
the sequence $\{S_k \alpha\}$. Finally, we extend our results to the
discrepancy of $\{S_k\}$ for general random walks $S_k$ without arithmetic
conditions on $X_1$, assuming only a mild polynomial rate on the weak
convergence of $\{S_k\}$ to the uniform distribution.
