  We study sums of a random multiplicative function; this is an example, of
number-theoretic interest, of sums of products of independent random variables
(chaoses). Using martingale methods, we establish a normal approximation for
the sum over those n \leq x with k distinct prime factors, provided that k =
o(log log x) as x \rightarrow \infty. We estimate the fourth moments of these
sums, and use a conditioning argument to show that if k is of the order of
magnitude of log log x then the analogous normal limit theorem does not hold.
The methods extend to treat the sum over those n \leq x with at most k distinct
prime factors, and in particular the sum over all n \leq x. We also treat a
substantially generalised notion of random multiplicative function.
