NONCONVENTIONAL LIMITS OF RANDOM SEQUENCES RELATED TO PARTITIONS OF INTEGERS

We deal with a sequence of integer-valued random variables {Z(N)}(N=1)(infinity) which is related to restricted partitions, or representations, of positive integers. We observe that Z(N) = X-1 + center dot center dot center dot + X-N for independent and bounded random variables X-j's, so Z(N) has finite mean EZ(N) and variance VarZ(N). We want to find the limit distribution of (Z) over cap (N) = (Z(N) - EZ(N))/root VarZ(N) as N -> infinity. While in many cases the limit distribution is normal, the main results established in this paper are that (Z) over cap (N) ->(d) Z(*), where Z(*) is a bounded random variable. We find explicitly the range of values of Z(*) and derive some properties of its distribution. The main tools used are moment generating functions, cumulant generating functions, moments, and cumulants of the random variables involved. Useful related topics are also discussed.