Limit distribution of a roundoff error

Let X] and {X} be the integer and the fractional parts of a random variable X. The conditional distribution function F_n(x)=P({X}≤x|X]=n) for an integer n is investigated. F_n for a large n is regarded as the distribution of a roundoff error in an extremal event. For most well-known continuous distributions, it is shown that F_n converges as n→∞ and three types of limit distributions appear as the limit distribution according to the tail behavior of F.