Limit distribution of a roundoff error

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