The Complete Convergence of Best Fit Decreasing

Consider a probability measure $\mu $ on $[0,1]$ and an independent sequence of random variables $X_1 , \cdots ,X_n , \cdots $ distributed according to $\mu $. No regularity assumptions are made on $\mu $. Denote by $F_n (X_1 , \cdots ,X_n )$ the number of unit-size bins that are used by First Fit Decreasing to pack $X_1 , \cdots ,X_n $. The existence of a constant $f(\mu )$ is proven such that for each $\varepsilon > 0$, we have \[\sum\limits_{n \geqq 1} {P\left( \left|n^{ - 1} F_n \left(X_1 , \cdots X_n \right) - f(\mu ) \right| \geqq \varepsilon \right)} < \infty .\]